Spatial Analysis of Place Cells
in the Rat Hippocampus
Julie Bower
BSc Cognitive Science
Session 2006/2007
The candidate confirms that the work submitted is their own and the appropriate credit
has been given where reference has been made to the work of others.
I understand that failure to attribute material which is obtained from another source may
be considered as plagiarism.
(Signature of student)________________________________
Acknowledgements
Thank you to my supervisor, Professor Tony Cohn, for his continuous support and feedback
during our weekly meetings as well as his prompt replies to my e-mails.
I would also like to thank my assessor Dr Brandon Bennett, for his constructive comments on
my mid-project report and during the progress meeting.
I would like to thank Yuri Dabaghian for providing the simulated place cell data.
I would also like to thank Phil, for proof-reading my report, and to Jamie, who taught me about
Leibniz.
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Table Of Contents
ACKNOWLEDGEMENTS ...................................................................................I
1. THE PROBLEM .............................................................................................1
1.1 Overview ...............................................................................................................................................1
1.2 Relevance to Cognitive Science ...........................................................................................................2
2. BACKGROUND .............................................................................................2
2.1 Place Cells in the Rat Hippocampus...................................................................................................2
2.2 Qualitative Spatial Reasoning .............................................................................................................5
2.3 Venn and Euler diagrams..................................................................................................................10
2.3.1 Computational Representation of Venn and Euler diagrams........................................................12
2.4 Graph Theory.....................................................................................................................................13
3. DESIGN AND IMPLEMENTATION ..............................................................14
3.1 Methodology .......................................................................................................................................14
3.2 Choice of Programming Language...................................................................................................14
3.3 Processing place cell data ..................................................................................................................14
3.4 Defining Place Field Relations ..........................................................................................................17
3.4.1Choice of calculus .........................................................................................................................17
3.4.2 Defining connectedness and RCC-5 relations. .............................................................................18
3.4.3 Determining accuracy of connectivity and RCC-5 relations ........................................................21
3.5 Defining characteristics of the environment ....................................................................................21
3.5.1 Linear Tracks, loops and Forks ....................................................................................................22
3.6 Representing the Data........................................................................................................................27
3.6.1 Qualitative Descriptions ...............................................................................................................28
3.6.2 Hand-drawn maps ........................................................................................................................29
3.6.3 Generating Venn and Euler diagrams...........................................................................................29
3.6.4 Complex environments.................................................................................................................31
4. RESULTS.....................................................................................................33
4.1 Reasoning............................................................................................................................................33
4.1.1 Connectivity and RCC-5 ..............................................................................................................33
4.1.2 Global properties ..........................................................................................................................33
4.2 Representation....................................................................................................................................34
4.2.1 Qualitative Descriptions ...............................................................................................................34
4.2.3 Hand-drawn diagrams ..................................................................................................................35
5. EVALUATION ..............................................................................................35
5.1 Reasoning............................................................................................................................................35
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5.1.1Processing the Data .......................................................................................................................35
5.1.2 Using RCC-5 ................................................................................................................................37
5.1.3 Detecting global properties ..........................................................................................................39
5.2 The topological map...........................................................................................................................39
5.2.1 Representing global properties .....................................................................................................39
5.2.2 Generating Venn and Euler Diagrams..........................................................................................40
5.2.3 Qualitative descriptions ................................................................................................................45
5.2.4 Hand-drawn drawn diagrams .......................................................................................................45
5.2.5 Appropriate choice of representation ...........................................................................................45
5.3 The effect of directional place cells ...................................................................................................46
5.4 How much spatial information can be inferred...............................................................................47
5.5 Future Extensions ..............................................................................................................................48
5.6 Conclusion ..........................................................................................................................................49
REFERENCES .................................................................................................51
APPENDIX A: REFLECTION ON PROJECT EXPERIENCE ..........................52
APPENDIX B: MINIMUM AIMS AND REQUIREMENTS .................................53
APPENDIX C: SIMULATED DATA LAYOUTS................................................54
APPENDIX D: TEST DATA LAYOUTS ...........................................................58
APPENDIX E: QUALITATIVE DESCRIPTIONS OF THE TEST AND
SIMULATED DATA..........................................................................................59
APPENDIX F: VENNMASTER AND DRAWEULER REPRESENTATIONS OF
TEST DATA .....................................................................................................72
APPENDIX G: VENNMASTER AND DRAWEULER REPRESENTATIONS OF
SIMULATED DATA..........................................................................................75
APPENDIX H: HAND-DRAWN REPRESENTATIONS OF SIMULATED DATA
.........................................................................................................................78
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1. The Problem
1.1 Overview
“Place cells” in the rat hippocampus have location-specific firing properties, and strongly fire only
when the rat is in a certain region of its environment, known as its “Place Field”. The information
encoded about the rat’s environment by the hippocampus appears to be chiefly topological in nature.
A key concept used in this project is the “Space Reconstructing Thought Experiment” (SRE), devised
by Dabaghian, Cohn and Frank [1], which is described below:
The experimenter is in a separate room from a rat whose place cells are being
recorded while it is free to run about in a maze. The experimenter receives real time
cell recordings and is free to analyse the data to extract information about the rat’s
environment and what the rat is doing.
The SRE brings about the question of how much information about the rat environment can be
inferred just from place cell activity data. As the reconstruction of the environment relies on the
topological information from the place cells, this project uses qualitative spatial reasoning techniques,
in particular the Region Connected Calculus (RCC), which is only concerned with relations between
regions as opposed to specific co-ordinates of regions in space.
These inferences made about the spatial configuration of the environment are then tested on test and
simulated place cell data. Many inferences can be made about the relation of place fields and also
about global properties of the environment using qualitative spatial reasoning. Inferences are made at
the local level, to detect which place fields are connected, and at the global level, to describe the
environment in terms of shape and significant features. However, the conclusions made from this
project are based on assumptions made about the data (for example, that it is complete) and about the
nature of the place field relations in the environment. These ideas are yet to be tested and verified with
real hippocampal data. The project minimum aims and requirements are described in Appendix B.
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1.2 Relevance to Cognitive Science
This project is interdisciplinary in nature, and reflects many topics studied in Cognitive Science.
Cognitive science is the study of cognitive processes, or the mind, by using a range of disciplines,
including computational methods, to test, hypothesize, and simulate brain processes. Although the
hippocampal encoding of place fields in the rat’s environment may be a low-level cognitive process,
this project draws upon many aspects and methods used in Cognitive science. The “cognitive map”, or
the idea of an “inner space” of an external spatial environment, is a strong concept used in
psychology. Neuroscientific research also plays an important role in terms of understanding the nature
of these hippocampal cells that configure the environment. This project uses these disciplines to
further understand the nature of these place cells, and uses computational methods to test hypotheses
about the place cells.
Cognitive Science also has a philosophical approach. In terms of this project, the concepts used in
qualitative spatial reasoning are related to ideas posed by Leibniz, who states that there is no absolute
location in space or time, but that objects and events are only relative to other objects and events. This
holds strong consequences on the nature of the universe and goes against the notion of an absolute
space with fixed co-ordinate positions [2].
2. Background
2.1 Place Cells in the Rat Hippocampus
It was discovered that rats could learn the layout of an environment that they had been exposed to
with or without a given reward. Rats could then use this knowledge to navigate towards a goal by
taking detours in complex mazes when direct routes were blocked or when more direct routes became
available. This led to the proposal that rats internalised a spatial representation of their environment,
or formed a “cognitive map” [3].
The hippocampus, which is a structure inside the temporal lobe of the brain, has been located as the
centre of spatial memory. As lesions in this area cause deficiencies in rat navigation and spatial
problem solving, it seems that without full hippocampal functioning, rats are unable to form a
cognitive map [4]. “Place Cells” are a specific type of pyramidal cell found in the CA1 and CA3
regions of the hippocampus. These place cells have very special firing properties; they fire strongly
when the rat is in a specific region of its environment, called its “place field”. In a given environment,
each place cell corresponds with a specific place field, and it fires most strongly when it is in the
centre of its field. The cells are virtually silent elsewhere; they hardly fire outside of its place field.
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Place fields of different place cells can also overlap. The location specific firing of place cells
suggests that it is these cells which are the basic units that form the cognitive map, and configure their
internal representation of space [5]. The place fields are initially established only after a few minutes
of being placed in a new environment [4]. Once the cognitive map is formed, the relation between
place cells and place fields stays stable and can be consistent over hours, sometimes days, and even
months [5]. It has been shown that these maps are recalled, and not recreated, by re-activating those
place cells involved in that specific cognitive map[4].
It seems that location can be determined from co-joint firing of many place cells. Cells with fields at
the location of rat will discharge most rapidly, and fire more slowly when they are further away from
the place field centre. When the rat is moving, place cells that fire closely together are likely to signify
close place fields in its environment. Muller suggests an average position of field centres can be taken
of active cells in a certain location. This can only be done on the assumption that each place cell has
only one place field in a given environment and that cell activity indicates proximity to a single place,
otherwise the accuracy is compromised. However, it is unlikely that one place cell has multiple fields
[4].
Place cell properties can support a “cognitive” graph, where stronger connections are made when
place fields are closer together, and place cells with overlapping place fields also strengthen [6].
Muller’s connected graph is automatically constructed during exploration and can explain detours and
shortcuts, as well as have directional considerations. Although there are advantages to this model,
such as being able to lesion certain parts of the graph, it suggests there are destination and goal cells
which do not seem likely to occur in the hippocampus [6].
Some place cells can be directional and fire in one direction only when the rat is moving along a
linear track. However, cells are not directionally selective in other environments and seem more
dependent on head direction[4] .Muller suggests that unidirectional place cell firing is for
behaviourally important regions such as foraging grounds, and directional cells are for paths
connecting such regions [4].
Place fields are evenly distributed throughout the environment but may be more common near walls
[4]. The shape of place fields tends to be elliptical or circular, but can be rectangular along walls
which may mean that rats treat boundaries differently from other regions. Most place cell firings have
a single maximum as opposed to local minima, which suggests that rats may represent different parts
of the environment at different resolutions [4]. O’Keefe and Brian showed that when stretching or
splitting walls of the rat’s environment, the rat’s corresponding place fields also stretch or split [7].
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If small changes are made in the environment, place cells tend to stay consistent and robust to the
original place field mapping. However, if the environment is radically changed, then partial or
complete remapping occurs. When a rat is placed in a different environment, place cells completely
remap; the location-specific firing changes dramatically and place cells may have different place
fields, or no place fields at all, as only about half of place cells are active in each environment [4].
There seems to be no correlation between which aspects of the environment are mapped by which
place cells. Partial re-mapping can also occur, where remapping only occurs locally. For example
when a barrier is placed in the environment, re-mapping occurs within the vicinity of the barrier, but
the relationships between cell firing and place fields elsewhere are preserved[4].
Place cells do not appear to signify absolute location because firing rates of place cells cannot be
predicted by looking at previous passes through the place field centre [5]. Place cells are also not a
topographic map of the environment [4]. Various theories such as the “local view” hypothesis have
tried to account for place cell activity. The local view hypothesis states that place cell activity is
determined by a unique, complex combination of sensory inputs from the outside world [7]. This can
be supported by findings that place cells rotate as a result of the rotation of external cues in a maze. In
addition, the rotation of one salient visual cue also causes place cells to rotate, suggesting that visual
cues are the main sensory modality that determines place field formation [5]. However, rats can form
reliable place fields in the dark and whilst deafened, so visual or auditory cues are not necessary for
the formation of place fields [5]. When lights are turned off in one session in a maze, and then turned
on again for the rat to see a cue has been rotated, its place cells do not necessarily rotate [7].
This evidence has led to the conclusion that place cells do not explicitly represent anything about the
external world in terms of events or objects [7], [8]. Proximal cues, such as vestibular (balance) and
proprioceptive (position of body parts) cues are also involved in place field formation, as place cells
rotate with respect to a rotated apparatus but remain fixed with respect to distal cues. In environments
which are symmetrical but have asymmetric distal cues, place cells can sometimes spontaneously
rotate [7]. Spinning the rat also causes disruption in place cell firing [5]. However, Knierim et al.
found that when there was a conflict between self-motion and external cues by testing rats in
microgravity, place cells were still able to form robust, spatially specific firing properties after an
extended period of adaptation than is usually required[9].
O’Keefe and Burgess’ study to show splitting and stretching of place fields led to their conclusion that
the distance from each wall of the environment makes an independent contribution to the firing of
each place cell. The results also showed that when a place field splits each half becomes directional.
However, it is the wall behind the rat, not the one which it is looking at, which controls cell firing [7].
McNaughton explains this with path integration, an intrinsic mechanism for updating spatial
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coordinates based on information from the rat’s motion, and that they only take absolute “position
fixes” when they actually bump into the walls [7]. When Gothard et al. created a “mismatch” between
the rats’ learnt environment and real-world co-ordinates by moving a reward site on a track, place
fields were controlled by a competitive interaction between path integration and external sensory
input [8].
Place cells are not exclusively sensory neurons under the control of the current environment because
when rats are in a darkened environment after being in the same environment with the lights on, place
cell firing and location of place fields remain the same and are not affected by the darkness. When a
rat is initially placed into a new environment that is dark, place cells exhibit altered firing properties
that continue after lights are turned on [5]. This evidence suggests that rats are also affected by their
experience and memory of their environment.
It seems that place fields are also behaviourally specific as place cell mapping changes if a
behavioural task changes. The location of place fields shifts depending on the location of water in a
number of cups, so it seems the task itself is represented [4], [5]. However, for some activities
behaviour is constant over time so place cell activity cannot be attributed to a specific behaviour. It
also seems that curiosity is enough for place cell establishment to occur; often no task is needed for
place field initialisation [4].
2.2 Qualitative Spatial Reasoning
Qualitative reasoning represents and reasons about commonsense information about the world without
using numerical computation. It makes this knowledge explicit and represents the underlying
abstractions used in science and engineering. These representations are symbolic and use a limited
vocabulary such as qualitative relationships between regions. Distinctions are made in quantity
spaces, but only those which are relevant and necessary to the application. After reasoning techniques
are applied, qualitative reasoning attempts to make predictions and explanations of systems
qualitatively, sometimes when quantitative descriptions may not be available or computationally
tractable.[10]
It seems that space is conceptually viewed in a qualitative manner, for example topologically.
Representing and reasoning about space qualitatively seems closer to how humans represent and
reason about space, especially as most natural language spatial expressions are purely qualitative.
Space is multi-dimensional and not adequately represented by single scalar quantities. Maths is too
abstract to be useful for common sense descriptions, yet on the other hand geometry does not provide
5
an appropriate level of abstraction. Qualitative Spatial Reasoning (QSR) tries to represent and reason
about space using extended spatial entities and its relations between them at the appropriate level of
abstraction desired. The methods employed must also have good computational properties. Some
significant spatial aspects that have been discussed in the literature are topology, orientation, shape
and distance. Applications of QSR include reasoning about physical systems such as scalar quantities,
semantics to natural language, in Geographical Information Systems and robotics.
Qualitative reasoning systems can represent scalar quantities, for example indicating the amount of
liquid in a tank, the height of a bouncing ball, or a more abstract quantity [10]. This can be
represented using the relations between the qualitative categories of numerical values, using the very
simple quantity space consisting of {+, 0, -} representing the two semi-open intervals of the real
number line and their dividing point. This emphasises the importance within Qualitative reasoning
that change is a continuous process; there cannot be a transition from + to – without going through 0.
One can compute with such a quantity space by using an addition table or other arithmetic operations
in a table, but some entries in such a table are ambiguous [11].
The RCC is a calculus created by Randell, Cui and Cohn [12]. It is, in its most general form, a firstorder topological approach to represent and reason about space. Regions, or extended spatial entities,
are used instead of traditional point-based mathematic approaches. This concept of using extended
entities in space is captured more intuitively as opposed to dimensionless points in geometry because
any real body in space will be a region and not just a single point [10]. These regions are non-empty
and regular which means they must all be of a uniform dimension, but this can be arbitrary. The
regions do not have to be internally connected; they can be multi-piece and have interior holes or
tunnels [10]. The RCC is built upon an original theory by Clarke which makes a distinction between
open and closed regions. Randell et al. [12] justify disregarding this distinction because such regions
occupy the same amount of space and they do not seem necessary to create a commonsense theory of
qualitative space. Introducing such a distinction also poses some “deep conceptual problems” [10].
The basis of the RCC is one primitive relation of “connectedness” between two regions, C(x,y), read
as “x connects with y”. Cohn et al. defines two regions as being connected if and only if their
topological closures share at least one point. C can be very expressive, and it is possible to define a
calculus with hundreds of distinctions [13]. There are various axioms that characterise the use of C,
for example that C is reflexive and symmetric. A large number of relations between two regions can
also be defined in terms of C (see Figure 1).
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Figure 1: Some relations defined in terms of C [10]
The relations P, PP, TPP, and NTTP are non-symmetrical and thus have converses Pi, PPi, TPPi and
NTTPi. Boolean functions can also be defined such as sum(x,y) which can then be used to defined
internal connectedness. There are also axioms that can specify properties of spatial regions such as the
axiom that every region has a Non-Tangential Proper Part.
A set of these 8 base relations, DC, EC, PO, TPP, TPPi, NTTP, NTTPi, and EQ, are denoted as the
RCC-8. These are provably JEPD (Jointly Exhaustive and Pairwise Disjoint), which means they are
mutually disjoint and provide an exhaustive classification of possible relations (See Figure 2).
Figure 2: The eight JEPD relations [10]
The RCC-5 is another set of JEPD base relations using the RCC theory, but does not take boundaries
into account. Instead its relations are DR, PO, EQ, PP, and PPi where DC and EC, which are used in
the RCC-8, are instead merged into DR (discrete). The RCC-5 can be used for regions with uncertain
boundaries because there is no tangential aspect and nothing to denote external connectedness.
The egg-yolk calculus can be used to account for indefinite of boundaries of regions. It adds another
binary relation to the RCC that x is crisper than region y. The yolk is minimum extension and the egg
is the maximum extension. The white is the area of indeterminacy. There are 46 JEPD relations that
can be naturally clustered into 13 groups when generalising the RCC-5 in this way and there are 252
JEPD relations between non crisp regions which can be clustered into 40 sets when using the RCC-8
[10].
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Egenhofer developed a different approach to topological relations by using nine possible intersections
between the interior, exterior and boundaries of two spatial entities. It is possible to use entities of
different dimensions or distinguish different degrees of intersection [14].
In QSR, a conceptual neighbourhood graph, or a continuity graph, can be built of the spatial relations.
As a key concept in qualitative reasoning is that change is continuous, a conceptual neighbourhood
can show which direct topological transitions are possible between relations in both directions,
showing its “neighbours” (see Figures 3 and 4). For example, the RCC-8 conceptual neighbourhood
shows that NTTP(x,y) cannot occur from PO(x,y), without either TPP(x,y) or EQ(x,y) happening inbetween. The conceptual neighbourhood of these RCC relations has to assume continuous translation
in terms of shape deformations and movement being continuous.
Figure 3: The RCC-8 conceptual neighbourhood [10]
Figure 4: The RCC-5 conceptual neighbourhood [10]
8
The most prevalent form of qualitative reasoning is based on the composition table [15] which is
made up of a JEPD set (see Figure 5). A compositional deduction is made based on the logical
properties of the relations R1(a,b) and R2(b,c), to infer the possible relations of R3(a,c). The relational
entries in the table are often disjunctive because of the qualitative nature of the calculus. For n JEPD
relations, a composition table of n x n relationships can be constructed. Each cell in a table
corresponds to a theorem [12] which means constructing a composition table can be complex and
difficult. Using a composition table can be a simple method for limited binary relations, but it is hard
to determine for which theories it is complete.
Figure 5: Composition table for RCC-8 [10]
Composition tables can be used in conjunction with conceptual neighbourhoods; however the
constructions of these were a “considerable challenge” as they were built manually [13]. However,
they can now be constructed automatically for the RCC by using an intuitionistic zero-order logic,
which is a weaker logic than classical first-order logic [10]. Every entry in the composition table
forms a connected sub-graph of the conceptual neighbourhood, for example, the entry involving DC
and PO needs the relation with EC. There has also been some success in constructing neighbourhood
graphs from information in composition tables using a constraint-based approach [10].
A simple qualitative spatial simulator can be built using conceptual neighbourhood diagram and a
composition table. The simulator takes an initial state and then branches out in a tree of successor
states by using the conceptual neighbourhood, and then forms the cross product of these sets. These
states can be checked for consistency by “triangle checking”; by checking every triple of relations for
consistency with the composition table [10].
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2.3 Venn and Euler diagrams
A set is a collection of distinct, abstract objects in which they have a common characteristic. If every
member of a set B belongs to a set A, then B is a subset of A. If every element of A is in B and every
element of B is in A, then A and B are equal [16]. The union of two sets consists of all members that
belong to either set or both. The intersection of two sets consists of all members that belong to both
sets.
A
A∩B
B
Figure 6: The intersection of set A and set B. The area enclosed within the box, but is outside the regions, is
the complement of the union of sets A and B.
Venn diagrams represent these sets as closed curved regions inside a box, which is the given universe.
Every region must be connected to every other region with finite points of intersection. These
diagrams therefore represent all the intersections between sets and all the possible relations between
them. With n curves in a Venn diagram, there must be exactly 2n regions, including the empty,
exterior region. For example in Figure 6 there are two curves, and 4 regions. Traditionally there are
three sets in a Venn diagram; however there can be larger number of sets [17]. If there are no
members within a region, this empty region is shaded in the Venn diagram.
Venn diagrams are a type of Euler diagram. Euler diagrams only show non-empty regions, and as a
result, represent certain relations very differently compared with Venn Diagrams (as shown in Figure
7). In addition, not all relations can be represented as Euler diagrams due to space restrictions [17].
Euler diagrams were used for syllogistic reasoning and can be used as a powerful reasoning tool; for
example, the inference “All A’s are C’s” can be made from drawing the premises ‘all A’s are B’s’ and
“all B’s are C’s” (see Figure 8). However there are some interpretational problems with this system
and more than one diagram is sometimes required to provide consistent interpretations [18].
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Venn Representation
Euler Representation
RCC-5 relation
PP(B,A)
B
A
DR(A,B)
A
B
EQ(A,B)
A,B
n/a
B
n/a
Figure 7: Venn and Euler representations of various relations. These are also labelled with their corresponding RCC-5
relations.
C
B
A
Figure 8: An Euler diagram expressing “All A’s are B’s” and “All B’s are C’s” by
showing A as a subset of B and B as a subset of C. This leads to the deduction “All A’s
are C’s” as A is also a subset of C.
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2.3.1 Computational Representation of Venn and Euler diagrams
There have recently been attempts to construct Venn and Euler diagrams for two or three sets, such as
the software DrawVenn [19]. There are also more powerful tools such as Venn Master and DrawEuler
which can construct such diagrams for a large number of sets. To maintain the consistency of complex
relations, optimisation methods are used such as evolutionary techniques with Venn Master, and
growth algorithms with DrawEuler. Strictly, they are not traditional Venn or Euler diagrams but
instead use these concepts to represent relations between sets.
RWD, a constraint editor, produces strict Euler diagrams that are used for reasoning purposes [20].
An initial layout of the diagram is produced by either a “Radialiser” or “Circuliser” method, and to
improve the comprehension of the diagram, nesting and clutter reducing algorithms are used.
Although these diagrams are traditional, accurately drawn Euler diagrams, only a limited number of
sets can be drawn of up to 5 sets, and these have restrictions upon the types of relations that can be
represented.
VennMaster
VennMaster is a visualisation approach for set relationships based on size-proportional Venn
diagrams that uses an evolutionary strategy for optimisation [21]. It is a java application for use with
GOMiner. The input for VennMaster is either a list or Go-miner file specifying which member names
belong to each set. Preliminary size-proportional regions representing these sets are constructed,
which are displayed as polygons with many edges and are therefore elliptical in shape. These sets are
then optimised within various parameters that can be adjusted using either an evolutionary or Particle
Swarm optimisation technique. Selecting each set with the cursor will display the name of the set and
the names of all members belonging to the set and these sets can also be dragged manually for user
interaction. After the optimisation, inconsistencies in overlaps may still occur; and these are displayed
in the “inconsistencies” tab. Inconsistent overlaps are shaded in grey, as with traditional Venn
diagrams, however not all inconsistencies are represented as such, as VennMaster allows empty and
non-connected regions unlike traditional Venn Diagrams (and like Euler diagrams). The
computational power required to perform the optimisations means that complex and large set relations
larger than twenty cannot be computed.
DrawEuler
DrawEuler draws area-proportional Euler diagrams for any number of sets using different
construction and layout algorithms to produce different diagrams[22].Venn diagrams can be generated
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showing all possible relations between a given number of sets, which are represented as polygons.
Euler diagrams can be generated after specifying relations between sets. DrawEuler first generates the
structure of its associated Venn diagram, and then compresses “unwanted” regions and updates the
algorithm. After construction, the area of each region can be altered and regions can be “compressed”
and deleted if they are empty. The result of manually compressing such regions may not produce the
same diagram compared with initially entering the set relations. Although there are different growth
algorithms to choose from, the diagrams do not differ significantly. Clicking on each region will
display the name of the region (i.e. the set name or the sets involved with the intersection). Regions
can be compressed and their sizes can be altered. The intentions of Chow and Ruskey were to produce
accurate area proportional diagrams without “regard for aesthetics” [22]. As a result, because regions
represented by DrawEuler tend to intersect at the centre of the diagram, after approximately ten sets
the individual regions cannot be identified and the diagram becomes difficult to decipher.
2.4 Graph Theory
Graph Theory uses concepts that have diagrammatic appeal in many areas of research including
topology, as it can be used as a model when representing binary relations [16]. A graph consists of
edges, vertices and a relation associating every edge with two vertices, which are the edge’s
endpoints. These vertices are not necessarily distinct; in a loop, the endpoints are equal as the vertex
has an edge from itself to itself [16]. Multiple edges can occur when the multiple edges have the same
pair of endpoints. Two graphs are isomorphic if they map directly onto each other with corresponding
vertices that preserve their edge relationships. Two vertices are adjacent to each other if they are both
endpoints of an edge. An adjacency matrix can be used to show the relationship between all vertices
in a graph. A cycle in a graph is a “closed path” from one node through to other nodes and then back
to itself.
A subgraph of a graph is a graph that consists of edges and nodes which are a subset of the graph.
Some types of subgraph may be useful to analyse specific parts of a graph. A clique in a graph is a
subgraph of that graph where every vertex in the clique is connected to every other vertex in the
clique. A smallest induced path is a subgraph of a graph that only contains edges between nodes in
which there are no “short-cuts” that can be taken by alternative edges [16]. A spanning tree in a graph
is a subgraph of the graph which contains an edge between all nodes of the graph, but where there are
no loops or cycles (see Figure 9). A minimum spanning tree in a graph with weighted edges is a
spanning tree that has the least total weight than other spanning trees. Figure 9 shows three graphs
which include the graphical concepts discussed.
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a.
b.
c.
Figure 9: Two isomorphic clique graphs a and b, each with 5 nodes and 9 edges. Each graph has one loop with a node
having the same edge to and from itself. Graph c is a spanning tree of a.
3. Design and Implementation
3.1 Methodology
The methodology chosen for this project is not a standard model such as one used in software
engineering development. The methodology is designed to investigate properties of hippocampal
place cells by using prototypes to test and then further develop concepts about rat place cell
properties, as well as infer spatial information from the rat place cell data. Various test and simulated
data are used to test the algorithms relating to place field relations and the rat’s environment, and are
then developed as part of an iterative process. The following sections reflect the progressive stages
undertaken. The definitions regarding place fields and place cells properties start from processing
basic level test data, and as more properties of place cells and environments are taken into
consideration, further definitions are developed to account for more complex environments.
