Deriving Sound Inference Rules
for Concept Diagrams
Peter Chapman, Gem Stapleton, John Howse
Ian Oliver
Visual Modelling Group, University of Brighton, UK
{p.b.chapman, g.e.stapleton, john.howse}@brighton.ac.uk
Nokia Services, Finland
[email protected]
Abstract—The process of designing and modelling an ontology
can be difficult, especially if the user finds the syntax to be
relatively inaccessible. Providing users with graphical syntax
with which they can model and visualise their ontology has
the potential to be helpful. Previously, we informally introduced
concept diagrams for ontology visualisation and modelling. We
present a case study comprising: (a) a set of axioms for an
ontology, and (b) a set of theorems that follow from the axioms,
together with their proofs. The proofs have been constructed so
that they are, in our opinion, of an intuitive style. From these
proofs, we derive a set of sound inference rules that can be used
to formally reason about ontologies following the same intuitive
style. This approach to designing inference rules differs from
previous efforts where the primary focus has been on obtaining
a set of sound and complete inference rules, rather than on
intuitiveness.
I. I NTRODUCTION
An ontology comprises a set of statements (called axioms)
that capture properties of individuals, concepts and roles.
Individuals represent particular elements of the modelled domain, with concepts and roles corresponding to classes and
binary relations, respectively. The primary (formal) notations
for ontology modelling are symbolic, such as description
logics or OWL [1]. Unfortunately, symbolic logics need not
be accessible to the broad range of users; for example, there
is ongoing work in biology concerning the gene ontology
[10]. To aid accessibility and understanding of an ontology,
standard ontology editors often provide visualisation support,
but it is rather limited. For example, Protégé includes a
plug-in visualisation package, OWLViz, that shows derived
hierarchical relationships between the concepts in the ontology
but does not show information about other binary relations,
for instance. These visualisations can be considered artefacts
created from the ontology and, thus, are not the primary
notation in which the ontology is created. This means that
the creator of the ontology needs to be fluent in symbolic
notations, which is not the case for many users who require
ontologies. Using diagrammatic logics for ontology specification has the potential to realise significant benefits. Currently,
some diagrammatic notations have been used for specifying
ontologies, but they are either not formalised [3] or do not
offer many of the benefits that good diagrammatic notations
afford [4]. Other work has looked at how to visualise an
instance of an ontology, as opposed to the ontology definition
[2].
Research on diagrammatic logics has progressed steadily,
since the early work of Shin [15]. The results most closely
related to this paper concern Kent’s constraint diagram notation [9]. Constraint diagrams were designed for formally
providing constraints on software models and give formal
syntax and semantics [6]. However, to make the notation
unambiguous, the ‘raw’ notation introduced by Kent had to
be augmented with reading trees to specify an ordering on
quantifiers, amongst other things. An evaluation of augmented
constraint diagrams was conducted in [16], which highlighted
various problems from both formal and usability perspectives.
In [12], we proposed ontology diagrams, which we have
renamed concept diagrams, for ontology modelling. Much of
the ‘atomic’ syntax of constraint diagrams, the syntax with
which diagrams are built, is shared by concept diagrams.
However, the manner in which concept diagrams use this
syntax is (sometimes subtly) different. Section II introduces
concept diagrams using a case study approach, and concludes
with an evaluation of concept diagrams. The syntax and
semantics are given in section III and section IV presents a
set of sound inference rules that are designed by examining
proofs presented in the case study.
II. C ONCEPT D IAGRAMS : A C ASE S TUDY
Our case study is a variation of the University of Manchester’s People Ontology [11]. This ontology relates people,
their pets and their vehicles. Owing to space restrictions, we
selectively present parts of this ontology, illustrating the core
syntax and semantics of concept diagrams. It is rich enough
to demonstrate reasoning tasks that one can perform. The
section concludes with an evaluation that identifies features
of concept diagrams that are deemed effective by cognitive
theories relating to diagrams and, in addition, teases out
improvements brought about over constraint diagrams.
We begin by presenting the axioms of the ontology.
Concept diagrams are based on Euler diagrams, which
employ closed curves to express subset and disjointness
relationships between concepts. The concept diagrams below
(which are also Euler diagrams) assert: (a) men are precisely
adult male people, (b) every van is a vehicle, and (c) every
driver is an adult:
adult
male
man
(a):
vehicle
adult
van
driver
(b):
(c):
(h) asserts that ABC1 is a white van. Finally, diagram (i) tells
us that Rex is an animal and a pet of Mick.
person
male
In (a), the shading asserts that the intersection of the sets adult,
male and person minus the set of men is empty.