3.2 Choice of Programming Language
The programming language chosen for this project is Prolog. Prolog is a Logic programming language
that is associated with artificial intelligence and language processing, and it can describe relations and
make queries about relations that hold [23]. Prolog is appropriate for this project in order to define
logical relationships between place fields in a naturalistic, qualitative manner. Prolog can capture this
with its logical nature and can easily translate qualitative spatial calculi relations such as the RCC-5
from first-order logic into Prolog, where inferences and representations can then be made about the rat
environment.
3.3 Processing place cell data
An existing simulator is used that creates place cell firing data for a given environment made up of
numerous place fields, each corresponding with a particular place cell (Y. Dabaghian, personal
14
communication). The data produced are individual files for each place cell, containing “time stamps” for
when each cell is firing. The behaviour of the simulated place cells are consistent with real cells, in that
they fire when the rat is in its place field, they are virtually silent elsewhere, and fire increasingly strongly
the closer it gets to its place field centre.
When a neuron, such as a place cell, is stimulated, it creates an electrophysical discharge, or “fires”.
When the place cell becomes increasingly active, for example when it is closer to its place field centre, it
fires at a more frequent rate compared to the rate of firing further from the place cell. This can also be
interpreted as the cell firing most “strongly” when the rat is in the cell’s corresponding place field, and the
cell’s greatest firing rate should occur at the time the rat is in the centre of the place field.
The rate of firing is calculated as a sum of occurrences of cell firings within a given time interval.
They are grouped together in temporal intervals, to give a continuous representation of cell firing as
opposed to individual discrete cell spike firing. 0.5 and 1 second temporal intervals are chosen because
the smaller the intervals, the more accurate the data will be. A 2 second interval may be too large in order
to detect the smaller differences in cell firing. Two temporal intervals are chosen in order to compare the
accuracy of these intervals. The place cell is considered to be firing during a given interval only if the cell
is firing on more occasions than a given threshold. If the firing rate is high enough (above the threshold),
there is enough stable cell activity to constitute the rat being in the corresponding place field (see Figure
10). A threshold is required to define the outer boundary of the place field to filter out low activity, noise
that may occur (such as from equipment) and to filter out spontaneous charging from the place cells [4]. If
two cells are both firing at a very low rate, without high enough thresholds, they may be declared as firing
together, and therefore their place fields as being connected, which may be undesirable(see Figure 10) .
As the desired relations between place fields are qualitative, quantitative data such as firing rates, at what
time the rat is in the centre of the place field or how long it takes to travel through the field is redundant
information; the desired information is simply whether the cell is firing or not. Various thresholds are
chosen for each interval in order to test and refine place field boundaries in numerous test data.
The thresholded data gives a list of intervals when each cell is firing over the threshold. The element in
the list is the lower bound of the time interval, for example:
firing(5, 1, 10, [1,2,3,9,10,11]) means that after thresholding above 10, cell 5 is firing, and that the rat is
in place field 5, between times 1-2, 2-3, 3-4, 9-10, 10-11 and 11-12 (the second argument, 1, defines the
interval size, hence all the times are 1 second long). When using this predicate to define relations
between cells, it is assumed that their firing rates have been grouped using the same interval and have a
mutual threshold.
15
a.
b.
Figure 10: a. A graph to show the firing rate of one place cell with respect to time. The rat is moving through the place cell’s
place field and the centre of the place field is at time = 5. With the defined threshold the place field boundaries are between t = 3
and t =7.
b. A graph to show the firing rats of two place cells with respect to time. Although they have mutual firing times, these are below
the threshold and therefore the Connected relation does not hold.
Variable thresholds are also used as opposed to a fixed threshold for each data. This is to account for
the fact that the place fields may be of varied sizes, as it has been shown that place fields may be of
different shapes when the rat is in more interesting parts of the environment. As a result some fields
which should be connected may not be as a result of having a low firing rate and having a firing rate
below the threshold out (see Figure 11). For each place cell, the threshold is taken as half of the
maximum firing rate. This is chosen with the assumption that the firing rate is uniformly Gaussian
shaped, and therefore even if the maximum firing rate of a place cell is very small, some data will be
taken from it as firing, unlike with a fixed threshold.
Figure 11: A graph to show the firing rats of two place cells with respect to time with different sized corresponding place
fields. Although the maximal firing rate of the second place cell occurs at the same time the first place cell fires, with a fixed
threshold only the first cell is considered to be firing at this time, and the two place fields will not be defined as connected.
16
3.4 Defining Place Field Relations
3.4.1Choice of calculus
Given place cell data, it is required to determine how the corresponding place fields are related to each
other in order to create a topological map of the rat’s environment. Reasoning about these place field
relations qualitatively seems appropriate as it seems that the hippocampus predominantly processes
topological information, as opposed to specified co-ordinates, about the environment. The exact location
of where the place fields are geometrically in the environment is not necessarily the most important
information; it matters more where place fields are in relation to each other and which possible paths the
rat can take, such as which place fields the rat has to travel through in order to be in other regions or place
fields in the environment.
The RCC-5 calculus can be used when reasoning about indeterminate regions, as it does not specify
tangential relations unlike RCC-8, and the specific boundary is less significant. RCC-5 is the chosen
calculus for reasoning with place fields because place fields are not well defined regions; even though
there is a designated threshold to determine a boundary for place fields, this is arbitrary, and the most
important information extracted will be whether place fields are part of or discrete to other place fields.
As place field locations are calculated from discrete place cell firing times, it seems unlikely that they are
standard geometric shapes that can have “edges” and tangential relations that are used in RCC-8. For the
tangential relations EC, TPP and NTPP to hold, there must be a single “point” the rat is at in which both
cells are firing, i.e. at the boundary of the place fields. Therefore it seems unjustified to use a calculus for
representing the rat, an extended entity, at a single point in space. Similarly, because the place may be
firing very weakly at locations not necessarily specified as the definite place field, it is unlikely that two
fields, even though they may occupy the same regions in space, will be defined as Equal(EQ) if there is
just one discrete time in which the cell is firing and the other is not. To a large extent the relations
depends on the choice of threshold; depending on the chosen threshold, for TPP relations,
“Neighbouring” place fields will either be Partially Overlapping, Discrete or a Proper Part depending on
how high or low the threshold is. As a result, it is more important to define the RCC-5 relations Partial
Overlap and Proper Part Of because these are used to understand how the rat is navigating throughout the
environment.
17
3.4.2 Defining connectedness and RCC-5 relations.
In qualitative spatial reasoning the predicate C(x,y) is used for two regions x and y which are connected
to each other. To define whether two place fields x and y are connected to each other, the following
predicate is used:
∀xy(Connected(x,y) ↔ ∃z (firing(x,i,t,z) & firing(y,i,t,z))).
This is equivalent to the statement, “x and y are connected if and only if there is a time when they are both
firing simultaneously” (Figure 12). If the rat in a location which causes two cells to fire, then that space
must be occupied by both place fields and the place fields must be connected. This also implies that
Connected(x,y) is symmetric; if x is connected to y, y is connected to x (both of which are standard
properties of the connected relation in the RCC calculi).
Figure 12: Two overlapping place fields
are both firing above the threshold. The
rat must be in both of these place fields
at times 3.5 – 4; therefore these regions
must hold a connection.
Further predicates are required in addition to the Connected predicate in order to convey further
relational information about the place fields. The RCC-5 relations need to be specified to infer
whether two place fields are overlapping, equal, or inside one another; the actual nature of the relation
between fields cannot be conveyed by calculating only whether there is just one time interval in which
they share a point in space. In order to extract further information about the place field relations, each
relation in the RCC-5 calculus is defined.
The relation Discrete(x,y) is interpreted as “x is discrete from y” and is specified using the original
definition of Discrete as used by Randell, Cui and Cohn [12]:
∀xy(Discrete(x,y) ↔ ¬ Connected(x,y)).
Which is translated as “x and y are discrete if and only if x is not connected to y”.
18
The relation Properpart(x,y) is defined as:
∀xyz (Properpart(x,y) ↔ (firing(x,i, h,z) firing(y,i,h z)) & ∃t (firing(y,i,h,t) & -firing(x,i,h,t)))).
Which is translated as “x is a proper part of y if and only if for all times that x is firing, y is firing at
the same time and there is at least one time which y fires when x does not fire”. Alternatively, the
relation Properpart(x,y) can be implemented in a similar manner with the definition:
∀xylmpq(Properpart(x,y) ↔ firinglist(x,l) & firinglist(y,m) & subset(l,m) & length(l,p) & length(m,q)
& greaterthan(q,p)).
Which reads as, “x is a proper part of y if and only if the list L of times that x is firing is a subset of
the list M of times z is firing, and that the size of m is greater than l”. The above definition uses the
following relations: firinglist(x,l) which reads as “l is a list of the times that x fires”, subset(x,y) which
reads as “x is a subset of y”, length(x,y) which reads as “the length of x is y”, and greaterthan(x,y)
which reads as “x is greater than y”.
The original definition for the relation Inverseproperpart(x,y) is used simply by calculating
Properpart(y,x) [12]:
∀xy(Inverseproperpart(x,y) ↔ Properpart(y,x)).
The relation for two equal place fields is defined as:
∀xyz(Equal(x,y) ↔ (firing(x,i,t,z) ↔ firing(y,i,t,z))).
Which translates as “x is equal to y if and only if every time x fires, y fires, and every time y fires, x
fires”.
As the RCC-5 relations are JEPD, the PartiallyOverlaps(x,y) relation (or PO(x,y)) can be specified as
the conjunction of the negation of the other 4 relations:
∀xy(PartiallyOverlaps(x,y) ↔ (Connected(x,y) & -Equal(x,y) & -Properpart(x,y) & Inverseproperpart(x,y))).
19
Which translates as: “x overlaps y if and only if x is connected to y, and they are not equal to each
other or a proper part of another”.
All five relations are now specified. However, this representation of the place field relations may be
missing further spatial information that is required to accurately represent the layout. For example, if
there is a connection between three cells x, y, and z such that Overlaps(x,y), Overlaps(y,z) and
Overlaps(x,z), the inference cannot be made whether there is a simultaneous intersection of the three
place fields, i.e. whether there is a region of space in which when the rat is in, all three cells
simultaneously fire. Figure 13 shows two possible layouts that hold given the overlapping relations.
Figure 13: Give the relations (x,y), overlaps(y,z) and overlaps(x,z), both layouts are possible interpretations, with the
first layout having an “intersection” in which all three cells fire simultaneously.
A three place relation can be calculated to determine whether such a triad has a region in which
simultaneous firing occurs from all fields:
∀xyz(TriadIntersection(x,y,z) ↔ ∃t (firing(x,i,h,t) & firing(y,i,h,t) & firing(z,i,h,t) & ¬(z=x) &
¬(z=y) & ¬(x=y))).
Which translates as: “Fields x, y and z occupy a common region of space if and only if there is a time
in which their corresponding place cells all fire”.
This can apply to any number of place cells which may all simultaneously fire, and therefore produce
overlapping place fields that occupy the same region of space. This can be applied to as a four, - fiveor even ten- place predicate named “SimultaneouslyFiring”, for example:
∀lmnopq(SimultaneouslyFiring(l,m,n,o,p,q) ↔ (∃t (firing(l,i,h,t) & firing(m,i,h,t) & firing(n,i,h,t) &
firing(o,i,h,t) & firing(p,i,h,t) & firing(q,i,h,t)) & ¬(l=m) & ¬(l=m) & ¬(l=n) & ¬(l=o) & ¬(l=p) &
¬(l=q) & ¬(m =n) & ¬(m=n) & ¬(m=o) & ¬(m=p) & ¬(m=q) & ¬(n=o) &¬(n=p) & ¬(n=q) & ¬(o=p)
& ¬(o=q) & ¬(p=q))).
Which translates as, “Cells l, m, n, o, p and q are simultaneously firing if and only if there is a
common time in which they all fire”.
20
This relation and the triad intersection are six-and three-place predicate relations of a more general
formula that any number of cells are simultaneously firing if and only if there is a common time in
which they all fire.
The “connected” predicate previously defined could also be viewed as a two-place predicate of a more
general definition of Connectivity, in that if and only if any number of cells are simultaneously firing,
then they are connected, for example:
∀xyzm(Connected(x,y,z,m) ↔ SimultaneouslyFiring(x,y,z,m)).
Which is translated as, “place fields x, y, z and m are connected if and only if their corresponding
place cells are spontaneously firing”.
3.4.3 Determining accuracy of connectivity and RCC-5 relations
Now that all relations have been defined, this is tested on Simulated place cell firing data provided for
an Arena, Circle track, W-Track, 8-Track, and Linear track. The accuracy of the RCC-5 relations
produced is calculated by comparing the results with a quantitative method of calculating topological
relations. The simulator creates the place cell firing data given a constructed layout of various place
fields, each which its place field centre defined by an x-y co-ordinate. The boundary of the place field
is constructed from calculating the radius of the place field circle as the standard deviation of the
probability it will fire at that distance from its centre. The results of calculating RCC-5 relations are
compared with the relations created from constructing the place fields as specifically sized 2-D
regions in co-ordinate space. The layout of the numbered place fields in each simulated environment
is displayed in Appendix C.
3.5 Defining characteristics of the environment
When given data regarding place cell firings, other spatial information may be useful to extract other than
the relation between place fields; compared with the local RCC-5 properties between place fields, more
global properties of the space being explored may be of interest. The general layout of the environment
that the rat is in when its place cell data is being recorded may be the most significant spatial information
that can be extracted. Once all the given RCC-5 relations between every place field are calculated, it will
be useful to tell the type of environment the rat may be in, such as a linear maze, an open arena, or even if
there are holes in the environment. There may be certain characteristics of the place fields that can signify
which of these aspects of the layout are present.
21
The environment layout is also important conceptually for rats, for example food is more likely to be
found in arenas, and linear paths are important for shuttling between regions for food [4]. Directional
place cells are more likely to be found in linear regions, especially when decision making is required [4],
so determining whether there is a forked environment is important when considering that a goal may be
along a certain path.
Distinctive types of environment can be detected; those which have linear paths where the width of
the path is approximately the width of the rat. When there are such defined paths then it is possible to
distinguish whether the rat is in a simple linear track, or whether the track is a loop, or whether the
track has a fork. When there are no such paths characteristic of maze layouts, then an “arena” can be
detected, where there is an undefined region in which the rat can move about more freely, and
therefore produces more erratic place field patterns. The assumption is made that there is no missing
place cell information; otherwise no inferences can be made about rat environment.
3.5.1 Linear Tracks, loops and Forks
When the simulated rat is in an environment such as a maze that is a linear path enclosed by walls,
there is only a specific path that the rat can travel, and therefore the place cell firing will be very
regular and predictable. The relations of place fields in such a linear path will be that of consecutive
overlapping place fields in a regular order. As the rat can only travel across a limited, narrow, linear
space, there will not be irregular or multiple overlapping of place fields as expected in an arena.
For example,
Figure 14 shows the layout of a simple maze in which the rat is placed into from the left hand side.
After an initialisation period, as the rat progresses along the track cell 1 will fire, there will be an
overlap between firing of cells 1 and 2, and then as the rat moves along into the centre of place field 2,
cell 2 will fire alone. This then shows that the corresponding fields 1, 2 and 3 overlap as the rat
progresses along the track. Topologically, the rat must pass through cell 2 to get from cell 1 to 3.
1
2
3
4
Figure 14: A simple linear track
22
The PartiallyOverlaps(x,y) relation is chosen because although other place fields may exist within the
region, for example fields that are a proper part of other fields (ProperPart(x,y)) and two fields that are
equal (Equal(x,y)), it is unlikely that two fields will be equal to each other (as discussed in section
3.4.1). In addition, the cells that continue the “chain” along the path are the ones that overlap; those
fields that are proper parts of the overlapping fields create a longer chain, but are not necessary to
travel through the maze (see Figure 15). If two cells are equal, then only one of the two fields are
needed to be part of a linear path.
5
1
2
3
4
6
Figure 15: A diagram showing a rat environment containing field relations PO(1,2), PO(1,3),PO(3,4),PO(4,6), PP(2,3), PP(5,4).
There only needs to be one path of overlapping place fields from field 1 to field 6 to detect a linear track, such as PO(1,3), PO(3,4),
PO(4,7). Other fields, such as those which are part of other fields can be “discarded” when detecting linear tracks.
The formal definition for detecting a linear path is:
∀xy(Linearpath(x,y) ↔ PartiallyOverlaps(x,y) v (PartiallyOverlaps(x,z) & Linearpath(z,y))).
Which is translated as “There is a linear path between place fields x and y is if there is a chain of
overlapping place fields from x to y”. This can account for the varying shapes of each place field and
topological distortions, for example if the environment is stretched such as with [8], the place fields
will stretch accordingly but the linear path will still exist as the relations will still be the same.
A simple linear track such as figures 14 and 15 can be defined as the only characteristic of the rat’s
environment if the linear path has a beginning and end at each side of the path, such as fields 1 and 4.
These “end fields” are defined as those which only overlap with one other cell:
∀xyz(Endfield(x) ↔ (PartiallyOverlaps(x,y) & PartiallyOverlaps(x,z)) (y=z)).
Once end fields are located, a linear track can be defined:
LinearTrack ↔ ∃xy(Linearpath(x,y) & Endfield(x) & Endfield(y) & ¬(x=y) & ∀z(Endfield(z)((z
=y) v (z=x)))).
Which is translated as, “The rat is exploring a linear track if and only if there exists only two end
fields and if there is a linear path between them.
23
Common maze layouts include forks in a path, such as a T-maze or a Y-maze [3], as illustrated in
Figure 16. These can also be modelled by determining whether there is a linear path between all
existing end fields. For example, the definition for the layout being a T-maze:
T-maze ↔ ∃xyz(Endfield(x) & Endfield(y) and Endfield(z) & Linearpath(x,y) & Linearpath(x,z) &
linearpath(y,z) & ∀p(Endfield(p) ((p=x) v (p=y) v (p=z)))).
Which is translated as, “the rat is in a T-maze if and only if there are three end fields and there is a
linear path between them”.
The number of forks in a forked environment is the number of end fields -2, such as with the T-maze,
there are three end fields and one fork. This concept can be applied to a more generally to define
forked environments of all sizes:
“There is a forked environment of n forks if and only if there are n-2 end fields and for all end fields
there is a linear path between them”.
Figure 16: A T maze (with place fields), a Y maze, and an H shaped maze (with two forks).
However, there may be occasions when a place field is classified as an end field when this is
undesired. For example, Figure 17 shows place field formations that may occur if the maze walls are
wider than the rat’s width. According to the current algorithm there are four “end” fields, and that the
layout is of a linear track with 2 forks. However the desired would require a linear track, or a “pseudolinear” track, to be detected. The most important aspect is the linear path between 1 and 6 and not the
fields that just branch off from the main path. This is making an assumption that place fields are
roughly the same size [4]; otherwise if cells 7 or 8 were very large then they could be classified as
being a fork themselves.
24
The relations member(x,y) and pathlist(x,y,l) are needed to define a “Naïve” Endfield, meaning “x is a
member of the list y” and “l is a list of overlapping place fields in between x and y” respectively. With
these further relations a “Naïve” Endfield is defined as:
∀xyzapl(Naïve(x) ↔ (Endfield(x) & PO(x,y) & Endfield(a) & Endfield(z) & linearpath(a,z) &
¬(X=Y) & ¬(X=Z) & pathlist(a,z,p) & member(y,p) & length(Z,L) & greaterthan(l,3))).
Which is translated as, “X is a naïve end field if and only if it partially overlaps with a place field that
is part of a linear path between two end fields, and that linear path contains more than 3 place fields”.
Therefore an addition is made to the algorithm such that “naïve” end fields are those which only
overlap with one other cell, but also the cell it overlaps with is either part of a loop, or part of a linear
track between other end fields that is longer than three fields. If the field was overlapping two fields
away from a linear path, then it is considered as far away from that path to be deemed a separate path
(and therefore part of a fork).
In addition a “real” end field needs to be defined using the original definition as an end field, but with
the addition that it is not “naïve”:
∀xy(RealEndfield(x) ↔ ((PartiallyOverlaps(x,y) & PartiallyOverlaps(x,z)) ((y=z) & Naïve(x)))).
Which is translated as, “x is a real end field if it overlaps with just one other end field and is not a
naïve end field.
7
1
2
3
4
5
6
8
Figure 17: Possible place field formations where fields 7 and 8 could be classified as “end fields”.
A “loop” can simply be defined as existing if there is a linear path from any place field from itself to
itself, without returning to any place field it has already visited. The whole environment is looped, or
circular, if there are no end fields and all the place fields are part of the circular maze. However, such
a maze must be of narrow walls and with a circular path; in order to distinguish this from something
such as a triad intersection, the size of place fields in the loop must be greater than three:
Loop ↔ ∀xpl(Linearpath(x, x) & pathlist(x,x,p) & length(p,l) & greaterthan(l,3)).
25
3.5.2 Undefined regions and arenas.
If the rat is in a larger region such as an arena with less interesting characteristics, the formation of
place fields are less regular and there are connections between many place fields. These arenas, other
than just being enclosed, can also feature in between linear tracks, which may mimic natural rat
environments, for example where there may be food stored.
Cliques
A clique of size N in a representation of place fields will exist when there are N place fields and that
all place fields are connected to each other. For example a clique of size three is represented as:
∀xyz (Clique([x, y, z])↔ (Connected(x, y) & Connected(x, z) & Connected(y, z))).
Which is translated as “x, y, and z are in a clique if they are all connected to each other”.
It may also be useful to define a maximal clique of place fields. For example, all connected place
fields will be a clique of size 2, but may be a smaller clique within a larger one. A maximal clique can
be defined by detecting whether adding a place field to the clique still results in a clique:
∀xy(MaximumClique(x) ↔ (clique(x) & ¬clique([x,y]))).
Which translates as, “x is a Maximum clique if and only if x is a clique and there is no place field that
can be added to the list of fields in the clique that will result in the new list also being a clique.
When a rat is in a large arena without walls to separate where the rat can explore, it is predicted that
there will exist cliques of a high size, as the rat will be moving about freely at all angles, forming
more erratic place fields. The prediction is made that if a clique of greater than 4 exists in the place
field relations, then the rat is likely to be in an arena or an undefined region:
Arena ↔ ∃pqrst(Clique([p,q,r,s,t])).
These more complex regions may also occur in more regular maze environments such as the starting
and end points of linear tracks. In these situations although there is an aspect of linearity, there are no
“end fields” and may be classified as a loop or an arena (such as with Figure 18). The end field
definition has a further formula added to it to account for this:
26
∀xyzpqt(CliqueEndField(X) ↔ (Clique([x,y,z]) & (PartiallyOverlaps(y,p) ((p=x) v (p=z))) &
(PartiallyOverlaps(z,q) ((p=y) v (p=x))) & ∃r(PartiallyOverlaps(x,r) & ¬(r=y) & ¬(r=z)) &
(PartiallyOverlaps(x,t) ((t=y) v (t=z) v (t=r)))).
Which is translated as, “X is a clique end field if it is a member of a clique of size three, in which it is
the only member of the clique that overlaps with another field”. This field definition can be
substituted for the end field definition for detecting linear tracks and naïve end fields.
Figure 18: A linear environment without “end fields”.
Six different test data were constructed to test the simple concepts used to detect used to detect global
properties of a rat’s environment. These are used to represent different layouts in addition to the
simulated data. The layouts are described below and are displayed diagrammatically in Appendix D.
Layout 1: linear track with two end fields
Layout 2: linear track with two end fields and 1 “naïve” end field
Layout 3: forked environment with 1 fork and three end fields
Layout 4: forked environment with 2 forks and four end fields
Layout 5: looped environment
Layout 6: linear track between two cliques of size 3.
3.6 Representing the Data
There are several ways of representing the data and the nature of the place fields given specific place
cells data. The three methods chosen differ in terms of what information is easily extracted from the
representation, but each highlight different aspects of place field relations. The data produced from
the most accurate threshold and temporal interval will be used for representing the data.
The following layouts are mutually exclusive from each other:
•
Single Loop
•
More than one loop connected by linear paths
•
Linear track with n forks
•
Loop(s) within a linear track
•
One Arena
27
•
Arenas connected by a linear path
These can be detected using the definitions from the previous section and are used when representing
the data.
3.6.1 Qualitative Descriptions
As the spatial relations between place fields are qualitative and do not specify exact points in space, it
seems commonsensical to represent and describe the rat environment, as an observer, in qualitative
natural language terms by describing the rat’s journey through place fields, as opposed to trying to
reconstruct visual information. This is especially justified as these linguistic relations will still be used
if deciphering a diagram of the rat’s environment.
To represent the topological map of the rat’s environment using qualitative sentences, the first step is
to describe the overall layout using one of the 6 categories listed in 3.6. A list of all end fields, naïve
end fields, cliques, relations in terms of which fields are connected to each other, and a list of all
RCC-5 relations that hold between place fields are used to construct the descriptions.
The descriptions should reflect a journey taken by the rat throughout the maze, describing any particular
characteristics of importance (such as end fields or naïve end fields), all of the RCC-5 relations between
the fields, and the general description of the layout. The “closed world assumption” is made in which
everything which is not connected is disconnected by default. Therefore it is unnecessary to specify which
cells are disconnected from each other.
The following guidelines are used to describe the different layouts:
•
Linear track with n forks: Start with one end field, or clique, and list the linear path from that
field to the other end field(s) or clique(s), including naïve end fields, and also describing any
fields that are proper parts of, or equal to fields that are in the linear path.
•
Single Loop: Start with an arbitrary field and describe the linear path from itself to itself,
including all RCC-5 relations and naïve end fields that are connected to the linear path.
•
One Arena: Describe the maximum cliques in the environment and the relation of place fields
within them and how they are connected to other regions.
28
3.6.2 Hand-drawn maps
Similar processes that are used for creating qualitative descriptions should be used to create hand-drawn
topological map of the environment. The layout of the environment should be taken into consideration
and a list of all end cells, cliques and relations between fields should be used.
When drawing the rat’s environment, the largest maximum cliques should be drawn first as these will be
the most complex part of the topological map. End fields should then be drawn, with overlapping place
fields as part of the linear path between these end fields, or within a loop are drawn. The other place fields
which are part of or equal to fields within the linear path are then added to the map as well as naïve end
fields, and any disconnected fields are added last.
3.6.3 Generating Venn and Euler diagrams
If each place field represents a region, or set containing all the times the corresponding cell fires, then
these sets can be represented as a Venn or Euler diagram, in which the intersection relates to RCC-5
Connections(as described in Figure 7). Both VennMaster and DrawEuler are chosen to represent the
topological map of the rat environment; this is because they can represent several sets and the relations
between sets. Both tools are used because they use different approaches; VennMaster aesthetically
presents regions topologically as Venn diagrams, where DrawEuler conveys semantic information using
Euler diagrams.
When representing place fields if there is no overlap between two fields, the overlap will still be
represented but shaded in grey. When using VennMaster, a list file must be created to specify data to
belong in each set (or place field) in the format
Object set
Or:
Time
field name
A list of all times each cell fires can be represented in this format. From this VennMaster will draw a
proportional Venn diagram. However, the firing data may very large for several cells or several time
intervals and may result in several large sets. As the qualitative relations do not require the fields to be of
a certain size, simplified data can be created from the place field relations by using the definitions of the
RCC-5 relations so that the sets are smaller but the relations are still represented.
When fields are overlapping each other there must be the same object in both fields but the sets cannot be
equal. When a field is a proper part of another cell all objects in that set must be in the other field but they
cannot be equal. When fields are discrete from each other then there cannot be a common object in both
29
sets. When fields are equal to each other their set list must be identical. The file for relations PO(a,b),
PO(b,c) and PO(c,a) between fields a, b, c can be written as:
1
a
2
a
2
b
3
b
4
b
4
c
5
c
6
c
6
a
The file must then be loaded in VennMaster as a list file. Either an evolutionary or a particle swarm
optimiser is chosen to produce the Venn diagram where each set represents a different place field (for
example Figure 19). Each set or intersection can be selected and the relevant sets are highlighted,
specifying the objects in each set, or category. The regions can be manually dragged and moved to
remove the grey intersections; any inconsistencies caused by this are highlighted in the “inconsistencies”
tab which specifies which overlaps need to occur.
a.
b.
Figure 19: a. The output diagram loading the data denoted in section 3.6.3 to describe the relations PO(a,b), PO(a,c),
PO(b,c). There is no “triad intersection” so this is shaded in grey.
b.The regions can be manually dragged to eliminate the grey area if possible.
30
An Euler diagram can be created using DrawEuler by specifying all the regions and set intersections,
or all the possible combinations of mutual place field firing at any given time. The regions, or place
fields, must have letter names (and not numerical names). If PO(a,b) holds between fields a and b,
then there will be a time when cell a is firing alone, a time when cell b is firing alone, and a time when
cells a and b will be firing together, by the definition of partial overlap. This data does not need to be
extracted from the RCC-5 relations but instead directly from the thresholded firing data. A list can be
produced of cells that fire at each given time interval after thresholding. Those fields which
simultaneously fire in the same time interval are connected. The input for DrawEuler will be the
combination of all place fields that fire in each time interval. Using the same example as in 3.6.3.1,
the input will be a, ab, b, bc, c, ca. A construction and growth algorithm then needs to be selected to
determine the structure and expansion method of the diagram. The Euler diagram is then displayed
(such as in Figure 20). As the choice of construction and layout algorithms do not produce
significantly different diagrams, only the Edwards layout and Differential growth algorithms are used
to represent the topological maps.
Figure 20: The output produced when entering sets
a, ab, b, bc, c and choosing the Edwards
Construction and Differential Growth layout
algorithms. Region a is denoted in black, b is in blue
and c is in red.
3.6.4 Complex environments
When the rat is exploring a complex environment such as in an arena where there may be many
cliques of place fields, it may be difficult to represent the place field relations when using all three
methods of representation. If there exists a clique of several place fields connected to other such
cliques, a hand-drawn map may not be able to express this accurately because of the limitations of
space and manual optimisation. Although a computer algorithm may be able to represent such
relations more accurately, the chosen tools can only represent a limited number of sets within one
diagram, and the number of relations between fields may not be able to be computed efficiently and
accurately.
31
In order to reduce the complexity of such representations, if there exists a clique of four or greater
place fields, and this itself is connected to at least one other clique of four or greater place fields, then
maximal cliques in the environment are to be represented as a single region. When representing these
regions visually, it will need to be specified that they are of a greater size than single place fields.
An addition is made to the qualitative description in which each maximal clique is numbered,
specifying which fields are a member of the clique, in the format clique(n,[m]), where n is the clique
number and m is the list of members in the clique n. A further addition is made by specifying which
maximal cliques each place field is in, in the format memberofclique(f,[c]), where f is the place field
number and c is the list of maximal cliques it belongs to. If these maximal cliques are connected to
other maximal cliques, then they can also be simplified and represented as a single region, which also
must be larger than the clique regions. A “clique of cliques” of a size of three cliques can be defined
as:
∀xyz(∃c(Cliqueofclique(c, [x,y,z]) ↔ (Clique(x,[m]) & Clique(y,[n]) & Clique(z,[p]) &
∃q(member(q,m) & member(q,n) & member(q,p)))).
Which is translated as, “there is a clique of cliques c containing cliques x, y, and z and there is a field
that is a member of all three cliques”
If there exist large, or numerous “cliques of cliques”, maximal cliques of cliques can also be found to
accurately represent the rat environment:
∀mnc(MaximumCliqueOfClique(c,[m]) ↔ Cliqueofclique(c,[m]) & ¬Cliqueofclique(c,[n,m])).
Which is translated as, “c is a maximal clique of cliques containing a list m of cliques as a member of
c if and only if c is a clique of cliques, and adding a place field n to the list m does not result in c still
being a clique of cliques”.
It may also be required that these maximal clique of cliques can further be reduced to a single region
if they form a clique with other maximal clique of cliques. If this is the case, then similar analysis is
done to visually represent such regions.
32
4. Results
4.1 Reasoning
4.1.1 Connectivity and RCC-5
The simulated arena environment has varying sizes of place fields, as well as fields that are discrete
from all fields and fields that were proper parts of other fields. As the other data has more regularly
sized place fields, and only Partially Overlapping relations holding between them, the accuracy of the
threshold and temporal interval varies more for the arena data. This is supported by the results for
accuracy of Connectivity relations not varying significantly for the W-, 8-, linear and circle track
between different threshold and interval choices, ranging from 95-99% for the variable threshold.
However, the results for the accuracy of place field relations in the arena environment varies
drastically depending on the type of threshold for the arena, as shown in Figure 21:. The optimal fixed
threshold is 10, however again this makes no significant difference to the accuracy for the W-, 8-,
linear and circle track. Although RCC-5 relations are calculated for all place fields, in the W-, 8-,
linear and circle tracks there were only partially overlapping fields, and this was correctly detected.
However, there were many fields that are proper parts of other fields in the arena, and although some
of these relations are detected, others are not, which suggests that the threshold is not perfect.
Threshold Interval(s)
10
10
Variable
Variable
0.5
1
0.5
1
Connectivity
relations
73%
47%
87%
88%
RCC-5
relations
58%
66%
68%
68%
PO
PP
DR
PPi
EQ
90%
59%
80%
80%
0%
86%
80%
80%
100%
0%
0%
0%
0%
86%
80%
80%
100%
100%
100%
100%
Figure 21: Accuracy of place field relations for the simulated arena data. Different thresholds and temporal intervals are
used.
To represent the place fields, the variable thresholded data is used as it produces more accurate data
than the fixed threshold. The chosen temporal interval for visualisation is at 1 second; as the
difference in accuracy is not significant, this interval is chosen simply because this reduces the size of
the data.
4.1.2 Global properties
The qualitative results for the test data are 100% accurate, in which all place field relations are
correct, and the correct global layout is detected, as well as end fields, clique end fields and naïve end
fields. However, for the simulated data, as many overlapping cliques were present, all environments
are defined as arenas.
33
4.2 Representation
4.2.1 Qualitative Descriptions
The output for qualitative descriptions of the test and simulated data are displayed in Appendix E.
For the simulated data, each maximal clique is represented. Each maximal clique of clique is named
and then placed as a member of a “superclique”, or a clique of maximal clique of cliques.
4.2.2 Generating Venn and Euler diagrams
The visual representation generated by the place field relations of the test data is shown in Appendix
F. The VennMaster representations prove to be relatively accurate when using the Particle Swarm
optimisation and computes a solution faster than using the Evolutionary algorithm. VennMaster
produces diagrams for layouts 1, 2, 3, and 6 that appear very similar to the original layout design (in
Appendix C). However, for the layouts which are not aesthetically pleasing such as for layout 4, the
regions can be dragged to look like the original layout, showing no inconsistencies. This is displayed
in Figure 22.
a.
b.
Figure 22: a. Output from VennMaster displaying test layout 4. The regions can be dragged to provide
an aesthically pleasing diagram without any grey regions or inconsistencies.
As each of the simulated data represents environments consisting of 50 fields or more, DrawEuler and
VennMaster are unable to represent the individual place field relations. However, as it is detected that
all simulated data are complex environments with several overlapping cliques, they are represented
visually as cliques of cliques. Appendix G shows the DrawEuler and VennMaster representations of
such cliques for each simulated data to show the general layout.
The accuacy of the Venn Master representations can be calculated by the amount of inconsistencies
and empty sets present in the diagram. Figure 23 shows some of these inaccuracies.
34
a.
Layout
Inconsistencies
Grey regions
Best algorithm
b.
Layout
Inconsistencies
Grey regions
Best algorithm
1
0
0
Particle
Swarm
Circle
1
32
Particle
Swarm
2
0
2
Particle
Swarm
Linear
1
26
Particle
Swarm
3
0
5
Particle
Swarm
W-track
1
13
Particle
Swarm
4
5
17
Particle
Swarm
23
Particle
Swarm
8-track
0
93
Particle
Swarm
6
0
2
Particle
Swarm
Arena
15
41
Particle
Swarm
Figure 23: The inaccuracies shown in the VennMaster representations of the test data (a) and the simulated data (b).
4.2.3 Hand-drawn diagrams
Hand-drawn diagrams are not produced for the test data as the test data is created from initial
drawings of the different layouts. The hand-drawn diagrams for the simulated data prove to be very
difficult and inaccurate as there are many complex field relations. As a result the best hand-drawn
representations are of the global shape of the environment, by representing cliques and cliques of
cliques (Appendix H). The hand-drawn diagrams accurately represent the same relations represented
by DrawEuler and VennMaster, and also prove to be more aesthetically pleasing. Some diagrams
prove to convey more topological information than the computer representations. For example the
representation of the linear track clearly shows the path was linear in comparison with the Arena, W
and 8 track representations.
5. Evaluation
5.1 Reasoning
5.1.1Processing the Data
Choice of interval grouping
The difference in the accuracy rate between the intervals of 1 and 0.5 seconds is not significant, with
the difference in accuracy rate being 1% using the variable threshold for the arena. This is because the
level in change of firing activity appears to be within the 0.5 – 1 second intervals, and reducing the
temporal interval would not result in accuracy of detecting connectivity between place cells.
However, a further implementation of the algorithm specifying 2 second intervals is made, resulting
35
in a drastic reduction in accuracy, which justifies the choice of using 0.5 and 1 second intervals. In
addition, the choice of 0.5 and 1 second intervals is further justified with the drastic difference of 26%
between the two intervals when calculating connectivity accuracy in the simulated arena data. The
data produced from selecting 1 second intervals is chosen for representation in order to reduce the size
of the data and increase computational speed. This is justified as the difference in accuracy is
negligible, and as the data files are so large this will be a benefit when working with real data.
Choice of threshold
The choice of threshold greatly affects the accuracy of the RCC-5 relations. As the place fields have
indeterminate boundaries, the threshold determines whether some fields are just partially overlapping,
or if it is reduced, part of a neighbouring place field. Although fields inside another field may have
some significance, it is unlikely that many fields will be part of another field. It seems that detecting
Connectivity is more important than other relations between place fields, because all connected fields
present in the W, 8, circle and linear track data are Partially Overlapping. However, the threshold still
determines whether fields are defined as connected or not, depending on how high the threshold is.
Using a fixed arbitrary threshold provides less accurate results than using a variable threshold. This is
because, although the actual size of the place fields is unimportant when reasoning about their
relations, they are of variable sizes in the simulated data for the arena. The optimal fixed threshold is
of 10 firings per 1 second and 0.5 second intervals, providing an accuracy rate of 73% and 47% for
the arena, and an accuracy rate of 99% for the linear track. If the place fields were all approximately
the same size, such as with the test data and the simulated data for the linear, W-, 8- and circle track
(as shown in Appendices B and C), then this would be a sufficient threshold. The results support this
with the reduced difference in accuracy rate between a fixed and variable threshold for these tracks.
However, because some place fields may be drastically larger than others, such as with the arena(see
Appendix C), choosing a high enough threshold to account for larger place fields results in the
threshold being higher than the maximum firing rate for drastically smaller place fields. Because it
takes less time for the rat to travel through a smaller place field such as field 7 in compared with field
41 in the arena, the maximal firing rate is much smaller. With a high threshold, some smaller place
fields are detected as not firing at all. If the threshold is low enough to account for these smaller place
fields, then a lot of the background, low level firing is included for the larger fields, increasing the
number of place field connections, thus reducing the accuracy of place field connections with them.
Using a variable threshold is more appropriate and improves the accuracy of defining RCC-5 relations
between place fields that are of variable sizes as it ensures that there are firing times for every place
cell that fires and takes into consideration the size of each place field. This method can also be used
with place fields which are regular and are of the same size. However, although the variable threshold
36
improves the accuracy rate of the simulated connections for the arena, many fields are incorrectly
detected as Partially Overlapping instead of being a Proper Part of a field, which suggests the
threshold is still not optimal. In addition, the method used is based on the assumption that the
Gaussian shaped firing rate is regular. If the firing rates of the place cells were more erratic, such as
with real data, which may be noisy and less predictable, then the choice of thresholding at half the
maximum firing rate may prove to be less accurate. Also, the difference in accuracy rate between an
environment with varied place field sizes in comparison to environments with more regular place field
sizes is 11%, which suggests that an alternative method could improve further still the accuracy of
detecting RCC-5 connections.
Choosing a rate of change in the firing data may result in thresholding out the lower level,
background firing and take data for only when the curve of the Gaussian becomes increasingly
steeper. This may require grouping firing rates using smaller intervals such as 0.5 or 0.25 seconds to
detect smaller changes within the data. As the boundaries for place fields are indeterminate and not
defined, using the egg-yolk calculus [10] may be an alternative method to determine connectivity
between place fields in which there are levels of connectivity depending on whether the region is part
of the “yolk” or the indeterminate “white” region. However, using the egg-yolk calculus still requires
an appropriate boundary to be determined, in which there has to be some cut-off point with field
connections. This highlights the problem of reasoning about undefined regions where it is difficult to
determine where place field regions begin and end.
5.1.2 Using RCC-5
Accuracy rate
Using RCC-5 relations with the simulated arena data provides an accuracy of 68% with the variable
threshold. RCC-5 is a justified calculus choice compared to just using the Connected predicate,
because some place field relations in the arena are Properpart of, (and Inverseproperpart of), as well
as Partially Overlapping. As all fields are assumed to be whole, there is no need for the convexity
relation. However, although there were some fields which were a proper part of others, this is a small
number in comparison to partially overlapping fields, and these only exist in the arena. In terms of
understanding global properties of the environment, and relations between the majority of place field
relations, it may be that the Connected relation is sufficient. It may be interesting to investigate
whether fields as proper parts of other fields only exist within arenas, and not with more regular
shaped environments, in which case it is justified to detect more than connectivity. In comparison
with RCC-8 calculus, RCC-5 seems more appropriate because as the majority of place field relations
37
are Partially Overlapping, it seems unlikely that there will be any tangential place field relations that
can be defined with the RCC-8.
Accuracy measure
The chosen measure of accuracy is a comparison with an alternative interpretation of place field
layout that uses an alternative threshold as its boundary of the place field. It uses the probability of the
firing rate at a distance from its place field centre to construct these diagrams as opposed to reasoning
with the firing data after having been thresholded and grouped into firing rates within intervals. The
assumption used when calculating the radius of the place fields is that there is a regular Gaussian
distribution of cell firing. When using real data, the knowledge of the co-ordinates of place field
regions and the probability of its firing rate will not be available as the rat will be moving around its
environment. The main information that can be taken with real data will be the global properties of
the environment such as its observed shape and layout. Therefore, there does not seem an appropriate
method of calculating the accuracy of field relations in a real situation. However, the lack of
alternative ways to present field relations further justifies the need and choice of a qualitative
approach to this problem.
Connectivity
The assumption is made that all cells corresponding to a place field in the environment do fire, and
fire most strongly the closer the rat approaches the place field centre. This assumption is important
when detecting global properties of the environment, such as a looped environment. If one place cell
does not fire, if there is missing data or just one poorly thresholded place cell, then this can have a
great effect on the reasoning of the type of environment the rat is in; for example in the simulated
Circle track, there are no times in which cells 29 and 32 fire simultaneously; and therefore appears to
be a linear track. Another example is with the simulated Arena; field 46 appears to be discrete from
any other field in the environment. In these cases these are not gaps in the track or environment and
the rat would travel from field 29 to 32 in the circle track, and the rat could not mysteriously appear in
field 46 without having travelled from another field. Similar problems may also occur when taking
place cells readings from a real rat hippocampus as only a limited sample of place cell firings may be
recorded, and thus there may be missing data. An alternative definition for connectivity could be used
to account for this problem, that place fields are connected if there is nothing firing in between. Also,
the problems associated with reasoning about indeterminate boundaries become less important, as two
place fields will still be considered connected as long as the rat travels from one to the other even if
the threshold is very high.
38
5.1.3 Detecting global properties
Some “end fields” in the simulated linear and W-track can also not be detected because they are
members of cliques that were also overlapping cliques. An alternative definition for an “end field”
could be a field which is connected to the least amount of cells, because if there are many cliques
formed in the environment, it is likely the ends of the track are to be least complex.
When representing the data as hand-drawn diagrams it may have been useful to know which fields
were intersection points at a fork or junction. A “junction” field cell could be defined as one which is
connected to the highest number of fields. Although this may not prove to be an actual junction in a
path, because cells within a clique will be defined as “junction” fields, it will certainly be an area of
interest as it will be in a region of high populated cells.
The definitions used are basic, conceptual ideas that are successfully applied to the test data, however,
because the relations between place fields are more complex than predicted, and many cliques are
present where many cells cover a similar space in the environment, it was incorrectly detected that the
rat was in an arena for all simulated data. The assumptions made about place field relations that do not
apply with the simulated data are unlikely to apply with real data.
These basic concepts of overlapping within a track could be used to improve on more complex data.
For example, instead of a linear path being that of fields overlapping other fields, a path could be that
of cliques overlapping cliques. This is supported by the fact that the representation of the linear track
is linear in shape with “ends” connected by cliques of cliques. The circle track also shows a similar
shape, in which if the ends were connected to each other, would be a circle track. Also, looking at the
results of the representations, there are more cliques of cliques of cliques in arenas compared to the
other environments.
5.2 The topological map
5.2.1 Representing global properties
As the visualisation tools are unable to represent the complexity and number of place fields that exist
in the simulated data, an alternative method for representation could have been to display certain
sections of each map. However, the assumption is made that every environment is continuous and that
the rat can explore all parts of the environment. Therefore, displaying only certain areas of the map
will result in problems with deciding on a “cut-off point” for place fields, and some relations between
fields that are in different diagrams will not be shown. This method would prove to be difficult to
39
visualise the entire environment and it may be impossible to join the sections together for non-simple,
linear environments, because the visualisation tools optimise the whole space for the given sets.
This supports the use of globally representing the environment. Displaying complex topological maps
as simplified regions which contain cliques is an alternative method that may provide more
information than representing every place field relation. However, enough properties must be shown
to show enough features of the environment, as the more general the representations become, there is
a loss of specific relations between place fields and there is no information about connectivity. This
method reduces the accuracy of the topological map to some extent, as finer detailed parts or aspects
of the environment such as smaller loops, forks or holes in the environment may be overlooked.
5.2.2 Generating Venn and Euler Diagrams
Both the VennMaster and DrawEuler programs are unable to represent the simulated place field
relations as the simulated data consists of 50 – 100 place fields and this is too large for the programs
to compute. However they are very useful for examining test data and smaller place field relations.
Current research is focused on constructing area-proportional Venn and Euler diagrams, which are not
needed for qualitative spatial representations. The limitations of such programs highlight the need for
a specialised visualisation program that can represent qualitative spatial relations that are not areaproportional and that can deal with large numbers of regions. The need for such computational
visualisation is also supported by the difficulty in trying to create visualizations manually as handdrawn diagrams. An algorithm using optimisation techniques may be the best approach to visualising
place field relations, and the fact that there is not an existing optimal algorithm for visualising place
fields highlights the difficulty of this optimisation problem.
VennMaster is a very powerful tool, but as it is not designed for qualitative relations, the input for the
diagram requires specific data be in each set. This is not taking advantage of relational information
and requires the reconstruction of qualitative information to represent these relations, which is not
entirely appropriate. However, the use of evolutionary algorithms and particle swarm intelligence is
an appropriate method to produce an optimal solution. The optimal method to represent all data is the
Particle Swarm algorithm, which produces more accurate diagrams and computes at a faster speed.
However, these diagrams produced are not always the most optimal solution as there are some
inconsistencies with missing relations, and some intersections as shaded in grey where there are no
objects in this region, as seen in traditional Venn diagrams. VennMaster does not produce traditional
Venn diagrams because they allow disconnected regions and do not show all the possible relations
between all regions. However this additional feature of “shading” allows for the expressiveness that
Venn diagrams provide.
40
These grey areas are not desirable because this shaded overlap does not occur topologically in space.
However, one can imagine these grey shaded areas to be part of just one of the existing regions, thus
eliminating the overlap (as described in Figure 24). However, this only applies for intersections of few
fields where there are no inconsistencies in the diagram. At times more optimal solutions that can
reduce the amount of shaded areas may be possible but unable to be represented by VennMaster.
a
b
c
Figure 24. a. A possible output from Venn Master showing PO(a,b) and PO(b,c) but ¬PO(a,c). The grey region represents the lack of members in
the intersection between a and c. This can be interpreted as the grey region “extending” one of the two regions a or c (20b, 20c).
The advantage of using VennMaster is that the regions can be manually dragged to provide a more
aesthetic diagram and as inconsistencies are displayed, these can also be addressed. This gives the
user freedom to adjust parts of the diagram with additional knowledge about characteristics of the
layout. However, for more complex diagrams, manually adjusting regions may create many
inconsistencies without improving the diagram.
In comparison to DrawEuler, VennMaster provides a more meaningful interpretation of place fields
and represents them topologically to form a topological map as it might be in the real environment.
However, VennMaster is only capable of representing 20 fields and with an increasing number of
fields and relations it performs less well. As a result large environments with many place fields cannot
be represented or the fields have to be simplified and represent one region as a group of fields (such as
cliques). However, although some aesthetic representations can be made that can easily be viewed,
often the collection of place fields does not convey much information. For example, with the test data,
the output does not easily show the existence of forks or linear tracks, possibly because of the
optimisation technique. Despite this, more information may be conveyed for complex data in
comparison with DrawEuler.
41
DrawEuler provides an abstract representation that is harder to interpret than VennMaster. However,
the relations between place fields are more logically sound as there are no inconsistencies, and any
empty intersections are “compressed”, reducing the need for grey areas that is used in VennMaster.
DrawEuler produces true, accurate Euler diagrams, but they are not aesthetically pleasing. As each
polygon must correspond with a letter (as opposed to a number), it is difficult to identify specific
regions, which becomes an increasing problem as the number of sets is increased. In addition, it seems
unlikely that place fields will be shaped as triangles or the angular polygons represented in
DrawEuler, as they are indeterminate regions which certainly will not have edges. Despite this,
relationships between fields can still be visualised and understood; for example Figure 20 can be
understood topologically as a relationship between three fields, and all RCC-5 relations represented in
DrawEuler are very similar to the original representations (see Figure 25). In order to travel from one
region to a connected region, if the diagram is to be understood topologically then the rat must reduce
itself to a point to travel throughout the maze (for example, see Figure 26). Even though this is an
impractical interpretation of the topological map, without the given constraint that the rat is an
extended object it is an accurate method for representing sets and relations. However, when an
increasing number of regions are added to the diagram, with the limited amount of space it becomes
harder to decipher, and it is impossible to comprehend diagrams consisting of more than 13 regions
(such as with Figure 27).
42
PP(A,B)
[b ab]
DR(A,B)
PO(A,B)
EQ(A,B)
[a b]
[a b ab]
[ab]
PPi(A,B)
[a ab]
Figure 25: the RCC-5 relations as represented by DrawEuler
43
Figure 26: A DrawEuler representation of a linear track
containing of 7 consecutive overlapping place fields between field
a (in black) and field g (in brown). Field b is represented in blue, c
in red, d in green, e in purple, and f in turquoise. To travel from
field c to d the rat must travel from the red region to the centre of
the environment and reduce itself to a point before travelling out
into the green region.
Figure 27: 13 field relations represented by DrawEuler.
44
5.2.3 Qualitative descriptions
Qualitative descriptions are the most successful method of representation and the easiest to produce.
However, although the specific binary relations between regions are useful, and also the existence of
specific fields of interest such as end fields and cliques, it does not give the overall properties of the
environment. By knowing the relations between fields, there is no overall impression of what the rat is
doing or where it is exploring. However, these have resulted in the most accurate representation of all
the place field relations in comparison with the generated diagrams. Although there are no global
properties that can be understood, these global properties may not be particularly observable from
other representations such as DrawEuler representations.
5.2.4 Hand-drawn drawn diagrams
Producing a manual representation of place field relations is very difficult, and produces diagrams
with questionable accuracy. Although producing diagrams of more general regions of cliques is much
easier, complex structures must be visualised mentally before drawing them. This task requires spatial
logical thinking and may require several attempts before an accurate representation. The fact that the
algorithms used in VennMaster cannot provide a perfect solution supports the incredible difficulty of
this task. However, as the diagrams are displayed topologically like VennMaster, but as Euler
diagrams (and without shaded regions), it conveys some information that neither software can
represent, for example with the linear track representation.
5.2.5 Appropriate choice of representation
When representing simpler diagrams, the three methods complement each other, but when the place
field relations become more complex with an abundance of cliques, hand-drawn diagrams and
computational representations become hard to produce. However, for simpler data using a program
such as VennMaster would be appropriate alongside qualitative descriptions, to improve the diagram
manually and use the descriptions to guide the user as to the general shape and significant aspects of
the topological map.
45
5.3 The effect of directional place cells
Figure 28: Possible Directional cells that could exist in a simple linear track. The arrows represent which direction the rat is
heading for the cell to fire; double headed arrows represent unidirectional cells.
Directional place cells tend to occur in linear paths and fire only when the rat is travelling in one
direction. If directional cells are present in the rat environment, then this may effect the representation
of the topological map. If there are several directional cells connected to each other, then this will give
the impression that there are two paths between two unidirectional cells (e.g. Figure 28). However,
research has shown it is more likely that a directional cell is “neighboured” by directional cells. It also
seems highly likely that there is a directional cell for a region of space when the rat is headed in one
direction, and no directional place cell that fires when the rat is headed in the other direction.
However, despite these differences in possible place cell configurations, a visual representation of
these cells will not show these place cells occupying the same region of space because they do not fire
at the same time as its corresponding place cell or unidirectional cell. If place cells exist in the data
they need to be identified in order to accurately represent their corresponding place fields. These
directional place cells require neighbouring place fields which fire when the rat goes in both
directions. If this assumption is made then the proposed definition could detect directional place cells:
X is a directional place cell if and only if its corresponding place field X is connected to field Y and X
is connected to field Z, and the rat only ever travels from Y to X to Z, and never travels from Z to X
to Y.
This can account for more than one directional place cell connected to each other, as long as they
eventually connect to unidirectional cells.
46
5.4 How much spatial information can be inferred
The limitations arising from representing and reasoning about place field relations are based on many
assumptions which have previously been discussed. There is an assumption made that the rat
hippocampus establishes place fields in the majority of the environment, and that all place cells
corresponding with these place fields do fire when the rat is in the place field. This is a reasonable
assumption to make, however it is an important one that must be acknowledged in order to make any
generalisations about the global properties of the environment. For example, one could not conclude
the rat was in a linear track if it was possible the rat could travel along a fork in the track without any
place cells firing. Another assumption made is that each place cell has only one place field; it is
assumed that if the place cell fires at two different times the rat is in the same location of space.
Otherwise, the topological layout of the place fields could be drastically different with a larger
number of fields. However, it is a reasonable assumption to make and is supported by current research
[4].
There may also be problems in which there are missing imperfect data, such as with real data. For
example, not all place cells may be sampled by recording equipment, or the place cell recordings
could be affected by noise. It may be that the alternative definition of Connectivity as outlined in
section 5.1 can account for some missing data, even if it may reduce the accuracy of some
connections. Using the current definition of Connectivity would result in disconnected place fields
with missing relations corresponding with the missing data. This would be an illogical representation
of place fields because the rat has to be continuously moving to travel between regions in the
environment.
The amount of spatial information that can be inferred depends entirely on the method of processing
the original firing data; as there is low level firing for each cell, only a concentrated, high rate of cell
firing is of interest. Place cells do fire outside of its place field, and there is some low-level cell firing
when the rat is in its corresponding place field, which calls for the need of a threshold; and although
the chosen threshold is satisfactory, the data analysis must be viewed within the limitations of
thresholding the data.