The statements made by the Euler diagram fragment
are limited, in that this fragment has the expressiveness
of monadic first order logic [17]. Concept diagrams use
additional syntax with which to make semantically richer
statements. For instance, arrows are used to define properties
of roles, and concept diagrams also use variables to allow
quantification over elements. An example illustrating the use
of arrows and variables can be seen in diagram (d), which
expresses that every animal is a pet of some set of people:
animal
animal
(d):
a
Þ
person
isPetOf
a
(g):
whiteThing
drives
ABC1
Mick
(h):
van
ABC1
animal
(i):
isPetOf
Rex
Mick
We have enough information to prove diagrammatically
some lemmas, culminating in proving that Mick is a white
van man. We use these examples to motivate the design of
inference rules for concept diagrams (section IV).
Our first diagrammatic proof, using the axioms of the people
ontology, establishes that Mick is a person.
person
More specifically, (d) asserts that if a is an animal then the
role isPetOf relates a to a (possibly empty) set of people, and
only people. Here, a is a free variable. When a is instantiated
as a particular element, e, the unlabelled curve represents the
image of isPetOf with its domain restricted to {e}. As animal
and person are not disjoint concepts – a person is an animal –
the curves representing these concepts are placed in separate
sub-diagrams, so that no inference can be made about the
relationship between them. A direct translation of (d), into
first order predicate logic, is
Lemma 1 Mick is a person:
animal
Proof From diagram (i) we deduce
that Rex is an animal:
(
)
animal(a) ⇒ animal(a) ∧ ∀p(isPetOf(a, p) ⇒ person(p)) .
From diagram (d) we deduce all of the individuals of which Rex is a pet
are people:
Since a is free, it is essentially quantifying over all elements;
by generalisation, the above formula is equivalent to the
sentence
From diagram (i) we
know Rex is a pet of
Mick:
(
(
))
∀a animal(a) ⇒ animal(a) ∧ ∀p(isPetOf(a, p) ⇒ person(p)) .
Now we will define the concepts of being a driver and a
white van man. The former is defined in diagram (e): p is a
driver if and only if p is a person who drives some vehicle.
The latter is defined in diagram (f): m is a white van man if
and only if m is a man who drives a white van.
(e):
driver
person
p
Û
man
(f):
whiteVanMan m
drives
vehicle
drives
van
p
m
Û
Mick
(1)
Rex
animal
person
isPetOf
(2)
Rex
animal
person
isPetOf
Rex
Mick
person
Therefore, Mick is a person, as required:
Mick
(4)
In the above proof, the deduction that the set of individuals of
which Rex is a pet are all people relied on pattern matching
diagram (d) and that derived at step (1). We believe it is
clear from the visualisations that one can make the given
deduction. The last step in the proof simply deletes syntax
from the diagram in the preceding step, thus weakening
information, to give the desired conclusion. Much of the
reasoning we shall demonstrate requires pattern matching and
syntax deletion.
whiteThing
In addition to the general properties for which we have
just defined axioms, we now introduce some further axioms
concerning individuals called Mick, ABC1, and Rex. Diagram
(g) expresses that Mick is male and drives ABC1. Diagram
(3)
adult
Lemma 2 Mick is an adult:
Mick
Proof From diagram (b) we
know that all vans are vehicles so we deduce, from
diagram (h):
whiteThing
van
vehicle
man
ABC1
(5)
(6)
Therefore Mick drives
some white thing which
is a van:
(7)
By diagram (f), we conclude that Mick is a
white van man:
vehicle
Therefore, ABC1 is a vehicle:
ABC1
male
From diagram (g), we
therefore deduce:
Now, ABC1 is a
particular
vehicle.