As shown with the results of visualising place fields using computer tools, there are many accurate
topological representations of various relations between place fields. As qualitative spatial reasoning
does not represent fixed points in space that regions hold, but instead the relations between other
objects, it makes it hard to generalise in terms of global environmental properties. For example, the
definitions used to detect linear tracks and forks may incorrectly define a vast number of different
47
layouts as linear. For example, although there is one single path that the rat travels through, a “U
track” as shown in Figure 29, it may be incorrect to define this track as a “linear track” because the rat
has to turn two corners, which may be neurologically different for the rat in comparison with a
straight line. In addition, the definition used for detecting forks does not specify the difference for
what may be a “Y” maze, a “T” maze or a “W” track as they all have just one fork. These problems
need to be addressed by specifying aspects of a maze in which a rat is turning a corner. Further
analysis of existing data will be needed in order to determine firing properties that are characteristic of
this behaviour.
Figure 29: A U track, Y-maze, T-maze and W-Track
5.5 Future Extensions
The most important application of this project would be to apply the definitions to real data, and to see
the effect of the complexity of the data on the definitions chosen. As the data in this project is
simulated, it proves difficult to make any firm conclusions as to the real nature of place field relations.
Further investigation into the processing of place cell data may be required for real data, and more
sophisticated thresholding such as using the Egg-Yolk calculus or using a rate of change may be
required.
There are other aspects of place field relations that can be investigated. For example, it may be
interesting to represent different paths that the rat takes on its journey through the environment.
However, a problem may be that significant aspects of the maze that the rat does not travel through
may be overlooked. Nonetheless, this could be an alternative method of representing complex field
relations. Other ways that complex and large numbers of field relations could be represented instead
of as cliques of cliques could be representing the field relations as a spanning tree or as an induced
path. However, a minimum spanning tree does not reduce the amount of place fields in the
representation, and an induced path may not drastically reduce the number of place field relations,
depending on the complexity and linearity of the environment.
48
When the rat is placed in a new environment, remapping occurs. This remapping could be modelled
using the conceptual neighbourhood used with RCC relations. Also, topological distortions may also
be interesting to model. When a rat environment such as a linear track is expanded or contracted, the
rat’s place fields correspondingly expand or contract. These topological distortions may be interesting
to model to further examine the relations between these fields. Further investigation could also be
made as to whether place fields distort in a similar fashion after distortions in environments such as
arenas and forked environments.
5.6 Conclusion
In order to summarise this project, the minimum aims and requirements are re-visited:
Minimum requirement 1: Convert output data from an existing simulator of rat place cell firing into a
symbolic form
The existing simulator is familiarised with in order to convert the output of cell firings for the
different environments into a Prolog format. The firing data is then processed as described in section
3.3 by grouping the firings into temporal intervals and then using two methods to threshold the data.
Minimum Requirement 2: Make inferences from the symbolic form of place cell data to represent a
topological map of the environment being explored by a simulated rat.
Connectivity and the RCC-5 relations are defined for the place fields by using the place cell data as
described in section 3.4. The most appropriate threshold and temporal interval grouping are those that
produce the most accurate relations between place fields. Accurate relations between place fields in
the simulated data were detected, with an accuracy rate of 99% for the circle, W-, 8- and linear track.
Although the detected relations in the arena were not as accurate, using a variable threshold improves
the accuracy to account for variable place field sizes.
Requirement 3: Display the topological map in an appropriate form.
The map is displayed in three methods as qualitative descriptions, hand-drawn diagrams and computer
representations. As there were complex relations in the simulated data, simple relations larger
collections of place fields were displayed. The test data produced complete, correct qualitative data,
and few inconsistencies were present in the test VennMaster representation. The appropriateness of
each representation is discussed in the evaluation chapter (Section 5), where it was concluded that
DrawEuler did not convey much information about the topological layout of the rat environment, but
VennMaster, used with the other methods, could be used to complement each other to understand
more about the rat environment.
49
Several extensions to this project have been made:
Extension 1: Global properties of the environment.
Detecting the layout of the environment was not required for the representation of the topological
map, but was a form of further reasoning that could be useful for non-visual representations of the
map and for aiding manual visual representations of the map.
Extension 2:Limitations.
Although the limitations of the design were discussed as part of evaluating the project, some
assumptions made were addressed. For example, the impact of directional place cells on the
topological map was discussed with a suggested algorithm to detect such cells.
Extension 3: Representing complex data
As the existing computer tools were unable to represent large complex field relations, further
reasoning was made to detect “cliques of cliques” and to detect connections between these regions,
which may also have formed further cliques of “cliques of cliques”. The existence of these structures
in the data seemed to play some significance on the global properties of the layout, as shown by the
visual representation of these simpler regions of clique structures.
Extension 4: The connectivity relation
Alternative methods of defining connectivity were discussed. The idea of several fields connecting
extended the two-place connectivity relation into a “Simultaneously firing” relation containing any
number of places. This was successfully implemented and was also used to detect cliques, such as the
“triad intersection” predicate. A further notion of two fields connecting if there is nothing firing in
between was also introduced.
50
References
[1]Dabaghian, Cohn and Frank, “Topological Coding in Hippocampus” Quantitative Biology, 2007,
Accessed 10th April 2007, URL = <http://arxiv.org/abs/q-bio.OT/0702052 2007>
[2]Gottfried Willhelm Leibniz: Philosophical Texts, Oxford University Press, 1998, Woolhouse and
Francks (ed. and trans.)
[3] O’Keefe and Nadel, The hippocampus as a cognitive map, Oxford University Press, 1978
[4]Muller, “A Quarter Century of Place Cells”, Neuron, 17, 979-99, 1996
[5] Best and White, “Hippocampal cellular activity: A brief history of space”, The National Academy
of Sciences, 1998
[6] Burgess and O’Keefe, “Cognitive Graphs, Resistive Grids, and the Hippocampal Representation
of Space”, J.Gen.Physiol, 107, 659-662, 1996
[7] McNaughton, “Cognitive cartography”, Nature, 1996
[8] Gothard, Skaggs and McNaughton, “Dynamics of Mismatch Correction in the Hippocampal
Ensemble Code for Space: Interaction between Path Integration and Environmental Cues”, The
Journal of Neuroscience, 1996
[9] Knierim, McNaughton and Poe, “Three-dimensional spatial selectivity of hippocampal neurons
during space flight”, Nature neuroscience, 2000
[10]Cohn, Bennett, Gooday and Gotts, “Qualitative Spatial Representation and Reasoning with the
Region Connection Calculus”. Geoinformatica, 1, 1-4, 1997
[11] Cohn, “Calculi for Qualitative Spatial Reasoning”, Artificial Intelligence and Symbolic
Mathematical Computation, pp 124-143, Springer Verlag, 1996.
[12] Randell, Cui and Cohn. “A Spatial Logic based on Regions and Connection”, 3rd Int. Conf. on
Knowledge Representation and Reasoning, Morgan Kaufman, 1992.
[13] Cohn, A.G. “The Challenge of Qualitative Spatial Reasoning”, Computing Surveys, 27 , pp 323326, 1995.
[14] Renz, Qualitative Spatial Reasoning with Topological Information, Springer, 2002.
[15] Cohn and Hazarika, “Qualitative Spatial Representation and Reasoning: An Overview”,
Fundamenta Informaticae, 2001
[16] West, Introduction to Graph Theory, 2nd Edition, Prentice Hall, 1999
[17] Ruskey, “A survey of Venn diagrams”, Electronic Journal of Combinatorics, 4, 1997
[18]Shin, Sun-Joo, Lemon, Oliver, "Diagrams", The Stanford Encyclopedia of Philosophy (Spring 2007
Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/spr2007/entries/diagrams/>.
[19] Chow and Ruskey, “Drawing Area-Proportional Venn and Euler Diagrams”,Graph Drawing,
Springer Berlin/ Heidelberg, 2004
[20] Flower, Fish, and Howse, “Euler diagram generation”, Journal of Theoretical Computer Science,
2005.
[21] Kestler, Müller, Gress and Buchholz, “Generalized Venn diagrams: a new method of visualizing
complex genetic set relations”, Bioinformatics, 21: 8, pp 1592–1595, 2005
[22] Chow and Ruskey, “Towards a General Solution to Drawing Area-Proportional Euler Diagrams”,
Electronic Notes in Theoretical Computer Science, 134, pp 3-18, 2005
[23] Tamir and Kandel, “Logic programming and the execution model of Prolog”, Information
Sciences - Applications, 4, 3, pp 167- 191, 1995.
51
Appendix A: Reflection on Project Experience
The most enjoyable aspect of this project was its subject matter. I was pleased to find a project highly
relevant to my degree course, Cognitive Science, and I found the topic of place cells really fascinating
as I am particularly interested in neuroscience. This underlying enthusiasm for the project provided
the motivation to complete deadlines and stick to the time schedule. I would advise any students about
to undertake a final year project to investigate carefully areas of interest to them before choosing a
project; although there may have been easier projects, it needs to be captivating enough to continue
the standard of work throughout the whole academic year.
In choosing to undertake this project there was an element of risk involved, in that starting the
implementation relied on receiving simulated data elsewhere. As a result, at the beginning of the
project I was unsure of the project direction. For this reason it was hard to create a time schedule for
the project. In addition, the nature of this project is very open ended and has many possibilities; the
methodology and aims of the project were very sketchy to begin with and were only further refined
after an increased knowledge of place cells. A well-structured project from the beginning would have
been advantageous but this problem was unavoidable. Although methodology accounted for
developmental aspect of the project, I feel that at times I could have been better organised within the
each aspect of the project.
Background research for an Artificial Intelligence project is crucial in order to understand the nature
of what the project entails and to understand relevant background research from other disciplines such
as Biology. I found that allowing myself to conduct background reading throughout the whole of the
first semester better prepared me for the implementation of the project and I allowed myself enough
time to fully comprehend the literature and understand difficult topics such as the neurological
properties of the place cells. The wide range of reading I undertook required me to write a concise
background reading chapter, linking concepts together. I would recommend for future students to
take the mid-term project report seriously as it could mean a completed report chapter written before
the start of the second semester, and it also allows feedback from the assessor. Although I needed to
add more sections to my background reading chapter after the mid-project report, I did not drastically
change much of what I wrote originally other than the constructive comments I received, and this
reduced the work load when coming to write the rest of the report.
I feel that time management is one of the most important aspects of a large project as this. Although I
managed to keep within the time schedule that I had created for the academic year, it was impossible
to predict unforeseen circumstances outside of the project as well as aspects of the project which
would take longer than others. As the milestones in the schedule were still quite vague, there was not
52
as strict a schedule that could be followed to. I would recommend to any person conducting such a
project to write a detailed time plan and stick to it; and if necessary to amend it as time passes so that
each part of the project has sufficient time spent on it. I feel that my choice in programming language
may have delayed the time schedule slightly. Although I still think that Prolog is the best language to
test concepts about place cells, I did not take into account the fact that the simulated data was in
Matlab. This required me to learn some Matlab to do some basic processing so that I understood the
data files, and then to convert the Matlab files into Prolog. In addition, although I have programmed
in Prolog before, there are languages in which I am more proficient at programming in. A language
such as Java may have outweighed the cost of an inappropriate language for the project with the
benefit of less time spent on learning about the language.
I found that my meetings with my supervisor were very useful; firstly to ensure sufficient progress
was undertaken, but also to explore ideas and concepts that I used in my implementation. These
discussions helped me develop methods that I could test on the data. Students should not undervalue
their supervisors; if your supervisor has time to see you, they are your most valuable resource. They
are the only people that know what your project is about and will be the most likely person that can
help you.
Overall, I thoroughly enjoyed this project, and I am happy with how it went. Although I would have
liked to have tested my methods on real data to wrap up the project, I would not have had time to fully
analyse the data and write up the report. I have learnt many lessons about time management and
organisation and I am pleased to have been involved in an interdisciplinary project that has relevance
to current research.
Appendix B: Minimum Aims and requirements
Minimum requirement 1: Convert output data from an existing simulator of rat place cell firing into a
symbolic form.
Minimum Requirement 2: Make inferences from the symbolic form of place cell data to represent a
topological map of the environment being explored by a simulated rat.
Requirement 3: Display the topological map in an appropriate form.
53
Appendix C: Simulated data layouts
The following diagrams have been constructed in Matlab, and show the different environments that
correspond with the place cell firing data. Each place field is numbered and corresponds with a place
cell.
Arena
54
Circle Track
55
8-track
56
W-track
Linear Track
57
Appendix D: Test data layouts
These have been hand constructed and the relations that hold between the numbered place fields are
used as the input for creating qualitative descriptions and visual representations of the environment.
Layout 1:
1
2
3
4
Layout 2:
4
1
2
3
5
6
Layout 3:
1
2
3
6
7
4
5
Layout 4:
5
10
4
9
6
3
2
7
8
11
1
12
Layout 5:
2
1
11
3
4
10
5
9
Layout 6:
6
7
8
1
3
4
5
7
6
2
8
58
Appendix E: Qualitative descriptions of the test and simulated data.
Test Layout 1:
lineartrackbetween, [1, 4]
forks, 0
endfields, [1,4]
partiallyOverlaps(3, 4).
partiallyOverlaps(2, 3).
partiallyOverlaps(1, 2).
Test Layout 2:
lineartrackbetween[1, 6]
forks, 0
endfields, [1,6]
naïve, [4]
partiallyOverlaps(1, 2).
partiallyOverlaps(2, 3).
partiallyOverlaps(3, 4).
partiallyOverlaps(3, 5).
partiallyOverlaps(5, 6).
Test Layout3:
lineartrackbetween, [1, 5, 7],
forks, 1
endfields, [1,5,7]
partiallyOverlaps(6, 7).
partiallyOverlaps(4, 5).
partiallyOverlaps(3, 6).
partiallyOverlaps(3, 4).
partiallyOverlaps(2, 3).
partiallyOverlaps(1, 2).
Layout4:
lineartrackbetween[1, 5, 10, 12],
forks, 2
endfields, [1,5,10,12]
partiallyOverlaps(11, 12).
partiallyOverlaps(9, 10).
partiallyOverlaps(8, 11).
partiallyOverlaps(8, 9).
partiallyOverlaps(7, 8).
partiallyOverlaps(6, 7).
partiallyOverlaps(4, 5).
partiallyOverlaps(3, 6).
partiallyOverlaps(3, 4).
partiallyOverlaps(2, 3).
partiallyOverlaps(1, 2).
Test Layout5:
loop[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1]
[][]
partiallyOverlaps(1, 2).
partiallyOverlaps(1, 11).
partiallyOverlaps(2, 3).
partiallyOverlaps(3, 4).
partiallyOverlaps(4, 5).
partiallyOverlaps(5, 6).
partiallyOverlaps(6, 7).
partiallyOverlaps(7, 8).
partiallyOverlaps(8, 9).
partiallyOverlaps(9, 10).
partiallyOverlaps(10, 11).
Test Layout 6:
lineartrack[]
cliques[[1, 2, 3], [6, 7, 8]]
cliqueend, [3,8]
partiallyOverlaps(1, 2).
partiallyOverlaps(1, 3).
partiallyOverlaps(2, 3).
partiallyOverlaps(3, 1).
partiallyOverlaps(3, 4).
partiallyOverlaps(4, 5).
partiallyOverlaps(5, 6).
partiallyOverlaps(6, 7).
partiallyOverlaps(6, 8).
partiallyOverlaps(7, 8).
Arena:
Arena[]
partiallyOverlaps(43, 48).
partiallyOverlaps(43, 47).
partiallyOverlaps(43, 45).
partiallyOverlaps(42, 50).
partiallyOverlaps(41, 50).
partiallyOverlaps(41, 49).
partiallyOverlaps(41, 44).
partiallyOverlaps(41, 42).
partiallyOverlaps(40, 45).
partiallyOverlaps(39, 42).
partiallyOverlaps(37, 44).
partiallyOverlaps(37, 41).
partiallyOverlaps(37, 38).
partiallyOverlaps(35, 41).
partiallyOverlaps(35, 38).
partiallyOverlaps(35, 37).
partiallyOverlaps(34, 47).
partiallyOverlaps(34, 44).
partiallyOverlaps(34, 41).
partiallyOverlaps(34, 38).
partiallyOverlaps(34, 37).
partiallyOverlaps(34, 35).
partiallyOverlaps(33, 47).
partiallyOverlaps(33, 43).
partiallyOverlaps(32, 41).
partiallyOverlaps(32, 37).
partiallyOverlaps(31, 50).
partiallyOverlaps(31, 42).
partiallyOverlaps(31, 41).
partiallyOverlaps(30, 46).
partiallyOverlaps(30, 36).
partiallyOverlaps(30, 31).
partiallyOverlaps(29, 47).
partiallyOverlaps(29, 34).
partiallyOverlaps(28, 42).
partiallyOverlaps(28, 39).
partiallyOverlaps(26, 44).
partiallyOverlaps(26, 37).
partiallyOverlaps(26, 32).
partiallyOverlaps(26, 31).
partiallyOverlaps(25, 49).
partiallyOverlaps(25, 41).
partiallyOverlaps(25, 35).
partiallyOverlaps(24, 47).
partiallyOverlaps(24, 43).
partiallyOverlaps(24, 33).
partiallyOverlaps(24, 29).
partiallyOverlaps(22, 36).
partiallyOverlaps(22, 30).
partiallyOverlaps(21, 50).
partiallyOverlaps(21, 42).
partiallyOverlaps(21, 41).
partiallyOverlaps(21, 36).
partiallyOverlaps(21, 31).
partiallyOverlaps(21, 30).
partiallyOverlaps(20, 27).
partiallyOverlaps(19, 49).
partiallyOverlaps(19, 41).
partiallyOverlaps(19, 37).
partiallyOverlaps(19, 34).
partiallyOverlaps(19, 32).
partiallyOverlaps(19, 26).
partiallyOverlaps(19, 25).
partiallyOverlaps(18, 44).
partiallyOverlaps(18, 41).
partiallyOverlaps(18, 37).
partiallyOverlaps(18, 26).
partiallyOverlaps(17, 41).
partiallyOverlaps(17, 30).
partiallyOverlaps(17, 21).
partiallyOverlaps(16, 48).
partiallyOverlaps(16, 47).
partiallyOverlaps(16, 43).
partiallyOverlaps(16, 24).
partiallyOverlaps(15, 45).
partiallyOverlaps(15, 43).
partiallyOverlaps(14, 48).
partiallyOverlaps(14, 44).
partiallyOverlaps(14, 41).
partiallyOverlaps(14, 36).
partiallyOverlaps(14, 31).
partiallyOverlaps(14, 30).
partiallyOverlaps(14, 26).
partiallyOverlaps(14, 21).
partiallyOverlaps(14, 18).
partiallyOverlaps(14, 16).
partiallyOverlaps(13, 41).
partiallyOverlaps(13, 14).
partiallyOverlaps(12, 47).
partiallyOverlaps(12, 44).
partiallyOverlaps(12, 41).
partiallyOverlaps(12, 37).
partiallyOverlaps(12, 34).
partiallyOverlaps(12, 29).
partiallyOverlaps(12, 26).
partiallyOverlaps(12, 19).
partiallyOverlaps(12, 18).
partiallyOverlaps(11, 39).
partiallyOverlaps(10, 50).
partiallyOverlaps(10, 42).
partiallyOverlaps(10, 41).
partiallyOverlaps(10, 36).
partiallyOverlaps(10, 31).
partiallyOverlaps(10, 30).
partiallyOverlaps(10, 21).
partiallyOverlaps(10, 17).
partiallyOverlaps(10, 14).
partiallyOverlaps(10, 13).
partiallyOverlaps(9, 45).
partiallyOverlaps(9, 36).
partiallyOverlaps(9, 14).
partiallyOverlaps(8, 48).
partiallyOverlaps(8, 36).
partiallyOverlaps(8, 16).
partiallyOverlaps(8, 14).
partiallyOverlaps(8, 10).
partiallyOverlaps(8, 9).
partiallyOverlaps(7, 41).
partiallyOverlaps(7, 31).
partiallyOverlaps(7, 30).
partiallyOverlaps(7, 21).
partiallyOverlaps(7, 14).
partiallyOverlaps(6, 47).
partiallyOverlaps(5, 50).
partiallyOverlaps(5, 42).
partiallyOverlaps(5, 41).
partiallyOverlaps(5, 39).
partiallyOverlaps(5, 31).
partiallyOverlaps(5, 28).
partiallyOverlaps(5, 21).
partiallyOverlaps(5, 17).
partiallyOverlaps(5, 10).
partiallyOverlaps(4, 45).
partiallyOverlaps(4, 43).
partiallyOverlaps(4, 15).
partiallyOverlaps(4, 9).
partiallyOverlaps(3, 45).
partiallyOverlaps(3, 40).
59
partiallyOverlaps(3, 15).
partiallyOverlaps(3, 9).
partiallyOverlaps(3, 4).
partiallyOverlaps(2, 42).
partiallyOverlaps(2, 39).
partiallyOverlaps(2, 31).
partiallyOverlaps(2, 30).
partiallyOverlaps(2, 28).
partiallyOverlaps(2, 21).
partiallyOverlaps(2, 17).
partiallyOverlaps(2, 10).
partiallyOverlaps(2, 5).
partiallyOverlaps(1, 36).
partiallyOverlaps(1, 27).
partiallyOverlaps(1, 22).
partiallyOverlaps(1, 20).
properPart(6, 29).
properPart(7, 10).
properPart(11, 2).
properPart(23, 40).
properPart(26, 41).
clique(101, [1, 20, 27]).
clique(102, [1, 22, 36]).
clique(103, [2, 5, 10]).
clique(104, [2, 5, 17]).
clique(105, [2, 5, 21]).
clique(106, [2, 5, 28]).
clique(107, [2, 5, 31]).
clique(108, [2, 5, 39]).
clique(109, [2, 5, 42]).
clique(110, [2, 10, 17]).
clique(111, [2, 10, 21]).
clique(112, [2, 10, 30]).
clique(113, [2, 10, 31]).
clique(114, [2, 10, 42]).
clique(115, [2, 17, 21]).
clique(116, [2, 17, 30]).
clique(117, [2, 21, 30]).
clique(118, [2, 21, 31]).
clique(119, [2, 21, 42]).
clique(120, [2, 28, 39]).
clique(121, [2, 28, 42]).
clique(122, [2, 30, 31]).
clique(123, [2, 31, 42]).
clique(124, [2, 39, 42]).
clique(125, [3, 4, 9]).
clique(126, [3, 4, 15]).
clique(127, [3, 4, 45]).
clique(128, [3, 9, 45]).
clique(129, [3, 15, 45]).
clique(130, [3, 40, 45]).
clique(131, [4, 9, 45, 3]).
clique(132, [4, 15, 43]).
clique(133, [4, 15, 45, 3]).
clique(134, [4, 43, 45]).
clique(135, [5, 10, 17, 2]).
clique(136, [5, 10, 21, 2]).
clique(137, [5, 10, 31, 2]).
clique(138, [5, 10, 41]).
clique(139, [5, 10, 42, 2]).
clique(140, [5, 10, 50]).
clique(141, [5, 17, 21, 2]).
clique(142, [5, 17, 41]).
clique(143, [5, 21, 31, 2]).
clique(144, [5, 21, 41]).
clique(145, [5, 21, 42, 2]).
clique(146, [5, 21, 50]).
clique(147, [5, 28, 39, 2]).
clique(148, [5, 28, 42, 2]).
clique(149, [5, 31, 41]).
clique(150, [5, 31, 42, 2]).
clique(151, [5, 31, 50]).
clique(152, [5, 39, 42, 2]).
clique(153, [5, 41, 42]).
clique(154, [5, 41, 50]).
clique(155, [5, 42, 50]).
clique(156, [7, 14, 21]).
clique(157, [7, 14, 30]).
clique(158, [7, 14, 31]).
clique(159, [7, 14, 41]).
clique(160, [7, 21, 30]).
clique(161, [7, 21, 31]).
clique(162, [7, 21, 41]).
clique(163, [7, 30, 31]).
clique(164, [7, 31, 41]).
clique(165, [8, 9, 14]).
clique(166, [8, 9, 36]).
clique(167, [8, 10, 14]).
clique(168, [8, 10, 36]).
clique(169, [8, 14, 16]).
clique(170, [8, 14, 36]).
clique(171, [8, 14, 48]).
clique(172, [8, 16, 48]).
clique(173, [9, 14, 36, 8]).
clique(174, [10, 13, 14]).
clique(175, [10, 13, 41]).
clique(176, [10, 14, 21]).
clique(177, [10, 14, 30]).
clique(178, [10, 14, 31]).
clique(179, [10, 14, 36, 8]).
clique(180, [10, 14, 41]).
clique(181, [10, 17, 21, 5, 2]).
clique(182, [10, 17, 30, 2]).
clique(183, [10, 17, 41, 5]).
clique(184, [10, 21, 30, 2]).
clique(185, [10, 21, 31, 5, 2]).
clique(186, [10, 21, 36]).
clique(187, [10, 21, 41, 5]).
clique(188, [10, 21, 42, 5, 2]).
clique(189, [10, 21, 50, 5]).
clique(190, [10, 30, 31, 2]).
clique(191, [10, 30, 36]).
clique(192, [10, 31, 41, 5]).
clique(193, [10, 31, 42, 5, 2]).
clique(194, [10, 31, 50, 5]).
clique(195, [10, 41, 42, 5]).
clique(196, [10, 41, 50, 5]).
clique(197, [10, 42, 50, 5]).
clique(198, [12, 18, 26]).
clique(199, [12, 18, 37]).
clique(200, [12, 18, 41]).
clique(201, [12, 18, 44]).
clique(202, [12, 19, 26]).
clique(203, [12, 19, 34]).
clique(204, [12, 19, 37]).
clique(205, [12, 19, 41]).
clique(206, [12, 26, 37]).
clique(207, [12, 26, 44]).
clique(208, [12, 29, 34]).
clique(209, [12, 29, 47]).
clique(210, [12, 34, 37]).
clique(211, [12, 34, 41]).
clique(212, [12, 34, 44]).
clique(213, [12, 34, 47]).
clique(214, [12, 37, 41]).
clique(215, [12, 37, 44]).
clique(216, [12, 41, 44]).
clique(217, [13, 14, 41, 10]).
clique(218, [14, 16, 48, 8]).
clique(219, [14, 18, 26]).
clique(220, [14, 18, 41]).
clique(221, [14, 18, 44]).
clique(222, [14, 21, 30, 10]).
clique(223, [14, 21, 31, 10]).
clique(224, [14, 21, 36, 10]).
clique(225, [14, 21, 41, 10]).
clique(226, [14, 26, 31]).
clique(227, [14, 26, 44]).
clique(228, [14, 30, 31, 10]).
clique(229, [14, 30, 36, 10]).
clique(230, [14, 31, 41, 10]).
clique(231, [14, 41, 44]).
clique(232, [15, 43, 45, 4]).
clique(233, [16, 24, 43]).
clique(234, [16, 24, 47]).
clique(235, [16, 43, 47]).
clique(236, [16, 43, 48]).
clique(237, [17, 21, 30, 10, 2]).
clique(238, [17, 21, 41, 10, 5]).
clique(239, [18, 26, 37, 12]).
clique(240, [18, 26, 44, 14]).
clique(241, [18, 37, 41, 12]).
clique(242, [18, 37, 44, 12]).
clique(243, [18, 41, 44, 14]).
clique(244, [19, 25, 41]).
clique(245, [19, 25, 49]).
clique(246, [19, 26, 32]).
clique(247, [19, 26, 37, 12]).
clique(248, [19, 32, 37]).
clique(249, [19, 32, 41]).
clique(250, [19, 34, 37, 12]).
clique(251, [19, 34, 41, 12]).
clique(252, [19, 37, 41, 12]).
clique(253, [19, 41, 49]).
clique(254, [21, 30, 31, 14, 10]).
clique(255, [21, 30, 36, 14, 10]).
clique(256, [21, 31, 41, 14, 10]).
clique(257, [21, 31, 42, 10, 5, 2]).
clique(258, [21, 31, 50, 10, 5]).
clique(259, [21, 41, 42, 10, 5]).
clique(260, [21, 41, 50, 10, 5]).
clique(261, [21, 42, 50, 10, 5]).
clique(262, [22, 30, 36]).
clique(263, [24, 29, 47]).
clique(264, [24, 33, 43]).
clique(265, [24, 33, 47]).
clique(266, [24, 43, 47, 16]).
clique(267, [25, 35, 41]).
clique(268, [25, 41, 49, 19]).
clique(269, [26, 32, 37, 19]).
clique(270, [26, 37, 44, 18, 12]).
clique(271, [28, 39, 42, 5, 2]).
clique(272, [29, 34, 47, 12]).
clique(273, [31, 41, 42, 21, 10, 5]).
clique(274, [31, 41, 50, 21, 10, 5]).
clique(275, [31, 42, 50, 21, 10, 5]).
clique(276, [32, 37, 41, 19]).
clique(277, [33, 43, 47, 24]).
clique(278, [34, 35, 37]).
clique(279, [34, 35, 38]).
clique(280, [34, 35, 41]).
clique(281, [34, 37, 38]).
clique(282, [34, 37, 41, 19, 12]).
clique(283, [34, 37, 44, 12]).
clique(284, [34, 41, 44, 12]).
clique(285, [35, 37, 38, 34]).
clique(286, [35, 37, 41, 34]).
clique(287, [37, 41, 44, 34, 12]).
clique(288, [41, 42, 50, 31, 21, 10, 5]).
clique(401, [1, 20, 27]).
clique(402, [3, 40, 45]).
clique(403, [4, 9, 45, 3]).
clique(404, [4, 15, 45, 3]).
clique(405, [8, 16, 48]).
clique(406, [15, 43, 45, 4]).
clique(407, [16, 43, 48]).
clique(408, [24, 29, 47]).
clique(409, [24, 43, 47, 16]).
clique(410, [33, 43, 47, 24]).
superclique(201,[181, 185, 188, 193, 237,
238, 254, 255, 256, 257, 258, 259, 260, 261,
273, 274, 275, 288]).
60
superclique(202, [288, 274, 273, 260, 259,
256, 238, 282, 287]).
superclique(203, [287, 270, 282]).
superclique(204, [288, 275, 274, 273, 261,
260, 259, 258, 257, 238, 237, 193, 188, 271,
181, 185]).
W track
Arena[]
partiallyOverlaps(66, 67).
partiallyOverlaps(65, 70).
partiallyOverlaps(65, 66).
partiallyOverlaps(62, 69).
partiallyOverlaps(61, 68).
partiallyOverlaps(60, 64).
partiallyOverlaps(59, 61).
partiallyOverlaps(58, 69).
partiallyOverlaps(58, 62).
partiallyOverlaps(57, 66).
partiallyOverlaps(56, 68).
partiallyOverlaps(56, 59).
partiallyOverlaps(55, 62).
partiallyOverlaps(55, 58).
partiallyOverlaps(54, 64).
partiallyOverlaps(54, 60).
partiallyOverlaps(53, 67).
partiallyOverlaps(53, 66).
partiallyOverlaps(53, 65).
partiallyOverlaps(52, 62).
partiallyOverlaps(51, 67).
partiallyOverlaps(51, 66).
partiallyOverlaps(51, 57).
partiallyOverlaps(51, 53).
partiallyOverlaps(50, 61).
partiallyOverlaps(50, 59).
partiallyOverlaps(48, 62).
partiallyOverlaps(48, 58).
partiallyOverlaps(48, 55).
partiallyOverlaps(48, 52).
partiallyOverlaps(47, 64).
partiallyOverlaps(47, 60).
partiallyOverlaps(47, 54).
partiallyOverlaps(46, 62).
partiallyOverlaps(46, 58).
partiallyOverlaps(46, 48).
partiallyOverlaps(45, 61).
partiallyOverlaps(45, 59).
partiallyOverlaps(45, 56).
partiallyOverlaps(45, 50).
partiallyOverlaps(44, 52).
partiallyOverlaps(44, 49).
partiallyOverlaps(43, 67).
partiallyOverlaps(43, 66).
partiallyOverlaps(43, 57).
partiallyOverlaps(43, 53).
partiallyOverlaps(43, 51).
partiallyOverlaps(42, 64).
partiallyOverlaps(41, 61).
partiallyOverlaps(41, 59).
partiallyOverlaps(41, 50).
partiallyOverlaps(41, 45).
partiallyOverlaps(40, 52).
partiallyOverlaps(40, 49).
partiallyOverlaps(40, 44).
partiallyOverlaps(39, 61).
partiallyOverlaps(39, 59).
partiallyOverlaps(39, 50).
partiallyOverlaps(39, 45).
partiallyOverlaps(38, 57).
partiallyOverlaps(38, 51).
partiallyOverlaps(38, 50).
partiallyOverlaps(37, 59).
partiallyOverlaps(37, 56).
partiallyOverlaps(37, 50).
partiallyOverlaps(37, 45).
partiallyOverlaps(37, 38).
partiallyOverlaps(36, 69).
partiallyOverlaps(36, 62).
partiallyOverlaps(36, 58).
partiallyOverlaps(36, 55).
partiallyOverlaps(36, 46).
partiallyOverlaps(35, 70).
partiallyOverlaps(35, 57).
partiallyOverlaps(35, 50).
partiallyOverlaps(35, 45).
partiallyOverlaps(35, 39).
partiallyOverlaps(35, 38).
partiallyOverlaps(35, 37).
partiallyOverlaps(34, 66).
partiallyOverlaps(34, 65).
partiallyOverlaps(34, 53).
partiallyOverlaps(33, 70).
partiallyOverlaps(33, 66).
partiallyOverlaps(33, 57).
partiallyOverlaps(33, 51).
partiallyOverlaps(33, 50).
partiallyOverlaps(33, 43).
partiallyOverlaps(33, 38).
partiallyOverlaps(33, 35).
partiallyOverlaps(32, 69).
partiallyOverlaps(32, 62).
partiallyOverlaps(32, 58).
partiallyOverlaps(32, 55).
partiallyOverlaps(32, 52).
partiallyOverlaps(32, 48).
partiallyOverlaps(32, 46).
partiallyOverlaps(32, 36).
partiallyOverlaps(31, 49).
partiallyOverlaps(31, 44).
partiallyOverlaps(31, 40).
partiallyOverlaps(30, 69).
partiallyOverlaps(30, 62).
partiallyOverlaps(30, 46).
partiallyOverlaps(30, 36).
partiallyOverlaps(30, 32).
partiallyOverlaps(29, 64).
partiallyOverlaps(29, 60).
partiallyOverlaps(29, 54).
partiallyOverlaps(28, 68).
partiallyOverlaps(28, 61).
partiallyOverlaps(28, 59).
partiallyOverlaps(28, 56).
partiallyOverlaps(28, 50).
partiallyOverlaps(28, 45).
partiallyOverlaps(28, 41).
partiallyOverlaps(28, 39).
partiallyOverlaps(28, 37).
partiallyOverlaps(27, 68).
partiallyOverlaps(27, 56).
partiallyOverlaps(27, 37).
partiallyOverlaps(27, 28).
partiallyOverlaps(26, 49).
partiallyOverlaps(26, 44).
partiallyOverlaps(26, 40).
partiallyOverlaps(26, 31).
partiallyOverlaps(25, 63).
partiallyOverlaps(25, 41).
partiallyOverlaps(24, 57).
partiallyOverlaps(24, 51).
partiallyOverlaps(24, 50).
partiallyOverlaps(24, 45).
partiallyOverlaps(24, 39).
partiallyOverlaps(24, 38).
partiallyOverlaps(24, 37).
partiallyOverlaps(24, 35).
partiallyOverlaps(24, 33).
partiallyOverlaps(23, 64).
partiallyOverlaps(23, 54).
partiallyOverlaps(23, 42).
partiallyOverlaps(23, 29).
partiallyOverlaps(22, 49).
partiallyOverlaps(22, 44).
partiallyOverlaps(22, 40).
partiallyOverlaps(22, 31).
partiallyOverlaps(22, 26).
partiallyOverlaps(21, 62).
partiallyOverlaps(21, 52).
partiallyOverlaps(21, 48).
partiallyOverlaps(21, 46).
partiallyOverlaps(21, 40).
partiallyOverlaps(21, 36).
partiallyOverlaps(21, 32).
partiallyOverlaps(21, 30).
partiallyOverlaps(20, 60).
partiallyOverlaps(20, 54).
partiallyOverlaps(20, 47).
partiallyOverlaps(20, 29).
partiallyOverlaps(19, 70).
partiallyOverlaps(19, 66).
partiallyOverlaps(19, 57).
partiallyOverlaps(19, 51).
partiallyOverlaps(19, 50).
partiallyOverlaps(19, 45).
partiallyOverlaps(19, 39).
partiallyOverlaps(19, 38).
partiallyOverlaps(19, 35).
partiallyOverlaps(19, 33).
partiallyOverlaps(19, 24).
partiallyOverlaps(18, 61).
partiallyOverlaps(18, 59).
partiallyOverlaps(18, 50).
partiallyOverlaps(18, 45).
partiallyOverlaps(18, 39).
partiallyOverlaps(18, 37).
partiallyOverlaps(18, 35).
partiallyOverlaps(18, 28).
partiallyOverlaps(18, 24).
partiallyOverlaps(17, 63).
partiallyOverlaps(17, 41).
partiallyOverlaps(17, 25).
partiallyOverlaps(16, 57).
partiallyOverlaps(16, 56).
partiallyOverlaps(16, 50).
partiallyOverlaps(16, 45).
partiallyOverlaps(16, 39).
partiallyOverlaps(16, 38).
partiallyOverlaps(16, 37).
partiallyOverlaps(16, 35).
partiallyOverlaps(16, 33).
partiallyOverlaps(16, 28).
partiallyOverlaps(16, 24).
partiallyOverlaps(16, 19).
partiallyOverlaps(16, 18).
partiallyOverlaps(15, 68).
partiallyOverlaps(15, 61).
partiallyOverlaps(15, 59).
partiallyOverlaps(15, 56).
partiallyOverlaps(15, 45).
partiallyOverlaps(15, 37).
partiallyOverlaps(15, 28).
partiallyOverlaps(15, 27).
partiallyOverlaps(15, 18).
partiallyOverlaps(14, 69).
partiallyOverlaps(14, 62).
partiallyOverlaps(14, 58).
partiallyOverlaps(14, 55).
partiallyOverlaps(14, 52).
partiallyOverlaps(14, 48).
partiallyOverlaps(14, 46).
partiallyOverlaps(14, 44).
partiallyOverlaps(14, 40).
partiallyOverlaps(14, 36).
partiallyOverlaps(14, 32).
partiallyOverlaps(14, 30).
partiallyOverlaps(14, 21).
61
partiallyOverlaps(13, 59).
partiallyOverlaps(13, 56).
partiallyOverlaps(13, 45).
partiallyOverlaps(13, 38).
partiallyOverlaps(13, 37).
partiallyOverlaps(13, 28).
partiallyOverlaps(13, 27).
partiallyOverlaps(13, 18).
partiallyOverlaps(13, 16).
partiallyOverlaps(13, 15).
partiallyOverlaps(12, 62).
partiallyOverlaps(12, 55).
partiallyOverlaps(12, 52).
partiallyOverlaps(12, 48).
partiallyOverlaps(12, 46).
partiallyOverlaps(12, 44).
partiallyOverlaps(12, 40).
partiallyOverlaps(12, 32).
partiallyOverlaps(12, 21).
partiallyOverlaps(12, 14).
partiallyOverlaps(11, 69).
partiallyOverlaps(11, 62).
partiallyOverlaps(11, 58).
partiallyOverlaps(11, 55).
partiallyOverlaps(11, 48).
partiallyOverlaps(11, 32).
partiallyOverlaps(11, 14).
partiallyOverlaps(10, 61).
partiallyOverlaps(10, 59).
partiallyOverlaps(10, 50).
partiallyOverlaps(10, 45).
partiallyOverlaps(10, 41).
partiallyOverlaps(10, 39).
partiallyOverlaps(10, 28).
partiallyOverlaps(10, 25).
partiallyOverlaps(10, 18).
partiallyOverlaps(9, 61).
partiallyOverlaps(9, 59).
partiallyOverlaps(9, 45).
partiallyOverlaps(9, 41).
partiallyOverlaps(9, 28).
partiallyOverlaps(9, 25).
partiallyOverlaps(9, 10).
partiallyOverlaps(8, 49).
partiallyOverlaps(8, 44).
partiallyOverlaps(8, 40).
partiallyOverlaps(8, 31).
partiallyOverlaps(8, 26).
partiallyOverlaps(8, 22).
partiallyOverlaps(7, 64).
partiallyOverlaps(7, 60).
partiallyOverlaps(7, 54).
partiallyOverlaps(7, 47).
partiallyOverlaps(7, 29).
partiallyOverlaps(7, 20).
partiallyOverlaps(6, 49).
partiallyOverlaps(6, 44).
partiallyOverlaps(6, 26).
partiallyOverlaps(6, 8).
partiallyOverlaps(5, 49).
partiallyOverlaps(5, 44).
partiallyOverlaps(5, 31).
partiallyOverlaps(5, 26).
partiallyOverlaps(5, 22).
partiallyOverlaps(5, 8).
partiallyOverlaps(5, 6).
partiallyOverlaps(4, 64).
partiallyOverlaps(4, 60).
partiallyOverlaps(4, 54).
partiallyOverlaps(4, 29).
partiallyOverlaps(4, 23).
partiallyOverlaps(4, 20).
partiallyOverlaps(4, 7).
partiallyOverlaps(3, 69).
partiallyOverlaps(3, 62).
partiallyOverlaps(3, 48).
partiallyOverlaps(3, 46).
partiallyOverlaps(3, 36).
partiallyOverlaps(3, 32).
partiallyOverlaps(3, 30).
partiallyOverlaps(3, 21).
partiallyOverlaps(3, 14).
partiallyOverlaps(3, 12).
partiallyOverlaps(2, 68).
partiallyOverlaps(2, 61).
partiallyOverlaps(2, 56).
partiallyOverlaps(2, 45).
partiallyOverlaps(2, 37).
partiallyOverlaps(2, 28).
partiallyOverlaps(2, 27).
partiallyOverlaps(2, 18).
partiallyOverlaps(2, 16).
partiallyOverlaps(2, 15).
partiallyOverlaps(2, 13).
partiallyOverlaps(1, 70).
partiallyOverlaps(1, 67).
partiallyOverlaps(1, 66).
partiallyOverlaps(1, 65).
partiallyOverlaps(1, 57).
partiallyOverlaps(1, 53).
partiallyOverlaps(1, 51).
partiallyOverlaps(1, 43).
partiallyOverlaps(1, 38).
partiallyOverlaps(1, 35).
partiallyOverlaps(1, 33).
partiallyOverlaps(1, 24).
partiallyOverlaps(1, 19).
partiallyOverlaps(1, 16).
clique(101, [1, 16, 19]).
clique(102, [1, 16, 24]).
clique(103, [1, 16, 33]).
clique(104, [1, 16, 35]).
clique(105, [1, 16, 38]).
clique(106, [1, 16, 57]).
clique(107, [1, 19, 24]).
clique(108, [1, 19, 33]).
clique(109, [1, 19, 35]).
clique(110, [1, 19, 38]).
clique(111, [1, 19, 51]).
clique(112, [1, 19, 57]).
clique(113, [1, 19, 66]).
clique(114, [1, 19, 70]).
clique(115, [1, 24, 33]).
clique(116, [1, 24, 35]).
clique(117, [1, 24, 38]).
clique(118, [1, 24, 51]).
clique(119, [1, 24, 57]).
clique(120, [1, 33, 35]).
clique(121, [1, 33, 38]).
clique(122, [1, 33, 43]).
clique(123, [1, 33, 51]).
clique(124, [1, 33, 57]).
clique(125, [1, 33, 66]).
clique(126, [1, 33, 70]).
clique(127, [1, 35, 38]).
clique(128, [1, 35, 57]).
clique(129, [1, 35, 70]).
clique(130, [1, 38, 51]).
clique(131, [1, 38, 57]).
clique(132, [1, 43, 51]).
clique(133, [1, 43, 53]).
clique(134, [1, 43, 57]).
clique(135, [1, 43, 66]).
clique(136, [1, 43, 67]).
clique(137, [1, 51, 53]).
clique(138, [1, 51, 57]).
clique(139, [1, 51, 66]).
clique(140, [1, 51, 67]).
clique(141, [1, 53, 65]).
clique(142, [1, 53, 66]).
clique(143, [1, 53, 67]).
clique(144, [1, 57, 66]).
clique(145, [1, 65, 66]).
clique(146, [1, 65, 70]).
clique(147, [1, 66, 67]).
clique(148, [2, 13, 15]).
clique(149, [2, 13, 16]).
clique(150, [2, 13, 18]).
clique(151, [2, 13, 27]).
clique(152, [2, 13, 28]).
clique(153, [2, 13, 37]).
clique(154, [2, 13, 45]).
clique(155, [2, 13, 56]).
clique(156, [2, 15, 18]).
clique(157, [2, 15, 27]).
clique(158, [2, 15, 28]).
clique(159, [2, 15, 37]).
clique(160, [2, 15, 45]).
clique(161, [2, 15, 56]).
clique(162, [2, 15, 61]).
clique(163, [2, 15, 68]).
clique(164, [2, 16, 18]).
clique(165, [2, 16, 28]).
clique(166, [2, 16, 37]).
clique(167, [2, 16, 45]).
clique(168, [2, 16, 56]).
clique(169, [2, 18, 28]).
clique(170, [2, 18, 37]).
clique(171, [2, 18, 45]).
clique(172, [2, 18, 61]).
clique(173, [2, 27, 28]).
clique(174, [2, 27, 37]).
clique(175, [2, 27, 56]).
clique(176, [2, 27, 68]).
clique(177, [2, 28, 37]).
clique(178, [2, 28, 45]).
clique(179, [2, 28, 56]).
clique(180, [2, 28, 61]).
clique(181, [2, 28, 68]).
clique(182, [2, 37, 45]).
clique(183, [2, 37, 56]).
clique(184, [2, 45, 56]).
clique(185, [2, 45, 61]).
clique(186, [2, 56, 68]).
clique(187, [2, 61, 68]).
clique(188, [3, 12, 14]).
clique(189, [3, 12, 21]).
clique(190, [3, 12, 32]).
clique(191, [3, 12, 46]).
clique(192, [3, 12, 48]).
clique(193, [3, 12, 62]).
clique(194, [3, 14, 21]).
clique(195, [3, 14, 30]).
clique(196, [3, 14, 32]).
clique(197, [3, 14, 36]).
clique(198, [3, 14, 46]).
clique(199, [3, 14, 48]).
clique(200, [3, 14, 62]).
clique(201, [3, 14, 69]).
clique(202, [3, 21, 30]).
clique(203, [3, 21, 32]).
clique(204, [3, 21, 36]).
clique(205, [3, 21, 46]).
clique(206, [3, 21, 48]).
clique(207, [3, 21, 62]).
clique(208, [3, 30, 32]).
clique(209, [3, 30, 36]).
clique(210, [3, 30, 46]).
clique(211, [3, 30, 62]).
clique(212, [3, 30, 69]).
clique(213, [3, 32, 36]).
clique(214, [3, 32, 46]).
clique(215, [3, 32, 48]).
clique(216, [3, 32, 62]).
62
clique(217, [3, 32, 69]).
clique(218, [3, 36, 46]).
clique(219, [3, 36, 62]).
clique(220, [3, 36, 69]).
clique(221, [3, 46, 48]).
clique(222, [3, 46, 62]).
clique(223, [3, 48, 62]).
clique(224, [3, 62, 69]).
clique(225, [4, 7, 20]).
clique(226, [4, 7, 29]).
clique(227, [4, 7, 54]).
clique(228, [4, 7, 60]).
clique(229, [4, 7, 64]).
clique(230, [4, 20, 29]).
clique(231, [4, 20, 54]).
clique(232, [4, 20, 60]).
clique(233, [4, 23, 29]).
clique(234, [4, 23, 54]).
clique(235, [4, 23, 64]).
clique(236, [4, 29, 54]).
clique(237, [4, 29, 60]).
clique(238, [4, 29, 64]).
clique(239, [4, 54, 60]).
clique(240, [4, 54, 64]).
clique(241, [4, 60, 64]).
clique(242, [5, 6, 8]).
clique(243, [5, 6, 26]).
clique(244, [5, 6, 44]).
clique(245, [5, 6, 49]).
clique(246, [5, 8, 22]).
clique(247, [5, 8, 26]).
clique(248, [5, 8, 31]).
clique(249, [5, 8, 44]).
clique(250, [5, 8, 49]).
clique(251, [5, 22, 26]).
clique(252, [5, 22, 31]).
clique(253, [5, 22, 44]).
clique(254, [5, 22, 49]).
clique(255, [5, 26, 31]).
clique(256, [5, 26, 44]).
clique(257, [5, 26, 49]).
clique(258, [5, 31, 44]).
clique(259, [5, 31, 49]).
clique(260, [5, 44, 49]).
clique(261, [6, 8, 26, 5]).
clique(262, [6, 8, 44, 5]).
clique(263, [6, 8, 49, 5]).
clique(264, [6, 26, 44, 5]).
clique(265, [6, 26, 49, 5]).
clique(266, [6, 44, 49, 5]).
clique(267, [7, 20, 29, 4]).
clique(268, [7, 20, 47]).
clique(269, [7, 20, 54, 4]).
clique(270, [7, 20, 60, 4]).
clique(271, [7, 29, 54, 4]).
clique(272, [7, 29, 60, 4]).
clique(273, [7, 29, 64, 4]).
clique(274, [7, 47, 54]).
clique(275, [7, 47, 60]).
clique(276, [7, 47, 64]).
clique(277, [7, 54, 60, 4]).
clique(278, [7, 54, 64, 4]).
clique(279, [7, 60, 64, 4]).
clique(280, [8, 22, 26, 5]).
clique(281, [8, 22, 31, 5]).
clique(282, [8, 22, 40]).
clique(283, [8, 22, 44, 5]).
clique(284, [8, 22, 49, 5]).
clique(285, [8, 26, 31, 5]).
clique(286, [8, 26, 40]).
clique(287, [8, 26, 44, 6, 5]).
clique(288, [8, 26, 49, 6, 5]).
clique(289, [8, 31, 40]).
clique(290, [8, 31, 44, 5]).
clique(291, [8, 31, 49, 5]).
clique(292, [8, 40, 44]).
clique(293, [8, 40, 49]).
clique(294, [8, 44, 49, 6, 5]).
clique(295, [9, 10, 25]).
clique(296, [9, 10, 28]).
clique(297, [9, 10, 41]).
clique(298, [9, 10, 45]).
clique(299, [9, 10, 59]).
clique(300, [9, 10, 61]).
clique(301, [9, 25, 41]).
clique(302, [9, 28, 41]).
clique(303, [9, 28, 45]).
clique(304, [9, 28, 59]).
clique(305, [9, 28, 61]).
clique(306, [9, 41, 45]).
clique(307, [9, 41, 59]).
clique(308, [9, 41, 61]).
clique(309, [9, 45, 59]).
clique(310, [9, 45, 61]).
clique(311, [9, 59, 61]).
clique(312, [10, 18, 28]).
clique(313, [10, 18, 39]).
clique(314, [10, 18, 45]).
clique(315, [10, 18, 50]).
clique(316, [10, 18, 59]).
clique(317, [10, 18, 61]).
clique(318, [10, 25, 41, 9]).
clique(319, [10, 28, 39]).
clique(320, [10, 28, 41, 9]).
clique(321, [10, 28, 45, 9]).
clique(322, [10, 28, 50]).
clique(323, [10, 28, 59, 9]).
clique(324, [10, 28, 61, 9]).
clique(325, [10, 39, 45]).
clique(326, [10, 39, 50]).
clique(327, [10, 39, 59]).
clique(328, [10, 39, 61]).
clique(329, [10, 41, 45, 9]).
clique(330, [10, 41, 50]).
clique(331, [10, 41, 59, 9]).
clique(332, [10, 41, 61, 9]).
clique(333, [10, 45, 50]).
clique(334, [10, 45, 59, 9]).
clique(335, [10, 45, 61, 9]).
clique(336, [10, 50, 59]).
clique(337, [10, 50, 61]).
clique(338, [10, 59, 61, 9]).
clique(339, [11, 14, 32]).
clique(340, [11, 14, 48]).
clique(341, [11, 14, 55]).
clique(342, [11, 14, 58]).
clique(343, [11, 14, 62]).
clique(344, [11, 14, 69]).
clique(345, [11, 32, 48]).
clique(346, [11, 32, 55]).
clique(347, [11, 32, 58]).
clique(348, [11, 32, 62]).
clique(349, [11, 32, 69]).
clique(350, [11, 48, 55]).
clique(351, [11, 48, 58]).
clique(352, [11, 48, 62]).
clique(353, [11, 55, 58]).
clique(354, [11, 55, 62]).
clique(355, [11, 58, 62]).
clique(356, [11, 58, 69]).
clique(357, [11, 62, 69]).
clique(358, [12, 14, 21, 3]).
clique(359, [12, 14, 32, 3]).
clique(360, [12, 14, 40]).
clique(361, [12, 14, 44]).
clique(362, [12, 14, 46, 3]).
clique(363, [12, 14, 48, 3]).
clique(364, [12, 14, 52]).
clique(365, [12, 14, 55]).
clique(366, [12, 14, 62, 3]).
clique(367, [12, 21, 32, 3]).
clique(368, [12, 21, 40]).
clique(369, [12, 21, 46, 3]).
clique(370, [12, 21, 48, 3]).
clique(371, [12, 21, 52]).
clique(372, [12, 21, 62, 3]).
clique(373, [12, 32, 46, 3]).
clique(374, [12, 32, 48, 3]).
clique(375, [12, 32, 52]).
clique(376, [12, 32, 55]).
clique(377, [12, 32, 62, 3]).
clique(378, [12, 40, 44]).
clique(379, [12, 40, 52]).
clique(380, [12, 44, 52]).
clique(381, [12, 46, 48, 3]).
clique(382, [12, 46, 62, 3]).
clique(383, [12, 48, 52]).
clique(384, [12, 48, 55]).
clique(385, [12, 48, 62, 3]).
clique(386, [12, 52, 62]).
clique(387, [12, 55, 62]).
clique(388, [13, 15, 18, 2]).
clique(389, [13, 15, 27, 2]).
clique(390, [13, 15, 28, 2]).
clique(391, [13, 15, 37, 2]).
clique(392, [13, 15, 45, 2]).
clique(393, [13, 15, 56, 2]).
clique(394, [13, 15, 59]).
clique(395, [13, 16, 18, 2]).
clique(396, [13, 16, 28, 2]).
clique(397, [13, 16, 37, 2]).
clique(398, [13, 16, 38]).
clique(399, [13, 16, 45, 2]).
clique(400, [13, 16, 56, 2]).
clique(401, [13, 18, 28, 2]).
clique(402, [13, 18, 37, 2]).
clique(403, [13, 18, 45, 2]).
clique(404, [13, 18, 59]).
clique(405, [13, 27, 28, 2]).
clique(406, [13, 27, 37, 2]).
clique(407, [13, 27, 56, 2]).
clique(408, [13, 28, 37, 2]).
clique(409, [13, 28, 45, 2]).
clique(410, [13, 28, 56, 2]).
clique(411, [13, 28, 59]).
clique(412, [13, 37, 38]).
clique(413, [13, 37, 45, 2]).
clique(414, [13, 37, 56, 2]).
clique(415, [13, 37, 59]).
clique(416, [13, 45, 56, 2]).
clique(417, [13, 45, 59]).
clique(418, [13, 56, 59]).
clique(419, [14, 21, 30, 3]).
clique(420, [14, 21, 32, 12, 3]).
clique(421, [14, 21, 36, 3]).
clique(422, [14, 21, 40, 12]).
clique(423, [14, 21, 46, 12, 3]).
clique(424, [14, 21, 48, 12, 3]).
clique(425, [14, 21, 52, 12]).
clique(426, [14, 21, 62, 12, 3]).
clique(427, [14, 30, 32, 3]).
clique(428, [14, 30, 36, 3]).
clique(429, [14, 30, 46, 3]).
clique(430, [14, 30, 62, 3]).
clique(431, [14, 30, 69, 3]).
clique(432, [14, 32, 36, 3]).
clique(433, [14, 32, 46, 12, 3]).
clique(434, [14, 32, 48, 12, 3]).
clique(435, [14, 32, 52, 12]).
clique(436, [14, 32, 55, 12]).
clique(437, [14, 32, 58, 11]).
clique(438, [14, 32, 62, 12, 3]).
clique(439, [14, 32, 69, 11]).
clique(440, [14, 36, 46, 3]).
clique(441, [14, 36, 55]).