Therefore
Mick
drives some vehicle:
We can discard the
information that Mick
is male:
vehicle
drives
Mick
ABC1
male
(8)
vehicle
drives
(9)
Mick
Person
By lemma 1, Mick is
a person, thus:
vehicle
drives
(10)
Mick
driver
Hence, by diagram (e),
Mick is a driver:
(11)
Mick
adult
By diagram (c) drivers are
adults:
driver
Mick
(12)
Mick
(13)
adult
Hence, Mick is an adult, as
required:
Lemma 3 establishes that Mick is a man and follows from
lemmas 1 and 2, together with diagrams (a) and (g) (the
interested reader may like to attempt the proof):
man
Lemma 3 Mick is a man:
van
drives
Mick
ABC1
(16)
whiteThing
man
van
drives
Mick
(17)
whiteThing
whiteVanMan
Mick
(18)
A. Evaluation of Concept Diagrams
vehicle
drives
Mick
From diagram (h) we
have:
Mick
Concept diagrams have been designed by taking into account cognitive theories of what constitutes an effective diagrammatic notation. For instance, they build on Euler diagrams, and it is well-known that these diagrams bring with
them free-rides [13], which have been empirically evaluated [14]. Free-rides occur when a piece of information is
explicitly conveyed by a diagram that would, in symbolic
notations, typically need to be inferred. Concept diagrams have
free rides on top of those provided by the underlying Euler
diagram; for instance, the location of variables inside curves
can give ‘free’ information. The first time we see this is in
line (3) in the proof of Lemma 1, where Mick is placed inside
the unlabelled curve which, in turn, is drawn inside the person
curve; for free, we can see that Mick is a person. Free rides
also occur in relation to the arrows, as was discussed in [8].
Concept diagrams are also well-matched to their semantics [7]: syntactic containment (resp. disjointness of
curves/distinctness of dots) represents semantic containment
(resp. disjointness of sets/distinctness of elements), and arrows
represent directed binary relations; see [16] for a related
discussion around well-matchedness for the closely related
constraint diagram notation.
Another effective feature of concept diagrams is their use
of multiple universes: they use collections of interlinked
sub-diagrams, each of which is making assertions over the
universal set, to avoid problems associated with diagram
clutter and over-specificity, both of which can be real
problems with diagrammatic notations. The use of multiple
universes can be seen in (d), where the arrow connects two
sub-diagrams. Without the use of multiple universes, (d)
would instead be much more cluttered:
whiteVanMan
Theorem 1 Mick is a white
van man:
Proof We can discard
the information that
Mick is male from
diagram (g):
(j):
drives
ABC1
Mick
man
By lemma 3, Mick is a
man so we deduce
drives
Mick
animal
isPetOf
Mick
ABC1
animal
a
Þ
person
a
With regard to over-specificity, using multiple universes
allows one to avoid asserting distinctness of individuals. For
example, in (e), we have not asserted that p represents an
element distinct from that represented by the unlabelled dot
inside the vehicle curve; similar avoidances of over-specifying,
(15) by asserting distinctness, occur in (f), (g), and (i). We see the
use of multiple universes as a particular strength, especially
(14)
since over-specificity is a common problem with diagrammatic
notations [13].
In the context of inference tasks, the visual proofs we have
given are of intuitive style and each deduction step can be
proved sound. We argue that intuitiveness follows from the
syntactic properties of the diagrams reflecting the semantics.
For instance, because containment at the syntactic level reflects
containment at the semantic level, one can use intuition about
the semantics when manipulating the syntax in an inference
step. This is, perhaps, a primary advantage of reasoning with
a carefully designed diagrammatic logic. We use the proofs
presented in this section as a basis for a set of inference rules
that can be both formalised and proved sound. Thus, concept
diagrams are able to formally prove theorems in an intuitive
style.
With respect to Constraint Diagrams (CDs), concept diagrams share much of the same atomic syntax (curves, dots,
shading, and arrows). However, the manner in which these
pieces of syntax are used to make statements is subtly, but
significantly, different:
1) CDs do not use variables at all; in CDs, dots are of
two types in order to represent universal or existential
quantification. This leads to inherent ambiguity in the
notation, which was resolved using reading trees [6].
Concept diagrams do not need reading trees at all
and are unambiguous; we provide formal semantics in
section III.
2) In CDs, quantification can only occur within a socalled unitary diagram meaning that certain classes of
statement are impossible to construct (particularly with
regard to universal quantification over a disjunctive
statement) [16]. Sometimes these statements can be
converted into equivalent statements that can be defined,
but this restrictive nature of their syntax places a burden
on users. Concept diagrams do not suffer from these
problems because they use variables with quantification
occurring ‘outside of unitary diagrams’. Thus, concept
diagrams overcome expressiveness problems with CDs
identified in [16].
3) CDs are a first-order logic, whereas concept diagrams
are a second-order logic, despite using essentially the
same atomic syntax as CDs.