63
clique(442, [14, 36, 58]).
clique(443, [14, 36, 62, 3]).
clique(444, [14, 36, 69, 3]).
clique(445, [14, 40, 44, 12]).
clique(446, [14, 40, 52, 12]).
clique(447, [14, 44, 52, 12]).
clique(448, [14, 46, 48, 12, 3]).
clique(449, [14, 46, 58]).
clique(450, [14, 46, 62, 12, 3]).
clique(451, [14, 48, 52, 12]).
clique(452, [14, 48, 55, 12]).
clique(453, [14, 48, 58, 11]).
clique(454, [14, 48, 62, 12, 3]).
clique(455, [14, 52, 62, 12]).
clique(456, [14, 55, 58, 11]).
clique(457, [14, 55, 62, 12]).
clique(458, [14, 58, 62, 11]).
clique(459, [14, 58, 69, 11]).
clique(460, [14, 62, 69, 11]).
clique(461, [15, 18, 28, 13, 2]).
clique(462, [15, 18, 37, 13, 2]).
clique(463, [15, 18, 45, 13, 2]).
clique(464, [15, 18, 59, 13]).
clique(465, [15, 18, 61, 2]).
clique(466, [15, 27, 28, 13, 2]).
clique(467, [15, 27, 37, 13, 2]).
clique(468, [15, 27, 56, 13, 2]).
clique(469, [15, 27, 68, 2]).
clique(470, [15, 28, 37, 13, 2]).
clique(471, [15, 28, 45, 13, 2]).
clique(472, [15, 28, 56, 13, 2]).
clique(473, [15, 28, 59, 13]).
clique(474, [15, 28, 61, 2]).
clique(475, [15, 28, 68, 2]).
clique(476, [15, 37, 45, 13, 2]).
clique(477, [15, 37, 56, 13, 2]).
clique(478, [15, 37, 59, 13]).
clique(479, [15, 45, 56, 13, 2]).
clique(480, [15, 45, 59, 13]).
clique(481, [15, 45, 61, 2]).
clique(482, [15, 56, 59, 13]).
clique(483, [15, 56, 68, 2]).
clique(484, [15, 59, 61]).
clique(485, [15, 61, 68, 2]).
clique(486, [16, 18, 24]).
clique(487, [16, 18, 28, 13, 2]).
clique(488, [16, 18, 35]).
clique(489, [16, 18, 37, 13, 2]).
clique(490, [16, 18, 39]).
clique(491, [16, 18, 45, 13, 2]).
clique(492, [16, 18, 50]).
clique(493, [16, 19, 24, 1]).
clique(494, [16, 19, 33, 1]).
clique(495, [16, 19, 35, 1]).
clique(496, [16, 19, 38, 1]).
clique(497, [16, 19, 39]).
clique(498, [16, 19, 45]).
clique(499, [16, 19, 50]).
clique(500, [16, 19, 57, 1]).
clique(501, [16, 24, 33, 1]).
clique(502, [16, 24, 35, 1]).
clique(503, [16, 24, 37]).
clique(504, [16, 24, 38, 1]).
clique(505, [16, 24, 39]).
clique(506, [16, 24, 45]).
clique(507, [16, 24, 50]).
clique(508, [16, 24, 57, 1]).
clique(509, [16, 28, 37, 13, 2]).
clique(510, [16, 28, 39]).
clique(511, [16, 28, 45, 13, 2]).
clique(512, [16, 28, 50]).
clique(513, [16, 28, 56, 13, 2]).
clique(514, [16, 33, 35, 1]).
clique(515, [16, 33, 38, 1]).
clique(516, [16, 33, 50]).
clique(517, [16, 33, 57, 1]).
clique(518, [16, 35, 37]).
clique(519, [16, 35, 38, 1]).
clique(520, [16, 35, 39]).
clique(521, [16, 35, 45]).
clique(522, [16, 35, 50]).
clique(523, [16, 35, 57, 1]).
clique(524, [16, 37, 38, 13]).
clique(525, [16, 37, 45, 13, 2]).
clique(526, [16, 37, 50]).
clique(527, [16, 37, 56, 13, 2]).
clique(528, [16, 38, 50]).
clique(529, [16, 38, 57, 1]).
clique(530, [16, 39, 45]).
clique(531, [16, 39, 50]).
clique(532, [16, 45, 50]).
clique(533, [16, 45, 56, 13, 2]).
clique(534, [17, 25, 41]).
clique(535, [17, 25, 63]).
clique(536, [18, 24, 35, 16]).
clique(537, [18, 24, 37, 16]).
clique(538, [18, 24, 39, 16]).
clique(539, [18, 24, 45, 16]).
clique(540, [18, 24, 50, 16]).
clique(541, [18, 28, 37, 16, 13, 2]).
clique(542, [18, 28, 39, 16]).
clique(543, [18, 28, 45, 16, 13, 2]).
clique(544, [18, 28, 50, 16]).
clique(545, [18, 28, 59, 15, 13]).
clique(546, [18, 28, 61, 15, 2]).
clique(547, [18, 35, 37, 16]).
clique(548, [18, 35, 39, 16]).
clique(549, [18, 35, 45, 16]).
clique(550, [18, 35, 50, 16]).
clique(551, [18, 37, 45, 16, 13, 2]).
clique(552, [18, 37, 50, 16]).
clique(553, [18, 37, 59, 15, 13]).
clique(554, [18, 39, 45, 16]).
clique(555, [18, 39, 50, 16]).
clique(556, [18, 39, 59, 10]).
clique(557, [18, 39, 61, 10]).
clique(558, [18, 45, 50, 16]).
clique(559, [18, 45, 59, 15, 13]).
clique(560, [18, 45, 61, 15, 2]).
clique(561, [18, 50, 59, 10]).
clique(562, [18, 50, 61, 10]).
clique(563, [18, 59, 61, 15]).
clique(564, [19, 24, 33, 16, 1]).
clique(565, [19, 24, 35, 16, 1]).
clique(566, [19, 24, 38, 16, 1]).
clique(567, [19, 24, 39, 16]).
clique(568, [19, 24, 45, 16]).
clique(569, [19, 24, 50, 16]).
clique(570, [19, 24, 51, 1]).
clique(571, [19, 24, 57, 16, 1]).
clique(572, [19, 33, 35, 16, 1]).
clique(573, [19, 33, 38, 16, 1]).
clique(574, [19, 33, 50, 16]).
clique(575, [19, 33, 51, 1]).
clique(576, [19, 33, 57, 16, 1]).
clique(577, [19, 33, 66, 1]).
clique(578, [19, 33, 70, 1]).
clique(579, [19, 35, 38, 16, 1]).
clique(580, [19, 35, 39, 16]).
clique(581, [19, 35, 45, 16]).
clique(582, [19, 35, 50, 16]).
clique(583, [19, 35, 57, 16, 1]).
clique(584, [19, 35, 70, 1]).
clique(585, [19, 38, 50, 16]).
clique(586, [19, 38, 51, 1]).
clique(587, [19, 38, 57, 16, 1]).
clique(588, [19, 39, 45, 16]).
clique(589, [19, 39, 50, 16]).
clique(590, [19, 45, 50, 16]).
clique(591, [19, 51, 57, 1]).
clique(592, [19, 51, 66, 1]).
clique(593, [19, 57, 66, 1]).
clique(594, [20, 29, 54, 7, 4]).
clique(595, [20, 29, 60, 7, 4]).
clique(596, [20, 47, 54, 7]).
clique(597, [20, 47, 60, 7]).
clique(598, [20, 54, 60, 7, 4]).
clique(599, [21, 30, 32, 14, 3]).
clique(600, [21, 30, 36, 14, 3]).
clique(601, [21, 30, 46, 14, 3]).
clique(602, [21, 30, 62, 14, 3]).
clique(603, [21, 32, 36, 14, 3]).
clique(604, [21, 32, 46, 14, 12, 3]).
clique(605, [21, 32, 48, 14, 12, 3]).
clique(606, [21, 32, 52, 14, 12]).
clique(607, [21, 32, 62, 14, 12, 3]).
clique(608, [21, 36, 46, 14, 3]).
clique(609, [21, 36, 62, 14, 3]).
clique(610, [21, 40, 52, 14, 12]).
clique(611, [21, 46, 48, 14, 12, 3]).
clique(612, [21, 46, 62, 14, 12, 3]).
clique(613, [21, 48, 52, 14, 12]).
clique(614, [21, 48, 62, 14, 12, 3]).
clique(615, [21, 52, 62, 14, 12]).
clique(616, [22, 26, 31, 8, 5]).
clique(617, [22, 26, 40, 8]).
clique(618, [22, 26, 44, 8, 5]).
clique(619, [22, 26, 49, 8, 5]).
clique(620, [22, 31, 40, 8]).
clique(621, [22, 31, 44, 8, 5]).
clique(622, [22, 31, 49, 8, 5]).
clique(623, [22, 40, 44, 8]).
clique(624, [22, 40, 49, 8]).
clique(625, [22, 44, 49, 8, 5]).
clique(626, [23, 29, 54, 4]).
clique(627, [23, 29, 64, 4]).
clique(628, [23, 42, 64]).
clique(629, [23, 54, 64, 4]).
clique(630, [24, 33, 35, 19, 16, 1]).
clique(631, [24, 33, 38, 19, 16, 1]).
clique(632, [24, 33, 50, 19, 16]).
clique(633, [24, 33, 51, 19, 1]).
clique(634, [24, 33, 57, 19, 16, 1]).
clique(635, [24, 35, 37, 18, 16]).
clique(636, [24, 35, 38, 19, 16, 1]).
clique(637, [24, 35, 39, 19, 16]).
clique(638, [24, 35, 45, 19, 16]).
clique(639, [24, 35, 50, 19, 16]).
clique(640, [24, 35, 57, 19, 16, 1]).
clique(641, [24, 37, 38, 16]).
clique(642, [24, 37, 45, 18, 16]).
clique(643, [24, 37, 50, 18, 16]).
clique(644, [24, 38, 50, 19, 16]).
clique(645, [24, 38, 51, 19, 1]).
clique(646, [24, 38, 57, 19, 16, 1]).
clique(647, [24, 39, 45, 19, 16]).
clique(648, [24, 39, 50, 19, 16]).
clique(649, [24, 45, 50, 19, 16]).
clique(650, [24, 51, 57, 19, 1]).
clique(651, [26, 31, 40, 22, 8]).
clique(652, [26, 31, 44, 22, 8, 5]).
clique(653, [26, 31, 49, 22, 8, 5]).
clique(654, [26, 40, 44, 22, 8]).
clique(655, [26, 40, 49, 22, 8]).
clique(656, [26, 44, 49, 22, 8, 5]).
clique(657, [27, 28, 37, 15, 13, 2]).
clique(658, [27, 28, 56, 15, 13, 2]).
clique(659, [27, 28, 68, 15, 2]).
clique(660, [27, 37, 56, 15, 13, 2]).
clique(661, [27, 56, 68, 15, 2]).
clique(662, [28, 37, 45, 18, 16, 13, 2]).
clique(663, [28, 37, 50, 18, 16]).
clique(664, [28, 37, 56, 27, 15, 13, 2]).
clique(665, [28, 37, 59, 18, 15, 13]).
clique(666, [28, 39, 45, 18, 16]).
64
clique(667, [28, 39, 50, 18, 16]).
clique(668, [28, 39, 59, 18, 10]).
clique(669, [28, 39, 61, 18, 10]).
clique(670, [28, 41, 45, 10, 9]).
clique(671, [28, 41, 50, 10]).
clique(672, [28, 41, 59, 10, 9]).
clique(673, [28, 41, 61, 10, 9]).
clique(674, [28, 45, 50, 18, 16]).
clique(675, [28, 45, 56, 16, 13, 2]).
clique(676, [28, 45, 59, 18, 15, 13]).
clique(677, [28, 45, 61, 18, 15, 2]).
clique(678, [28, 50, 59, 18, 10]).
clique(679, [28, 50, 61, 18, 10]).
clique(680, [28, 56, 59, 15, 13]).
clique(681, [28, 56, 68, 27, 15, 2]).
clique(682, [28, 59, 61, 18, 15]).
clique(683, [28, 61, 68, 15, 2]).
clique(684, [29, 54, 60, 20, 7, 4]).
clique(685, [29, 54, 64, 23, 4]).
clique(686, [29, 60, 64, 7, 4]).
clique(687, [30, 32, 36, 21, 14, 3]).
clique(688, [30, 32, 46, 21, 14, 3]).
clique(689, [30, 32, 62, 21, 14, 3]).
clique(690, [30, 32, 69, 14, 3]).
clique(691, [30, 36, 46, 21, 14, 3]).
clique(692, [30, 36, 62, 21, 14, 3]).
clique(693, [30, 36, 69, 14, 3]).
clique(694, [30, 46, 62, 21, 14, 3]).
clique(695, [30, 62, 69, 14, 3]).
clique(696, [31, 40, 44, 26, 22, 8]).
clique(697, [31, 40, 49, 26, 22, 8]).
clique(698, [31, 44, 49, 26, 22, 8, 5]).
clique(699, [32, 36, 46, 30, 21, 14, 3]).
clique(700, [32, 36, 55, 14]).
clique(701, [32, 36, 58, 14]).
clique(702, [32, 36, 62, 30, 21, 14, 3]).
clique(703, [32, 36, 69, 30, 14, 3]).
clique(704, [32, 46, 48, 21, 14, 12, 3]).
clique(705, [32, 46, 58, 14]).
clique(706, [32, 46, 62, 30, 21, 14, 3]).
clique(707, [32, 48, 52, 21, 14, 12]).
clique(708, [32, 48, 55, 14, 12]).
clique(709, [32, 48, 58, 14, 11]).
clique(710, [32, 48, 62, 21, 14, 12, 3]).
clique(711, [32, 52, 62, 21, 14, 12]).
clique(712, [32, 55, 58, 14, 11]).
clique(713, [32, 55, 62, 14, 12]).
clique(714, [32, 58, 62, 14, 11]).
clique(715, [32, 58, 69, 14, 11]).
clique(716, [32, 62, 69, 30, 14, 3]).
clique(717, [33, 35, 38, 24, 19, 16, 1]).
clique(718, [33, 35, 50, 24, 19, 16]).
clique(719, [33, 35, 57, 24, 19, 16, 1]).
clique(720, [33, 35, 70, 19, 1]).
clique(721, [33, 38, 50, 24, 19, 16]).
clique(722, [33, 38, 51, 24, 19, 1]).
clique(723, [33, 38, 57, 24, 19, 16, 1]).
clique(724, [33, 43, 51, 1]).
clique(725, [33, 43, 57, 1]).
clique(726, [33, 43, 66, 1]).
clique(727, [33, 51, 57, 24, 19, 1]).
clique(728, [33, 51, 66, 19, 1]).
clique(729, [33, 57, 66, 19, 1]).
clique(730, [34, 53, 65]).
clique(731, [34, 53, 66]).
clique(732, [34, 65, 66]).
clique(733, [35, 37, 38, 24, 16]).
clique(734, [35, 37, 45, 24, 18, 16]).
clique(735, [35, 37, 50, 24, 18, 16]).
clique(736, [35, 38, 50, 33, 24, 19, 16]).
clique(737, [35, 38, 57, 33, 24, 19, 16, 1]).
clique(738, [35, 39, 45, 24, 19, 16]).
clique(739, [35, 39, 50, 24, 19, 16]).
clique(740, [35, 45, 50, 24, 19, 16]).
clique(741, [36, 46, 58, 32, 14]).
clique(742, [36, 46, 62, 32, 30, 21, 14, 3]).
clique(743, [36, 55, 58, 32, 14]).
clique(744, [36, 55, 62, 32, 14]).
clique(745, [36, 58, 62, 32, 14]).
clique(746, [36, 58, 69, 32, 14]).
clique(747, [36, 62, 69, 32, 30, 14, 3]).
clique(748, [37, 38, 50, 35, 24, 16]).
clique(749, [37, 45, 50, 35, 24, 18, 16]).
clique(750, [37, 45, 56, 28, 16, 13, 2]).
clique(751, [37, 45, 59, 28, 18, 15, 13]).
clique(752, [37, 50, 59, 28, 18]).
clique(753, [37, 56, 59, 28, 15, 13]).
clique(754, [38, 51, 57, 33, 24, 19, 1]).
clique(755, [39, 45, 50, 35, 24, 19, 16]).
clique(756, [39, 45, 59, 28, 18, 10]).
clique(757, [39, 45, 61, 28, 18, 10]).
clique(758, [39, 50, 59, 28, 18, 10]).
clique(759, [39, 50, 61, 28, 18, 10]).
clique(760, [39, 59, 61, 28, 18, 10]).
clique(761, [40, 44, 49, 31, 26, 22, 8]).
clique(762, [40, 44, 52, 14, 12]).
clique(763, [41, 45, 50, 28, 10]).
clique(764, [41, 45, 59, 28, 10, 9]).
clique(765, [41, 45, 61, 28, 10, 9]).
clique(766, [41, 50, 59, 28, 10]).
clique(767, [41, 50, 61, 28, 10]).
clique(768, [41, 59, 61, 28, 10, 9]).
clique(769, [43, 51, 53, 1]).
clique(770, [43, 51, 57, 33, 1]).
clique(771, [43, 51, 66, 33, 1]).
clique(772, [43, 51, 67, 1]).
clique(773, [43, 53, 66, 1]).
clique(774, [43, 53, 67, 1]).
clique(775, [43, 57, 66, 33, 1]).
clique(776, [43, 66, 67, 1]).
clique(777, [45, 50, 59, 41, 28, 10]).
clique(778, [45, 50, 61, 41, 28, 10]).
clique(779, [45, 56, 59, 37, 28, 15, 13]).
clique(780, [45, 59, 61, 41, 28, 10, 9]).
clique(781, [46, 48, 58, 32, 14]).
clique(782, [46, 48, 62, 32, 21, 14, 12, 3]).
clique(783, [46, 58, 62, 36, 32, 14]).
clique(784, [47, 54, 60, 20, 7]).
clique(785, [47, 54, 64, 7]).
clique(786, [47, 60, 64, 7]).
clique(787, [48, 52, 62, 32, 21, 14, 12]).
clique(788, [48, 55, 58, 32, 14, 11]).
clique(789, [48, 55, 62, 32, 14, 12]).
clique(790, [48, 58, 62, 46, 32, 14]).
clique(791, [50, 59, 61, 45, 41, 28, 10]).
clique(792, [51, 53, 66, 43, 1]).
clique(793, [51, 53, 67, 43, 1]).
clique(794, [51, 57, 66, 43, 33, 1]).
clique(795, [51, 66, 67, 43, 1]).
clique(796, [53, 65, 66, 34]).
clique(797, [53, 66, 67, 51, 43, 1]).
clique(798, [54, 60, 64, 47, 7]).
clique(799, [55, 58, 62, 48, 32, 14, 11]).
clique(800, [58, 62, 69, 36, 32, 14]).
superclique(901,[287,288,294,616,618,619,
621,622,625,651,652,653,654,655,656,696,6
97,698,761]).
superclique(902,[761, 698, 697, 696, 656,
655, 654, 652, 651, 625, 621, 618, 294, 762,
287]).
superclique(903,[798, 784, 686, 685, 684,
598, 594, 595]).
superclique(904,[461, 462, 463, 466, 467,
468, 470, 471, 472,
476, 477, 479, 487, 489, 491, 509, 511, 513,
525, 527, 533, 541, 543, 545, 546, 551, 553,
559, 560, 635, 642, 643, 657, 658, 659, 660,
661, 662, 663, 664, 665, 666, 667, 668, 669,
670, 672, 673, 674, 675, 676, 677, 678, 679,
680, 681, 682, 683, 734, 735, 749, 750, 751,
752, 753, 756, 757, 758, 759, 760, 763, 764,
765, 766, 767, 768, 777, 778, 779, 780,
791]).
superclique(905,[666,777, 564, 565, 566,
571, 572, 573, 576,
579, 583, 587, 630, 631, 632, 634, 636,
637, 638, 639, 640, 644, 646, 647, 648,
649, 717, 718, 719, 721, 723, 733, 736,
737, 738, 739, 740, 748, 755]).
superclique(906,[423, 424, 426, 433, 434,
438, 448, 450, 454,
599, 600, 601, 602, 603, 604, 605, 606, 607,
608, 609, 610,
611, 612, 613, 614, 615, 687, 688, 689, 690,
691, 692, 693,
694, 695, 699, 702, 703, 704, 706, 707, 708,
709, 710, 711,
712, 713, 714, 715, 716, 741, 742, 743, 744,
745, 746, 747,
762, 781, 782, 783, 787, 788, 789, 790, 799,
800]).
Circle:
Arena[]
partiallyOverlaps(97, 99).
partiallyOverlaps(95, 100).
partiallyOverlaps(92, 96).
partiallyOverlaps(89, 96).
partiallyOverlaps(89, 92).
partiallyOverlaps(87, 90).
partiallyOverlaps(85, 90).
partiallyOverlaps(85, 87).
partiallyOverlaps(84, 99).
partiallyOverlaps(84, 97).
partiallyOverlaps(83, 93).
partiallyOverlaps(81, 99).
partiallyOverlaps(81, 97).
partiallyOverlaps(81, 94).
partiallyOverlaps(81, 84).
partiallyOverlaps(80, 91).
partiallyOverlaps(80, 83).
partiallyOverlaps(79, 99).
partiallyOverlaps(79, 97).
partiallyOverlaps(79, 84).
partiallyOverlaps(79, 81).
partiallyOverlaps(78, 96).
partiallyOverlaps(78, 92).
partiallyOverlaps(78, 89).
partiallyOverlaps(78, 86).
partiallyOverlaps(77, 100).
partiallyOverlaps(77, 95).
partiallyOverlaps(76, 82).
partiallyOverlaps(75, 100).
partiallyOverlaps(75, 95).
partiallyOverlaps(75, 77).
partiallyOverlaps(74, 93).
partiallyOverlaps(73, 96).
partiallyOverlaps(73, 92).
partiallyOverlaps(73, 89).
partiallyOverlaps(72, 90).
partiallyOverlaps(72, 87).
partiallyOverlaps(72, 85).
65
partiallyOverlaps(71, 96).
partiallyOverlaps(71, 92).
partiallyOverlaps(71, 89).
partiallyOverlaps(71, 73).
partiallyOverlaps(70, 100).
partiallyOverlaps(70, 95).
partiallyOverlaps(70, 77).
partiallyOverlaps(70, 75).
partiallyOverlaps(69, 100).
partiallyOverlaps(69, 95).
partiallyOverlaps(69, 94).
partiallyOverlaps(69, 77).
partiallyOverlaps(69, 75).
partiallyOverlaps(69, 70).
partiallyOverlaps(68, 90).
partiallyOverlaps(68, 87).
partiallyOverlaps(68, 85).
partiallyOverlaps(68, 72).
partiallyOverlaps(67, 88).
partiallyOverlaps(67, 80).
partiallyOverlaps(65, 80).
partiallyOverlaps(65, 73).
partiallyOverlaps(65, 71).
partiallyOverlaps(64, 100).
partiallyOverlaps(64, 77).
partiallyOverlaps(64, 75).
partiallyOverlaps(64, 70).
partiallyOverlaps(63, 98).
partiallyOverlaps(63, 90).
partiallyOverlaps(63, 87).
partiallyOverlaps(63, 85).
partiallyOverlaps(63, 72).
partiallyOverlaps(63, 68).
partiallyOverlaps(62, 93).
partiallyOverlaps(62, 74).
partiallyOverlaps(61, 88).
partiallyOverlaps(61, 80).
partiallyOverlaps(61, 67).
partiallyOverlaps(60, 91).
partiallyOverlaps(60, 80).
partiallyOverlaps(59, 90).
partiallyOverlaps(59, 87).
partiallyOverlaps(59, 85).
partiallyOverlaps(59, 72).
partiallyOverlaps(59, 68).
partiallyOverlaps(59, 63).
partiallyOverlaps(58, 89).
partiallyOverlaps(58, 86).
partiallyOverlaps(58, 78).
partiallyOverlaps(57, 94).
partiallyOverlaps(57, 69).
partiallyOverlaps(56, 100).
partiallyOverlaps(56, 95).
partiallyOverlaps(56, 94).
partiallyOverlaps(56, 77).
partiallyOverlaps(56, 75).
partiallyOverlaps(56, 70).
partiallyOverlaps(56, 69).
partiallyOverlaps(56, 57).
partiallyOverlaps(55, 82).
partiallyOverlaps(55, 76).
partiallyOverlaps(54, 100).
partiallyOverlaps(54, 75).
partiallyOverlaps(54, 70).
partiallyOverlaps(54, 66).
partiallyOverlaps(54, 64).
partiallyOverlaps(53, 96).
partiallyOverlaps(53, 92).
partiallyOverlaps(53, 89).
partiallyOverlaps(53, 78).
partiallyOverlaps(53, 73).
partiallyOverlaps(53, 71).
partiallyOverlaps(52, 91).
partiallyOverlaps(52, 80).
partiallyOverlaps(52, 71).
partiallyOverlaps(52, 65).
partiallyOverlaps(52, 60).
partiallyOverlaps(51, 98).
partiallyOverlaps(51, 87).
partiallyOverlaps(51, 85).
partiallyOverlaps(51, 72).
partiallyOverlaps(51, 63).
partiallyOverlaps(50, 91).
partiallyOverlaps(50, 80).
partiallyOverlaps(50, 60).
partiallyOverlaps(49, 100).
partiallyOverlaps(49, 77).
partiallyOverlaps(49, 75).
partiallyOverlaps(49, 70).
partiallyOverlaps(49, 64).
partiallyOverlaps(49, 54).
partiallyOverlaps(48, 93).
partiallyOverlaps(48, 74).
partiallyOverlaps(48, 62).
partiallyOverlaps(47, 82).
partiallyOverlaps(47, 76).
partiallyOverlaps(47, 55).
partiallyOverlaps(46, 90).
partiallyOverlaps(46, 87).
partiallyOverlaps(46, 85).
partiallyOverlaps(46, 72).
partiallyOverlaps(46, 68).
partiallyOverlaps(46, 63).
partiallyOverlaps(46, 59).
partiallyOverlaps(46, 51).
partiallyOverlaps(45, 89).
partiallyOverlaps(45, 86).
partiallyOverlaps(45, 78).
partiallyOverlaps(45, 58).
partiallyOverlaps(45, 55).
partiallyOverlaps(45, 53).
partiallyOverlaps(43, 88).
partiallyOverlaps(43, 61).
partiallyOverlaps(43, 50).
partiallyOverlaps(42, 91).
partiallyOverlaps(42, 88).
partiallyOverlaps(42, 67).
partiallyOverlaps(42, 61).
partiallyOverlaps(42, 60).
partiallyOverlaps(42, 50).
partiallyOverlaps(42, 43).
partiallyOverlaps(41, 99).
partiallyOverlaps(41, 97).
partiallyOverlaps(41, 84).
partiallyOverlaps(41, 81).
partiallyOverlaps(41, 79).
partiallyOverlaps(41, 44).
partiallyOverlaps(40, 90).
partiallyOverlaps(40, 87).
partiallyOverlaps(40, 85).
partiallyOverlaps(40, 72).
partiallyOverlaps(40, 68).
partiallyOverlaps(40, 63).
partiallyOverlaps(40, 59).
partiallyOverlaps(40, 51).
partiallyOverlaps(40, 46).
partiallyOverlaps(38, 90).
partiallyOverlaps(38, 87).
partiallyOverlaps(38, 85).
partiallyOverlaps(38, 72).
partiallyOverlaps(38, 68).
partiallyOverlaps(38, 59).
partiallyOverlaps(38, 46).
partiallyOverlaps(38, 40).
partiallyOverlaps(37, 93).
partiallyOverlaps(37, 74).
partiallyOverlaps(37, 48).
partiallyOverlaps(36, 93).
partiallyOverlaps(36, 83).
partiallyOverlaps(36, 80).
partiallyOverlaps(36, 67).
partiallyOverlaps(35, 91).
partiallyOverlaps(35, 80).
partiallyOverlaps(35, 65).
partiallyOverlaps(35, 60).
partiallyOverlaps(35, 52).
partiallyOverlaps(34, 90).
partiallyOverlaps(34, 87).
partiallyOverlaps(34, 85).
partiallyOverlaps(34, 68).
partiallyOverlaps(34, 59).
partiallyOverlaps(34, 46).
partiallyOverlaps(34, 40).
partiallyOverlaps(34, 38).
partiallyOverlaps(33, 97).
partiallyOverlaps(33, 44).
partiallyOverlaps(33, 41).
partiallyOverlaps(32, 39).
partiallyOverlaps(31, 87).
partiallyOverlaps(31, 85).
partiallyOverlaps(31, 72).
partiallyOverlaps(31, 63).
partiallyOverlaps(31, 59).
partiallyOverlaps(31, 51).
partiallyOverlaps(31, 46).
partiallyOverlaps(31, 40).
partiallyOverlaps(30, 88).
partiallyOverlaps(30, 67).
partiallyOverlaps(30, 61).
partiallyOverlaps(30, 43).
partiallyOverlaps(30, 42).
partiallyOverlaps(29, 98).
partiallyOverlaps(28, 91).
partiallyOverlaps(28, 88).
partiallyOverlaps(28, 61).
partiallyOverlaps(28, 60).
partiallyOverlaps(28, 50).
partiallyOverlaps(28, 43).
partiallyOverlaps(28, 42).
partiallyOverlaps(28, 30).
partiallyOverlaps(27, 98).
partiallyOverlaps(27, 90).
partiallyOverlaps(27, 87).
partiallyOverlaps(27, 85).
partiallyOverlaps(27, 72).
partiallyOverlaps(27, 68).
partiallyOverlaps(27, 63).
partiallyOverlaps(27, 59).
partiallyOverlaps(27, 51).
partiallyOverlaps(27, 46).
partiallyOverlaps(27, 40).
partiallyOverlaps(27, 38).
partiallyOverlaps(27, 34).
partiallyOverlaps(27, 31).
partiallyOverlaps(26, 82).
partiallyOverlaps(26, 76).
partiallyOverlaps(26, 55).
partiallyOverlaps(26, 47).
partiallyOverlaps(25, 86).
partiallyOverlaps(25, 82).
partiallyOverlaps(25, 76).
partiallyOverlaps(25, 58).
partiallyOverlaps(25, 55).
partiallyOverlaps(25, 45).
partiallyOverlaps(24, 88).
partiallyOverlaps(24, 67).
partiallyOverlaps(24, 61).
partiallyOverlaps(24, 43).
partiallyOverlaps(24, 42).
partiallyOverlaps(24, 30).
partiallyOverlaps(24, 28).
partiallyOverlaps(23, 98).
partiallyOverlaps(23, 51).
partiallyOverlaps(23, 31).
partiallyOverlaps(23, 29).
66
partiallyOverlaps(23, 27).
partiallyOverlaps(22, 96).
partiallyOverlaps(22, 92).
partiallyOverlaps(22, 80).
partiallyOverlaps(22, 73).
partiallyOverlaps(22, 71).
partiallyOverlaps(22, 65).
partiallyOverlaps(22, 60).
partiallyOverlaps(22, 52).
partiallyOverlaps(22, 35).
partiallyOverlaps(21, 99).
partiallyOverlaps(21, 97).
partiallyOverlaps(21, 94).
partiallyOverlaps(21, 84).
partiallyOverlaps(21, 81).
partiallyOverlaps(21, 79).
partiallyOverlaps(21, 57).
partiallyOverlaps(21, 41).
partiallyOverlaps(20, 66).
partiallyOverlaps(20, 39).
partiallyOverlaps(20, 32).
partiallyOverlaps(19, 99).
partiallyOverlaps(19, 94).
partiallyOverlaps(19, 84).
partiallyOverlaps(19, 81).
partiallyOverlaps(19, 79).
partiallyOverlaps(19, 57).
partiallyOverlaps(19, 56).
partiallyOverlaps(19, 21).
partiallyOverlaps(18, 93).
partiallyOverlaps(18, 74).
partiallyOverlaps(18, 62).
partiallyOverlaps(18, 48).
partiallyOverlaps(18, 37).
partiallyOverlaps(17, 93).
partiallyOverlaps(17, 83).
partiallyOverlaps(17, 80).
partiallyOverlaps(17, 67).
partiallyOverlaps(17, 36).
partiallyOverlaps(16, 62).
partiallyOverlaps(16, 44).
partiallyOverlaps(16, 41).
partiallyOverlaps(16, 33).
partiallyOverlaps(15, 96).
partiallyOverlaps(15, 92).
partiallyOverlaps(15, 89).
partiallyOverlaps(15, 86).
partiallyOverlaps(15, 78).
partiallyOverlaps(15, 73).
partiallyOverlaps(15, 71).
partiallyOverlaps(15, 53).
partiallyOverlaps(15, 45).
partiallyOverlaps(14, 66).
partiallyOverlaps(14, 54).
partiallyOverlaps(14, 39).
partiallyOverlaps(14, 20).
partiallyOverlaps(13, 82).
partiallyOverlaps(13, 76).
partiallyOverlaps(13, 55).
partiallyOverlaps(13, 47).
partiallyOverlaps(13, 26).
partiallyOverlaps(13, 25).
partiallyOverlaps(12, 99).
partiallyOverlaps(12, 97).
partiallyOverlaps(12, 94).
partiallyOverlaps(12, 84).
partiallyOverlaps(12, 81).
partiallyOverlaps(12, 79).
partiallyOverlaps(12, 57).
partiallyOverlaps(12, 41).
partiallyOverlaps(12, 21).
partiallyOverlaps(12, 19).
partiallyOverlaps(11, 47).
partiallyOverlaps(11, 38).
partiallyOverlaps(11, 34).
partiallyOverlaps(11, 26).
partiallyOverlaps(10, 86).
partiallyOverlaps(10, 82).
partiallyOverlaps(10, 78).
partiallyOverlaps(10, 58).
partiallyOverlaps(10, 55).
partiallyOverlaps(10, 45).
partiallyOverlaps(10, 25).
partiallyOverlaps(10, 13).
partiallyOverlaps(9, 94).
partiallyOverlaps(9, 69).
partiallyOverlaps(9, 57).
partiallyOverlaps(9, 56).
partiallyOverlaps(8, 91).
partiallyOverlaps(8, 80).
partiallyOverlaps(8, 65).
partiallyOverlaps(8, 60).
partiallyOverlaps(8, 52).
partiallyOverlaps(8, 50).
partiallyOverlaps(8, 35).
partiallyOverlaps(8, 28).
partiallyOverlaps(8, 22).
partiallyOverlaps(7, 93).
partiallyOverlaps(7, 83).
partiallyOverlaps(7, 74).
partiallyOverlaps(7, 48).
partiallyOverlaps(7, 37).
partiallyOverlaps(7, 18).
partiallyOverlaps(7, 17).
partiallyOverlaps(6, 98).
partiallyOverlaps(6, 72).
partiallyOverlaps(6, 63).
partiallyOverlaps(6, 51).
partiallyOverlaps(6, 31).
partiallyOverlaps(6, 29).
partiallyOverlaps(6, 27).
partiallyOverlaps(6, 23).
partiallyOverlaps(5, 100).
partiallyOverlaps(5, 95).
partiallyOverlaps(5, 77).
partiallyOverlaps(5, 75).
partiallyOverlaps(5, 70).
partiallyOverlaps(5, 69).
partiallyOverlaps(5, 56).
partiallyOverlaps(5, 9).
partiallyOverlaps(4, 97).
partiallyOverlaps(4, 44).
partiallyOverlaps(4, 41).
partiallyOverlaps(4, 33).
partiallyOverlaps(4, 16).
partiallyOverlaps(3, 94).
partiallyOverlaps(3, 77).
partiallyOverlaps(3, 69).
partiallyOverlaps(3, 57).
partiallyOverlaps(3, 56).
partiallyOverlaps(3, 21).
partiallyOverlaps(3, 19).
partiallyOverlaps(3, 12).
partiallyOverlaps(3, 9).
partiallyOverlaps(2, 90).
partiallyOverlaps(2, 87).
partiallyOverlaps(2, 85).
partiallyOverlaps(2, 72).
partiallyOverlaps(2, 68).
partiallyOverlaps(2, 63).
partiallyOverlaps(2, 59).
partiallyOverlaps(2, 51).
partiallyOverlaps(2, 46).
partiallyOverlaps(2, 40).
partiallyOverlaps(2, 38).
partiallyOverlaps(2, 34).
partiallyOverlaps(2, 31).
partiallyOverlaps(2, 27).
partiallyOverlaps(2, 6).
partiallyOverlaps(1, 93).
partiallyOverlaps(1, 83).
partiallyOverlaps(1, 80).
partiallyOverlaps(1, 74).
partiallyOverlaps(1, 48).
partiallyOverlaps(1, 37).
partiallyOverlaps(1, 36).
partiallyOverlaps(1, 17).
partiallyOverlaps(1, 7).
clique(101, [1, 7, 17]).
clique(102, [1, 7, 37]).
clique(103, [1, 7, 48]).
clique(104, [1, 7, 74]).
clique(105, [1, 7, 83]).
clique(106, [1, 7, 93]).
clique(107, [1, 17, 36]).
clique(108, [1, 17, 80]).
clique(109, [1, 17, 83]).
clique(110, [1, 17, 93]).
clique(111, [1, 36, 80]).
clique(112, [1, 36, 83]).
clique(113, [1, 36, 93]).
clique(114, [1, 37, 48]).
clique(115, [1, 37, 74]).
clique(116, [1, 37, 93]).
clique(117, [1, 48, 74]).
clique(118, [1, 48, 93]).
clique(119, [1, 74, 93]).
clique(120, [1, 80, 83]).
clique(121, [1, 83, 93]).
clique(122, [2, 6, 27]).
clique(123, [2, 6, 31]).
clique(124, [2, 6, 51]).
clique(125, [2, 6, 63]).
clique(126, [2, 6, 72]).
clique(127, [2, 27, 31]).
clique(128, [2, 27, 34]).
clique(129, [2, 27, 38]).
clique(130, [2, 27, 40]).
clique(131, [2, 27, 46]).
clique(132, [2, 27, 51]).
clique(133, [2, 27, 59]).
clique(134, [2, 27, 63]).
clique(135, [2, 27, 68]).
clique(136, [2, 27, 72]).
clique(137, [2, 27, 85]).
clique(138, [2, 27, 87]).
clique(139, [2, 27, 90]).
clique(140, [2, 31, 40]).
clique(141, [2, 31, 46]).
clique(142, [2, 31, 51]).
clique(143, [2, 31, 59]).
clique(144, [2, 31, 63]).
clique(145, [2, 31, 72]).
clique(146, [2, 31, 85]).
clique(147, [2, 31, 87]).
clique(148, [2, 34, 38]).
clique(149, [2, 34, 40]).
clique(150, [2, 34, 46]).
clique(151, [2, 34, 59]).
clique(152, [2, 34, 68]).
clique(153, [2, 34, 85]).
clique(154, [2, 34, 87]).
clique(155, [2, 34, 90]).
clique(156, [2, 38, 40]).
clique(157, [2, 38, 46]).
clique(158, [2, 38, 59]).
clique(159, [2, 38, 68]).
clique(160, [2, 38, 72]).
clique(161, [2, 38, 85]).
clique(162, [2, 38, 87]).
clique(163, [2, 38, 90]).
clique(164, [2, 40, 46]).
clique(165, [2, 40, 51]).
clique(166, [2, 40, 59]).
67
clique(167, [2, 40, 63]).
clique(168, [2, 40, 68]).
clique(169, [2, 40, 72]).
clique(170, [2, 40, 85]).
clique(171, [2, 40, 87]).
clique(172, [2, 40, 90]).
clique(173, [2, 46, 51]).
clique(174, [2, 46, 59]).
clique(175, [2, 46, 63]).
clique(176, [2, 46, 68]).
clique(177, [2, 46, 72]).
clique(178, [2, 46, 85]).
clique(179, [2, 46, 87]).
clique(180, [2, 46, 90]).
clique(181, [2, 51, 63]).
clique(182, [2, 51, 72]).
clique(183, [2, 51, 85]).
clique(184, [2, 51, 87]).
clique(185, [2, 59, 63]).
clique(186, [2, 59, 68]).
clique(187, [2, 59, 72]).
clique(188, [2, 59, 85]).
clique(189, [2, 59, 87]).
clique(190, [2, 59, 90]).
clique(191, [2, 63, 68]).
clique(192, [2, 63, 72]).
clique(193, [2, 63, 85]).
clique(194, [2, 63, 87]).
clique(195, [2, 63, 90]).
clique(196, [2, 68, 72]).
clique(197, [2, 68, 85]).
clique(198, [2, 68, 87]).
clique(199, [2, 68, 90]).
clique(200, [2, 72, 85]).
clique(201, [2, 72, 87]).
clique(202, [2, 72, 90]).
clique(203, [2, 85, 87]).
clique(204, [2, 85, 90]).
clique(205, [2, 87, 90]).
clique(206, [3, 9, 56]).
clique(207, [3, 9, 57]).
clique(208, [3, 9, 69]).
clique(209, [3, 9, 94]).
clique(210, [3, 12, 19]).
clique(211, [3, 12, 21]).
clique(212, [3, 12, 57]).
clique(213, [3, 12, 94]).
clique(214, [3, 19, 21]).
clique(215, [3, 19, 56]).
clique(216, [3, 19, 57]).
clique(217, [3, 19, 94]).
clique(218, [3, 21, 57]).
clique(219, [3, 21, 94]).
clique(220, [3, 56, 57]).
clique(221, [3, 56, 69]).
clique(222, [3, 56, 77]).
clique(223, [3, 56, 94]).
clique(224, [3, 57, 69]).
clique(225, [3, 57, 94]).
clique(226, [3, 69, 77]).
clique(227, [3, 69, 94]).
clique(228, [4, 16, 33]).
clique(229, [4, 16, 41]).
clique(230, [4, 16, 44]).
clique(231, [4, 33, 41]).
clique(232, [4, 33, 44]).
clique(233, [4, 33, 97]).
clique(234, [4, 41, 44]).
clique(235, [4, 41, 97]).
clique(236, [5, 9, 56]).
clique(237, [5, 9, 69]).
clique(238, [5, 56, 69]).
clique(239, [5, 56, 70]).
clique(240, [5, 56, 75]).
clique(241, [5, 56, 77]).
clique(242, [5, 56, 95]).
clique(243, [5, 56, 100]).
clique(244, [5, 69, 70]).
clique(245, [5, 69, 75]).
clique(246, [5, 69, 77]).
clique(247, [5, 69, 95]).
clique(248, [5, 69, 100]).
clique(249, [5, 70, 75]).
clique(250, [5, 70, 77]).
clique(251, [5, 70, 95]).
clique(252, [5, 70, 100]).
clique(253, [5, 75, 77]).
clique(254, [5, 75, 95]).
clique(255, [5, 75, 100]).
clique(256, [5, 77, 95]).
clique(257, [5, 77, 100]).
clique(258, [5, 95, 100]).
clique(259, [6, 23, 27]).
clique(260, [6, 23, 29]).
clique(261, [6, 23, 31]).
clique(262, [6, 23, 51]).
clique(263, [6, 23, 98]).
clique(264, [6, 27, 31, 2]).
clique(265, [6, 27, 51, 2]).
clique(266, [6, 27, 63, 2]).
clique(267, [6, 27, 72, 2]).
clique(268, [6, 27, 98]).
clique(269, [6, 29, 98]).
clique(270, [6, 31, 51, 2]).
clique(271, [6, 31, 63, 2]).
clique(272, [6, 31, 72, 2]).
clique(273, [6, 51, 63, 2]).
clique(274, [6, 51, 72, 2]).
clique(275, [6, 51, 98]).
clique(276, [6, 63, 72, 2]).
clique(277, [6, 63, 98]).
clique(278, [7, 17, 83, 1]).
clique(279, [7, 17, 93, 1]).
clique(280, [7, 18, 37]).
clique(281, [7, 18, 48]).
clique(282, [7, 18, 74]).
clique(283, [7, 18, 93]).
clique(284, [7, 37, 48, 1]).
clique(285, [7, 37, 74, 1]).
clique(286, [7, 37, 93, 1]).
clique(287, [7, 48, 74, 1]).
clique(288, [7, 48, 93, 1]).
clique(289, [7, 74, 93, 1]).
clique(290, [7, 83, 93, 1]).
clique(291, [8, 22, 35]).
clique(292, [8, 22, 52]).
clique(293, [8, 22, 60]).
clique(294, [8, 22, 65]).
clique(295, [8, 22, 80]).
clique(296, [8, 28, 50]).
clique(297, [8, 28, 60]).
clique(298, [8, 28, 91]).
clique(299, [8, 35, 52]).
clique(300, [8, 35, 60]).
clique(301, [8, 35, 65]).
clique(302, [8, 35, 80]).
clique(303, [8, 35, 91]).
clique(304, [8, 50, 60]).
clique(305, [8, 50, 80]).
clique(306, [8, 50, 91]).
clique(307, [8, 52, 60]).
clique(308, [8, 52, 65]).
clique(309, [8, 52, 80]).
clique(310, [8, 52, 91]).
clique(311, [8, 60, 80]).
clique(312, [8, 60, 91]).
clique(313, [8, 65, 80]).
clique(314, [8, 80, 91]).
clique(315, [9, 56, 57, 3]).
clique(316, [9, 56, 69, 5]).
clique(317, [9, 56, 94, 3]).
clique(318, [9, 57, 69, 3]).
clique(319, [9, 57, 94, 3]).
clique(320, [9, 69, 94, 3]).
clique(321, [10, 13, 25]).
clique(322, [10, 13, 55]).
clique(323, [10, 13, 82]).
clique(324, [10, 25, 45]).
clique(325, [10, 25, 55]).
clique(326, [10, 25, 58]).
clique(327, [10, 25, 82]).
clique(328, [10, 25, 86]).
clique(329, [10, 45, 55]).
clique(330, [10, 45, 58]).
clique(331, [10, 45, 78]).
clique(332, [10, 45, 86]).
clique(333, [10, 55, 82]).
clique(334, [10, 58, 78]).
clique(335, [10, 58, 86]).
clique(336, [10, 78, 86]).
clique(337, [11, 26, 47]).
clique(338, [11, 34, 38]).
clique(339, [12, 19, 21, 3]).
clique(340, [12, 19, 57, 3]).
clique(341, [12, 19, 79]).
clique(342, [12, 19, 81]).
clique(343, [12, 19, 84]).
clique(344, [12, 19, 94, 3]).
clique(345, [12, 19, 99]).
clique(346, [12, 21, 41]).
clique(347, [12, 21, 57, 3]).
clique(348, [12, 21, 79]).
clique(349, [12, 21, 81]).
clique(350, [12, 21, 84]).
clique(351, [12, 21, 94, 3]).
clique(352, [12, 21, 97]).
clique(353, [12, 21, 99]).
clique(354, [12, 41, 79]).
clique(355, [12, 41, 81]).
clique(356, [12, 41, 84]).
clique(357, [12, 41, 97]).
clique(358, [12, 41, 99]).
clique(359, [12, 57, 94, 3]).
clique(360, [12, 79, 81]).
clique(361, [12, 79, 84]).
clique(362, [12, 79, 97]).
clique(363, [12, 79, 99]).
clique(364, [12, 81, 84]).
clique(365, [12, 81, 94]).
clique(366, [12, 81, 97]).
clique(367, [12, 81, 99]).
clique(368, [12, 84, 97]).
clique(369, [12, 84, 99]).
clique(370, [12, 97, 99]).
clique(371, [13, 25, 55, 10]).
clique(372, [13, 25, 76]).
clique(373, [13, 25, 82, 10]).
clique(374, [13, 26, 47]).
clique(375, [13, 26, 55]).
clique(376, [13, 26, 76]).
clique(377, [13, 26, 82]).
clique(378, [13, 47, 55]).
clique(379, [13, 47, 76]).
clique(380, [13, 47, 82]).
clique(381, [13, 55, 76]).
clique(382, [13, 55, 82, 10]).
clique(383, [13, 76, 82]).
clique(384, [14, 20, 39]).
clique(385, [14, 20, 66]).
clique(386, [14, 54, 66]).
clique(387, [15, 45, 53]).
clique(388, [15, 45, 78]).
clique(389, [15, 45, 86]).
clique(390, [15, 45, 89]).
clique(391, [15, 53, 71]).
68
clique(392, [15, 53, 73]).
clique(393, [15, 53, 78]).
clique(394, [15, 53, 89]).
clique(395, [15, 53, 92]).
clique(396, [15, 53, 96]).
clique(397, [15, 71, 73]).
clique(398, [15, 71, 89]).
clique(399, [15, 71, 92]).
clique(400, [15, 71, 96]).
clique(401, [15, 73, 89]).
clique(402, [15, 73, 92]).
clique(403, [15, 73, 96]).
clique(404, [15, 78, 86]).
clique(405, [15, 78, 89]).
clique(406, [15, 78, 92]).
clique(407, [15, 78, 96]).
clique(408, [15, 89, 92]).
clique(409, [15, 89, 96]).
clique(410, [15, 92, 96]).
clique(411, [16, 33, 41, 4]).
clique(412, [16, 33, 44, 4]).
clique(413, [16, 41, 44, 4]).
clique(414, [17, 36, 67]).
clique(415, [17, 36, 80, 1]).
clique(416, [17, 36, 83, 1]).
clique(417, [17, 36, 93, 1]).
clique(418, [17, 67, 80]).
clique(419, [17, 80, 83, 1]).
clique(420, [17, 83, 93, 7, 1]).
clique(421, [18, 37, 48, 7]).
clique(422, [18, 37, 74, 7]).
clique(423, [18, 37, 93, 7]).
clique(424, [18, 48, 62]).
clique(425, [18, 48, 74, 7]).
clique(426, [18, 48, 93, 7]).
clique(427, [18, 62, 74]).
clique(428, [18, 62, 93]).
clique(429, [18, 74, 93, 7]).
clique(430, [19, 21, 57, 12, 3]).
clique(431, [19, 21, 79, 12]).
clique(432, [19, 21, 81, 12]).
clique(433, [19, 21, 84, 12]).
clique(434, [19, 21, 94, 12, 3]).
clique(435, [19, 21, 99, 12]).
clique(436, [19, 56, 57, 3]).
clique(437, [19, 56, 94, 3]).
clique(438, [19, 57, 94, 12, 3]).
clique(439, [19, 79, 81, 12]).
clique(440, [19, 79, 84, 12]).
clique(441, [19, 79, 99, 12]).
clique(442, [19, 81, 84, 12]).
clique(443, [19, 81, 94, 12]).
clique(444, [19, 81, 99, 12]).
clique(445, [19, 84, 99, 12]).
clique(446, [20, 32, 39]).
clique(447, [21, 41, 79, 12]).
clique(448, [21, 41, 81, 12]).
clique(449, [21, 41, 84, 12]).
clique(450, [21, 41, 97, 12]).
clique(451, [21, 41, 99, 12]).
clique(452, [21, 57, 94, 19, 12, 3]).
clique(453, [21, 79, 81, 19, 12]).
clique(454, [21, 79, 84, 19, 12]).
clique(455, [21, 79, 97, 12]).
clique(456, [21, 79, 99, 19, 12]).
clique(457, [21, 81, 84, 19, 12]).
clique(458, [21, 81, 94, 19, 12]).
clique(459, [21, 81, 97, 12]).
clique(460, [21, 81, 99, 19, 12]).
clique(461, [21, 84, 97, 12]).
clique(462, [21, 84, 99, 19, 12]).
clique(463, [21, 97, 99, 12]).
clique(464, [22, 35, 52, 8]).
clique(465, [22, 35, 60, 8]).
clique(466, [22, 35, 65, 8]).
clique(467, [22, 35, 80, 8]).
clique(468, [22, 52, 60, 8]).
clique(469, [22, 52, 65, 8]).
clique(470, [22, 52, 71]).
clique(471, [22, 52, 80, 8]).
clique(472, [22, 60, 80, 8]).
clique(473, [22, 65, 71]).
clique(474, [22, 65, 73]).
clique(475, [22, 65, 80, 8]).
clique(476, [22, 71, 73]).
clique(477, [22, 71, 92]).
clique(478, [22, 71, 96]).
clique(479, [22, 73, 92]).
clique(480, [22, 73, 96]).
clique(481, [22, 92, 96]).
clique(482, [23, 27, 31, 6]).
clique(483, [23, 27, 51, 6]).
clique(484, [23, 27, 98, 6]).
clique(485, [23, 29, 98, 6]).
clique(486, [23, 31, 51, 6]).
clique(487, [23, 51, 98, 6]).
clique(488, [24, 28, 30]).
clique(489, [24, 28, 42]).
clique(490, [24, 28, 43]).
clique(491, [24, 28, 61]).
clique(492, [24, 28, 88]).
clique(493, [24, 30, 42]).
clique(494, [24, 30, 43]).
clique(495, [24, 30, 61]).
clique(496, [24, 30, 67]).
clique(497, [24, 30, 88]).
clique(498, [24, 42, 43]).
clique(499, [24, 42, 61]).
clique(500, [24, 42, 67]).
clique(501, [24, 42, 88]).
clique(502, [24, 43, 61]).
clique(503, [24, 43, 88]).
clique(504, [24, 61, 67]).
clique(505, [24, 61, 88]).
clique(506, [24, 67, 88]).
clique(507, [25, 45, 55, 10]).
clique(508, [25, 45, 58, 10]).
clique(509, [25, 45, 86, 10]).
clique(510, [25, 55, 76, 13]).
clique(511, [25, 55, 82, 13, 10]).
clique(512, [25, 58, 86, 10]).
clique(513, [25, 76, 82, 13]).
clique(514, [26, 47, 55, 13]).
clique(515, [26, 47, 76, 13]).
clique(516, [26, 47, 82, 13]).
clique(517, [26, 55, 76, 13]).
clique(518, [26, 55, 82, 13]).
clique(519, [26, 76, 82, 13]).
clique(520, [27, 31, 40, 2]).
clique(521, [27, 31, 46, 2]).
clique(522, [27, 31, 51, 23, 6]).
clique(523, [27, 31, 59, 2]).
clique(524, [27, 31, 63, 6, 2]).
clique(525, [27, 31, 72, 6, 2]).
clique(526, [27, 31, 85, 2]).
clique(527, [27, 31, 87, 2]).
clique(528, [27, 34, 38, 2]).
clique(529, [27, 34, 40, 2]).
clique(530, [27, 34, 46, 2]).
clique(531, [27, 34, 59, 2]).
clique(532, [27, 34, 68, 2]).
clique(533, [27, 34, 85, 2]).
clique(534, [27, 34, 87, 2]).
clique(535, [27, 34, 90, 2]).
clique(536, [27, 38, 40, 2]).
clique(537, [27, 38, 46, 2]).
clique(538, [27, 38, 59, 2]).
clique(539, [27, 38, 68, 2]).
clique(540, [27, 38, 72, 2]).
clique(541, [27, 38, 85, 2]).
clique(542, [27, 38, 87, 2]).
clique(543, [27, 38, 90, 2]).
clique(544, [27, 40, 46, 2]).
clique(545, [27, 40, 51, 2]).
clique(546, [27, 40, 59, 2]).
clique(547, [27, 40, 63, 2]).
clique(548, [27, 40, 68, 2]).
clique(549, [27, 40, 72, 2]).
clique(550, [27, 40, 85, 2]).
clique(551, [27, 40, 87, 2]).
clique(552, [27, 40, 90, 2]).
clique(553, [27, 46, 51, 2]).
clique(554, [27, 46, 59, 2]).
clique(555, [27, 46, 63, 2]).
clique(556, [27, 46, 68, 2]).
clique(557, [27, 46, 72, 2]).
clique(558, [27, 46, 85, 2]).
clique(559, [27, 46, 87, 2]).
clique(560, [27, 46, 90, 2]).
clique(561, [27, 51, 63, 6, 2]).
clique(562, [27, 51, 72, 6, 2]).
clique(563, [27, 51, 85, 2]).
clique(564, [27, 51, 87, 2]).
clique(565, [27, 51, 98, 23, 6]).
clique(566, [27, 59, 63, 2]).
clique(567, [27, 59, 68, 2]).
clique(568, [27, 59, 72, 2]).
clique(569, [27, 59, 85, 2]).
clique(570, [27, 59, 87, 2]).
clique(571, [27, 59, 90, 2]).
clique(572, [27, 63, 68, 2]).
clique(573, [27, 63, 72, 6, 2]).
clique(574, [27, 63, 85, 2]).
clique(575, [27, 63, 87, 2]).
clique(576, [27, 63, 90, 2]).
clique(577, [27, 63, 98, 6]).
clique(578, [27, 68, 72, 2]).
clique(579, [27, 68, 85, 2]).
clique(580, [27, 68, 87, 2]).
clique(581, [27, 68, 90, 2]).
clique(582, [27, 72, 85, 2]).
clique(583, [27, 72, 87, 2]).
clique(584, [27, 72, 90, 2]).
clique(585, [27, 85, 87, 2]).
clique(586, [27, 85, 90, 2]).
clique(587, [27, 87, 90, 2]).
clique(588, [28, 30, 42, 24]).
clique(589, [28, 30, 43, 24]).
clique(590, [28, 30, 61, 24]).
clique(591, [28, 30, 88, 24]).
clique(592, [28, 42, 43, 24]).
clique(593, [28, 42, 50]).
clique(594, [28, 42, 60]).
clique(595, [28, 42, 61, 24]).
clique(596, [28, 42, 88, 24]).
clique(597, [28, 42, 91]).
clique(598, [28, 43, 50]).
clique(599, [28, 43, 61, 24]).
clique(600, [28, 43, 88, 24]).
clique(601, [28, 50, 60, 8]).
clique(602, [28, 50, 91, 8]).
clique(603, [28, 60, 91, 8]).
clique(604, [28, 61, 88, 24]).
clique(605, [30, 42, 43, 28, 24]).
clique(606, [30, 42, 61, 28, 24]).
clique(607, [30, 42, 67, 24]).
clique(608, [30, 42, 88, 28, 24]).
clique(609, [30, 43, 61, 28, 24]).
clique(610, [30, 43, 88, 28, 24]).
clique(611, [30, 61, 67, 24]).
clique(612, [30, 61, 88, 28, 24]).
clique(613, [30, 67, 88, 24]).
clique(614, [31, 40, 46, 27, 2]).
clique(615, [31, 40, 51, 27, 2]).
clique(616, [31, 40, 59, 27, 2]).
69
clique(617, [31, 40, 63, 27, 2]).
clique(618, [31, 40, 72, 27, 2]).
clique(619, [31, 40, 85, 27, 2]).
clique(620, [31, 40, 87, 27, 2]).
clique(621, [31, 46, 51, 27, 2]).
clique(622, [31, 46, 59, 27, 2]).
clique(623, [31, 46, 63, 27, 2]).
clique(624, [31, 46, 72, 27, 2]).
clique(625, [31, 46, 85, 27, 2]).
clique(626, [31, 46, 87, 27, 2]).
clique(627, [31, 51, 63, 27, 6, 2]).
clique(628, [31, 51, 72, 27, 6, 2]).
clique(629, [31, 51, 85, 27, 2]).
clique(630, [31, 51, 87, 27, 2]).
clique(631, [31, 59, 63, 27, 2]).
clique(632, [31, 59, 72, 27, 2]).
clique(633, [31, 59, 85, 27, 2]).
clique(634, [31, 59, 87, 27, 2]).
clique(635, [31, 63, 72, 27, 6, 2]).
clique(636, [31, 63, 85, 27, 2]).
clique(637, [31, 63, 87, 27, 2]).
clique(638, [31, 72, 85, 27, 2]).
clique(639, [31, 72, 87, 27, 2]).
clique(640, [31, 85, 87, 27, 2]).
clique(641, [33, 41, 44, 16, 4]).
clique(642, [33, 41, 97, 4]).
clique(643, [34, 38, 40, 27, 2]).
clique(644, [34, 38, 46, 27, 2]).
clique(645, [34, 38, 59, 27, 2]).
clique(646, [34, 38, 68, 27, 2]).
clique(647, [34, 38, 85, 27, 2]).
clique(648, [34, 38, 87, 27, 2]).
clique(649, [34, 38, 90, 27, 2]).
clique(650, [34, 40, 46, 27, 2]).
clique(651, [34, 40, 59, 27, 2]).
clique(652, [34, 40, 68, 27, 2]).
clique(653, [34, 40, 85, 27, 2]).
clique(654, [34, 40, 87, 27, 2]).
clique(655, [34, 40, 90, 27, 2]).
clique(656, [34, 46, 59, 27, 2]).
clique(657, [34, 46, 68, 27, 2]).
clique(658, [34, 46, 85, 27, 2]).
clique(659, [34, 46, 87, 27, 2]).
clique(660, [34, 46, 90, 27, 2]).
clique(661, [34, 59, 68, 27, 2]).
clique(662, [34, 59, 85, 27, 2]).
clique(663, [34, 59, 87, 27, 2]).
clique(664, [34, 59, 90, 27, 2]).
clique(665, [34, 68, 85, 27, 2]).
clique(666, [34, 68, 87, 27, 2]).
clique(667, [34, 68, 90, 27, 2]).
clique(668, [34, 85, 87, 27, 2]).
clique(669, [34, 85, 90, 27, 2]).
clique(670, [34, 87, 90, 27, 2]).
clique(671, [35, 52, 60, 22, 8]).
clique(672, [35, 52, 65, 22, 8]).
clique(673, [35, 52, 80, 22, 8]).
clique(674, [35, 52, 91, 8]).
clique(675, [35, 60, 80, 22, 8]).
clique(676, [35, 60, 91, 8]).
clique(677, [35, 65, 80, 22, 8]).
clique(678, [35, 80, 91, 8]).
clique(679, [36, 67, 80, 17]).
clique(680, [36, 80, 83, 17, 1]).
clique(681, [36, 83, 93, 17, 1]).
clique(682, [37, 48, 74, 18, 7]).
clique(683, [37, 48, 93, 18, 7]).
clique(684, [37, 74, 93, 18, 7]).
clique(685, [38, 40, 46, 34, 27, 2]).
clique(686, [38, 40, 59, 34, 27, 2]).
clique(687, [38, 40, 68, 34, 27, 2]).
clique(688, [38, 40, 72, 27, 2]).
clique(689, [38, 40, 85, 34, 27, 2]).
clique(690, [38, 40, 87, 34, 27, 2]).
clique(691, [38, 40, 90, 34, 27, 2]).
clique(692, [38, 46, 59, 34, 27, 2]).
clique(693, [38, 46, 68, 34, 27, 2]).
clique(694, [38, 46, 72, 27, 2]).
clique(695, [38, 46, 85, 34, 27, 2]).
clique(696, [38, 46, 87, 34, 27, 2]).
clique(697, [38, 46, 90, 34, 27, 2]).
clique(698, [38, 59, 68, 34, 27, 2]).
clique(699, [38, 59, 72, 27, 2]).
clique(700, [38, 59, 85, 34, 27, 2]).
clique(701, [38, 59, 87, 34, 27, 2]).
clique(702, [38, 59, 90, 34, 27, 2]).
clique(703, [38, 68, 72, 27, 2]).
clique(704, [38, 68, 85, 34, 27, 2]).
clique(705, [38, 68, 87, 34, 27, 2]).
clique(706, [38, 68, 90, 34, 27, 2]).
clique(707, [38, 72, 85, 27, 2]).
clique(708, [38, 72, 87, 27, 2]).
clique(709, [38, 72, 90, 27, 2]).
clique(710, [38, 85, 87, 34, 27, 2]).
clique(711, [38, 85, 90, 34, 27, 2]).
clique(712, [38, 87, 90, 34, 27, 2]).
clique(713, [40, 46, 51, 31, 27, 2]).
clique(714, [40, 46, 59, 38, 34, 27, 2]).
clique(715, [40, 46, 63, 31, 27, 2]).
clique(716, [40, 46, 68, 38, 34, 27, 2]).
clique(717, [40, 46, 72, 38, 27, 2]).
clique(718, [40, 46, 85, 38, 34, 27, 2]).
clique(719, [40, 46, 87, 38, 34, 27, 2]).
clique(720, [40, 46, 90, 38, 34, 27, 2]).
clique(721, [40, 51, 63, 31, 27, 2]).
clique(722, [40, 51, 72, 31, 27, 2]).
clique(723, [40, 51, 85, 31, 27, 2]).
clique(724, [40, 51, 87, 31, 27, 2]).
clique(725, [40, 59, 63, 31, 27, 2]).
clique(726, [40, 59, 68, 38, 34, 27, 2]).
clique(727, [40, 59, 72, 38, 27, 2]).
clique(728, [40, 59, 85, 38, 34, 27, 2]).
clique(729, [40, 59, 87, 38, 34, 27, 2]).
clique(730, [40, 59, 90, 38, 34, 27, 2]).
clique(731, [40, 63, 68, 27, 2]).
clique(732, [40, 63, 72, 31, 27, 2]).
clique(733, [40, 63, 85, 31, 27, 2]).
clique(734, [40, 63, 87, 31, 27, 2]).
clique(735, [40, 63, 90, 27, 2]).
clique(736, [40, 68, 72, 38, 27, 2]).
clique(737, [40, 68, 85, 38, 34, 27, 2]).
clique(738, [40, 68, 87, 38, 34, 27, 2]).
clique(739, [40, 68, 90, 38, 34, 27, 2]).
clique(740, [40, 72, 85, 38, 27, 2]).
clique(741, [40, 72, 87, 38, 27, 2]).
clique(742, [40, 72, 90, 38, 27, 2]).
clique(743, [40, 85, 87, 38, 34, 27, 2]).
clique(744, [40, 85, 90, 38, 34, 27, 2]).
clique(745, [40, 87, 90, 38, 34, 27, 2]).
clique(746, [41, 79, 81, 21, 12]).
clique(747, [41, 79, 84, 21, 12]).
clique(748, [41, 79, 97, 21, 12]).
clique(749, [41, 79, 99, 21, 12]).
clique(750, [41, 81, 84, 21, 12]).
clique(751, [41, 81, 97, 21, 12]).
clique(752, [41, 81, 99, 21, 12]).
clique(753, [41, 84, 97, 21, 12]).
clique(754, [41, 84, 99, 21, 12]).
clique(755, [41, 97, 99, 21, 12]).
clique(756, [42, 43, 50, 28]).
clique(757, [42, 43, 61, 30, 28, 24]).
clique(758, [42, 43, 88, 30, 28, 24]).
clique(759, [42, 50, 60, 28]).
clique(760, [42, 50, 91, 28]).
clique(761, [42, 60, 91, 28]).
clique(762, [42, 61, 67, 30, 24]).
clique(763, [42, 61, 88, 30, 28, 24]).
clique(764, [42, 67, 88, 30, 24]).
clique(765, [43, 61, 88, 42, 30, 28, 24]).
clique(766, [45, 53, 78, 15]).
clique(767, [45, 53, 89, 15]).
clique(768, [45, 58, 78, 10]).
clique(769, [45, 58, 86, 25, 10]).
clique(770, [45, 58, 89]).
clique(771, [45, 78, 86, 15]).
clique(772, [45, 78, 89, 15]).
clique(773, [46, 51, 63, 40, 31, 27, 2]).
clique(774, [46, 51, 72, 40, 31, 27, 2]).
clique(775, [46, 51, 85, 40, 31, 27, 2]).
clique(776, [46, 51, 87, 40, 31, 27, 2]).
clique(777, [46, 59, 63, 40, 31, 27, 2]).
clique(778, [46, 59, 68, 40, 38, 34, 27, 2]).
clique(779, [46, 59, 72, 40, 38, 27, 2]).
clique(780, [46, 59, 85, 40, 38, 34, 27, 2]).
clique(781, [46, 59, 87, 40, 38, 34, 27, 2]).
clique(782, [46, 59, 90, 40, 38, 34, 27, 2]).
clique(783, [46, 63, 68, 40, 27, 2]).
clique(784, [46, 63, 72, 40, 31, 27, 2]).
clique(785, [46, 63, 85, 40, 31, 27, 2]).
clique(786, [46, 63, 87, 40, 31, 27, 2]).
clique(787, [46, 63, 90, 40, 27, 2]).
clique(788, [46, 68, 72, 40, 38, 27, 2]).
clique(789, [46, 68, 85, 40, 38, 34, 27, 2]).
clique(790, [46, 68, 87, 40, 38, 34, 27, 2]).
clique(791, [46, 68, 90, 40, 38, 34, 27, 2]).
clique(792, [46, 72, 85, 40, 38, 27, 2]).
clique(793, [46, 72, 87, 40, 38, 27, 2]).
clique(794, [46, 72, 90, 40, 38, 27, 2]).
clique(795, [46, 85, 87, 40, 38, 34, 27, 2]).
clique(796, [46, 85, 90, 40, 38, 34, 27, 2]).
clique(797, [46, 87, 90, 40, 38, 34, 27, 2]).
clique(798, [47, 55, 76, 26, 13]).
clique(799, [47, 55, 82, 26, 13]).
clique(800, [47, 76, 82, 26, 13]).
clique(801, [48, 62, 74, 18]).
clique(802, [48, 62, 93, 18]).
clique(803, [48, 74, 93, 37, 18, 7]).
clique(804, [49, 54, 64]).
clique(805, [49, 54, 70]).
clique(806, [49, 54, 75]).
clique(807, [49, 54, 100]).
clique(808, [49, 64, 70]).
clique(809, [49, 64, 75]).
clique(810, [49, 64, 77]).
clique(811, [49, 64, 100]).
clique(812, [49, 70, 75]).
clique(813, [49, 70, 77]).
clique(814, [49, 70, 100]).
clique(815, [49, 75, 77]).
clique(816, [49, 75, 100]).
clique(817, [49, 77, 100]).
clique(818, [50, 60, 80, 8]).
clique(819, [50, 60, 91, 42, 28]).
clique(820, [50, 80, 91, 8]).
clique(821, [51, 63, 72, 46, 40, 31, 27, 2]).
clique(822, [51, 63, 85, 46, 40, 31, 27, 2]).
clique(823, [51, 63, 87, 46, 40, 31, 27, 2]).
clique(824, [51, 63, 98, 27, 6]).
clique(825, [51, 72, 85, 46, 40, 31, 27, 2]).
clique(826, [51, 72, 87, 46, 40, 31, 27, 2]).
clique(827, [51, 85, 87, 46, 40, 31, 27, 2]).
clique(828, [52, 60, 80, 35, 22, 8]).
clique(829, [52, 60, 91, 35, 8]).
clique(830, [52, 65, 71, 22]).
clique(831, [52, 65, 80, 35, 22, 8]).
clique(832, [52, 80, 91, 35, 8]).
clique(833, [53, 71, 73, 15]).
clique(834, [53, 71, 89, 15]).
clique(835, [53, 71, 92, 15]).
clique(836, [53, 71, 96, 15]).
clique(837, [53, 73, 89, 15]).
clique(838, [53, 73, 92, 15]).
clique(839, [53, 73, 96, 15]).
clique(840, [53, 78, 89, 45, 15]).
clique(841, [53, 78, 92, 15]).
70
clique(842, [53, 78, 96, 15]).
clique(843, [53, 89, 92, 15]).
clique(844, [53, 89, 96, 15]).
clique(845, [53, 92, 96, 15]).
clique(846, [54, 64, 70, 49]).
clique(847, [54, 64, 75, 49]).
clique(848, [54, 64, 100, 49]).
clique(849, [54, 70, 75, 49]).
clique(850, [54, 70, 100, 49]).
clique(851, [54, 75, 100, 49]).
clique(852, [55, 76, 82, 47, 26, 13]).
clique(853, [56, 57, 69, 9, 3]).
clique(854, [56, 57, 94, 19, 3]).
clique(855, [56, 69, 70, 5]).
clique(856, [56, 69, 75, 5]).
clique(857, [56, 69, 77, 5]).
clique(858, [56, 69, 94, 9, 3]).
clique(859, [56, 69, 95, 5]).
clique(860, [56, 69, 100, 5]).
clique(861, [56, 70, 75, 5]).
clique(862, [56, 70, 77, 5]).
clique(863, [56, 70, 95, 5]).
clique(864, [56, 70, 100, 5]).
clique(865, [56, 75, 77, 5]).
clique(866, [56, 75, 95, 5]).
clique(867, [56, 75, 100, 5]).
clique(868, [56, 77, 95, 5]).
clique(869, [56, 77, 100, 5]).
clique(870, [56, 95, 100, 5]).
clique(871, [57, 69, 94, 56, 9, 3]).
clique(872, [58, 78, 86, 45, 10]).
clique(873, [58, 78, 89, 45]).
clique(874, [59, 63, 68, 46, 40, 27, 2]).
clique(875, [59, 63, 72, 46, 40, 31, 27, 2]).
clique(876, [59, 63, 85, 46, 40, 31, 27, 2]).
clique(877, [59, 63, 87, 46, 40, 31, 27, 2]).
clique(878, [59, 63, 90, 46, 40, 27, 2]).
clique(879, [59, 68, 72, 46, 40, 38, 27, 2]).
clique(880, [59, 68, 85, 46, 40, 38, 34, 27,
2]).
clique(881, [59, 68, 87, 46, 40, 38, 34, 27,
2]).
clique(882, [59, 68, 90, 46, 40, 38, 34, 27,
2]).
clique(883, [59, 72, 85, 46, 40, 38, 27, 2]).
clique(884, [59, 72, 87, 46, 40, 38, 27, 2]).
clique(885, [59, 72, 90, 46, 40, 38, 27, 2]).
clique(886, [59, 85, 87, 46, 40, 38, 34, 27,
2]).
clique(887, [59, 85, 90, 46, 40, 38, 34, 27,
2]).
clique(888, [59, 87, 90, 46, 40, 38, 34, 27,
2]).
clique(889, [60, 80, 91, 52, 35, 8]).
clique(890, [61, 67, 80]).
clique(891, [61, 67, 88, 42, 30, 24]).
clique(892, [62, 74, 93, 48, 18]).
clique(893, [63, 68, 72, 59, 46, 40, 27, 2]).
clique(894, [63, 68, 85, 59, 46, 40, 27, 2]).
clique(895, [63, 68, 87, 59, 46, 40, 27, 2]).
clique(896, [63, 68, 90, 59, 46, 40, 27, 2]).
clique(897, [63, 72, 85, 59, 46, 40, 31, 27,
2]).
clique(898, [63, 72, 87, 59, 46, 40, 31, 27,
2]).
clique(899, [63, 72, 90, 59, 46, 40, 27, 2]).
clique(900, [63, 85, 87, 59, 46, 40, 31, 27,
2]).
clique(901, [63, 85, 90, 59, 46, 40, 27, 2]).
clique(902, [63, 87, 90, 59, 46, 40, 27, 2]).
clique(903, [64, 70, 75, 54, 49]).
clique(904, [64, 70, 77, 49]).
clique(905, [64, 70, 100, 54, 49]).
clique(906, [64, 75, 77, 49]).
clique(907, [64, 75, 100, 54, 49]).
clique(908, [64, 77, 100, 49]).
clique(909, [65, 71, 73, 22]).
clique(910, [68, 72, 85, 63, 59, 46, 40, 27,
2]).
clique(911, [68, 72, 87, 63, 59, 46, 40, 27,
2]).
clique(912, [68, 72, 90, 63, 59, 46, 40, 27,
2]).
clique(913, [68, 85, 87, 63, 59, 46, 40, 27,
2]).
clique(914, [68, 85, 90, 63, 59, 46, 40, 27,
2]).
clique(915, [68, 87, 90, 63, 59, 46, 40, 27,
2]).
clique(916, [69, 70, 75, 56, 5]).
clique(917, [69, 70, 77, 56, 5]).
clique(918, [69, 70, 95, 56, 5]).
clique(919, [69, 70, 100, 56, 5]).
clique(920, [69, 75, 77, 56, 5]).
clique(921, [69, 75, 95, 56, 5]).
clique(922, [69, 75, 100, 56, 5]).
clique(923, [69, 77, 95, 56, 5]).
clique(924, [69, 77, 100, 56, 5]).
clique(925, [69, 95, 100, 56, 5]).
clique(926, [70, 75, 77, 69, 56, 5]).
clique(927, [70, 75, 95, 69, 56, 5]).
clique(928, [70, 75, 100, 69, 56, 5]).
clique(929, [70, 77, 95, 69, 56, 5]).
clique(930, [70, 77, 100, 69, 56, 5]).
clique(931, [70, 95, 100, 69, 56, 5]).
clique(932, [71, 73, 89, 53, 15]).
clique(933, [71, 73, 92, 53, 15]).
clique(934, [71, 73, 96, 53, 15]).
clique(935, [71, 89, 92, 53, 15]).
clique(936, [71, 89, 96, 53, 15]).
clique(937, [71, 92, 96, 53, 15]).
clique(938, [72, 85, 87, 68, 63, 59, 46, 40,
27, 2]).
clique(939, [72, 85, 90, 68, 63, 59, 46, 40,
27, 2]).
clique(940, [72, 87, 90, 68, 63, 59, 46, 40,
27, 2]).
clique(941, [73, 89, 92, 71, 53, 15]).
clique(942, [73, 89, 96, 71, 53, 15]).
clique(943, [73, 92, 96, 71, 53, 15]).
clique(944, [75, 77, 95, 70, 69, 56, 5]).
clique(945, [75, 77, 100, 70, 69, 56, 5]).
clique(946, [75, 95, 100, 70, 69, 56, 5]).
clique(947, [77, 95, 100, 75, 70, 69, 56, 5]).
clique(948, [78, 89, 92, 53, 15]).
clique(949, [78, 89, 96, 53, 15]).
clique(950, [78, 92, 96, 53, 15]).
clique(951, [79, 81, 84, 41, 21, 12]).
clique(952, [79, 81, 97, 41, 21, 12]).
clique(953, [79, 81, 99, 41, 21, 12]).
clique(954, [79, 84, 97, 41, 21, 12]).
clique(955, [79, 84, 99, 41, 21, 12]).
clique(956, [79, 97, 99, 41, 21, 12]).
clique(957, [81, 84, 97, 79, 41, 21, 12]).
clique(958, [81, 84, 99, 79, 41, 21, 12]).
clique(959, [81, 97, 99, 79, 41, 21, 12]).
clique(960, [84, 97, 99, 81, 79, 41, 21, 12]).
clique(961, [85, 87, 90, 72, 68, 63, 59, 46,
40, 27, 2]).
clique(962, [89, 92, 96, 78, 53, 15]).
superclique(1006,[910, 911, 912, 913, 914,
915, 938, 939, 940, 961, 894, 895, 896, 897,
898, 899, 900, 901, 902, 883, 884, 885, 886,
887, 888, 893, 882,881,
875,876,877,878,879,880, 874, 827, 826,
825, 824, 823, 822, 821, 797, 796, 795,794,
793,792, 791, 790, 789, 788, 787,786, 785,
784, 783, 782, 781, 780, 779,778, 777, 776,
775, 774, 773, 745, 744,743, 742, 741, 740,
739, 738, 737, 736,735, 734, 733, 732,731,
730, 729, 728,727, 726, 725, 724, 723, 722,
721, 720,719, 718, 717, 716, 715, 714, 713,
712,711, 710, 709, 708, 707, 706, 705,
704,703, 702, 701, 700, 699, 698, 697,
696,695, 694, 693, 692, 691, 690, 689,
688,687, 686, 685, 670, 669, 668, 667, 666,
665,
664,663,662,661,660,659,522,524,525,561,5
62,565,573,614,615,616,617,618,619,620,62
1,622,
623,624,625,626,627,628,629,630,631,632,6
33,634,635,636,637,638,639,640,643,644,64
5,646,647,648,649,650,651,652,653,654,655
,656,657,658]).
superclique(1007,[960, 959,958,956, 955,
954,953,952,951,755,754,753, 752, 751,
750, 749, 748, 747, 746, 462, 460, 458, 457,
456, 454, 453, 452, 438, 430, 434]).
superclique(1008, [891, 819, 765, 764, 763,
762, 758, 757, 612, 610, 609, 605, 606,
608]).
superclique(1009, [960, 959, 958, 957, 956,
955, 954, 953, 952, 951, 755, 754, 753, 752,
751, 750, 749, 748, 641, 746, 747]).
superclique(1010, [889, 832, 831, 829, 828,
677, 675, 671, 672, 673]).
superclique(1011, [892, 803, 684, 682, 420,
683]).
superclique(1012, [840, 769, 872]).
superclique(1013, [871, 858, 854, 452, 438,
853, 430, 434]).
superclique(1014, [947, 946, 945, 944, 931,
930, 928, 927, 926, 922, 919, 916, 903, 905,
907]).
superclique(1015, [947, 946, 945, 944, 931,
930, 929, 928, 927, 926, 925, 924, 923, 922,
921, 920, 919, 918, 916, 871, 858, 917, 853,
854]).
superclique(1016, [962, 950, 949, 948, 943,
942, 941, 937, 936, 935, 934, 932, 840,
933]).
superclique(1017, [640, 746, 747]).
superclique(1018, [640, 746, 748]).
superclique(1019, [640, 746, 749]).
superclique(1020, [640, 746, 750]).
superclique(1020, [640, 746, 751]).
superclique(1021, [640, 746, 752]).
superclique(1001,[511, 769, 872]).
superclique(1002,[511, 798, 799, 800, 852]).
superclique(1003,[420, 680, 681]).
superclique(1004,[892, 803, 684, 420, 681,
683]).
superclique(1005, [960, 959, 958, 957, 956,
955, 954, 953, 952, 951, 755, 754, 753, 752,
751, 750, 749, 748, 747, 746, 462, 460, 458,
457, 456, 454, 453, 452, 430, 434, 438]).
71
Appendix F: VennMaster and DrawEuler representations of test data
Layout 1:
Layout 2:
72
Layout 3:
Layout 4:
73
Layout 5:
Layout 6:
74
Appendix G: VennMaster and DrawEuler representations of simulated
data
Arena
75
W-track
8-track
76
Linear track
Circle Track
77
Appendix H: Hand-drawn representations of simulated data
The numbers refer to the cliques, and fields as represented in the qualitative descriptions.
8-track:
Arena:
78
Linear:
Circle:
W-track:
79