4) CDs do not use multiple universes. This means that CDs
can be cluttered and necessarily over-specify information.
5) In terms of reasoning, some sound (but not complete)
inference rules have been derived for CDs [5]. However,
the problems regarding quantification described above
mean that some seemingly intuitive manipulations of
the syntax that produce a ‘logical consequence’ are,
in fact, unsound. Moreover, sound reasoning can be
counter-intuitive. Again, we refer the reader to [16] for
an elaboration on this point. Concept diagrams do not
suffer the same kinds of problem.
III. S YNTAX AND S EMANTICS
As is typical with diagrammatic logics, we can formalise
concept diagrams using an abstract syntax together with a
model theoretic semantics. For ease of accessibility, we will
describe their concrete syntax, linking it with their semantics,
without presenting a full formalisation.
Concept diagrams, as is standard with any logic, are defined
over a specified vocabulary. In particular, we have the following pairwise disjoint sets: IN is a set of individual names,
VE is a set of variables that quantify over elements, AC is a
set of atomic concepts, VS is a set of variables that quantify
over sets, and R is a set of roles. The elements of these sets
are used to label diagrammatic syntax.
For example, in (d), the label a is chosen from VS, animal
is in AC and isPetOf is in R. Strictly speaking, the unlabelled
curve in (d) has a label in VS (this curve is an existentially
quantified subset of the universal set). We have adopted a
convention that any existentially quantified variable (from VE
or VS) where the respective existential quantifier is innermost
can be omitted, as can the quantifier, provided this does not
lead to ambiguity. This is, technically, an abuse of notation
but leads to less cluttered diagrams. The formal version of
(d), where a is still free, is:
exists P
(k):
animal
animal
a
Þ
person
isPetOf
P
a
Unlabelled dots, as in (e), are treated in the same way; so,
strictly speaking, the unlabelled dot in (e) technically has
a label, v say, chosen from VE and the diagram formally
includes ‘exists v’. Diagram (g) also abuses notation, as well as
using labels from IN : Mick and ABC1 both represent specific
individuals (constants). In the concrete syntax, to help readers
distinguish variables from other labels we place them in italics.
Concept diagrams are based on Euler diagrams. An Euler
diagram comprises a set of labelled, closed curves, each of
which represents a set. The curve labels are chosen from AC ∪
VS. The curves give rise to a set of regions in the diagram as
follows: given a subset of the curves, C, in Euler diagrams d,
the set of points in the plane interior to those curves in C but
exterior to the remaining curves in d is a region called a zone.
Zones may be shaded. For example, (a) has 9 zones (including
that outside all three curves), of which one is shaded.
Definition 1. A unitary diagram is a collection of Euler
diagrams, together, possibly, with additional syntax:
1) They may include spiders, which are trees whose vertices
are placed within some subset of the zones in the
collection of Euler diagrams1 . Each spider is labelled
by a unique element of IN ∪ VE.
1 In this paper, all spiders have their vertices placed within a single Euler
diagram component of a unitary diagram.
2) Spiders may be joined by =, to assert that the elements
the spiders represent are equal2 .
3) Unitary diagrams may include arrows which have a
source and a target within the unitary diagram. The
source and target can be curves or spiders (or one of
each). Each arrow has a unique label chosen from R.
We can now define concept diagrams:
Definition 2. Every unitary diagram is a concept diagram.
If d1 and d2 are concept diagrams then so are (d1 ∧ d2 ),
(d1 ∨ d2 ), (d1 ⇒ d2 ), (d1 ⇔ d2 ), and ¬d1 . If d1 is a concept
diagram and v is a variable in VE ∪ VS then ∃v d1 and ∀v d1
are concept diagrams.
At the concrete level, we use rectangles instead of brackets.
There are choices for the concrete representation of the logical
operators. For example, Shin connects diagrams with a line
segment to denote disjunction [15].
We also take a standard approach to defining the semantics
of concept diagrams. First, the vocabulary over which the logic
is defined is interpreted appropriately:
Definition 3. An interpretation, I = (U, ·I ), where
1) U is a non-empty set, called the universal set,
2) for each element, s, in IN , sI is an element of U ,
3) for each element, c, in AC, cI is a subset of U , and
4) for each role, r, in R, rI is a binary relation on U .
An extension of ·I to variables and zones ensures
1) for each element, e, in VE, eI is an element of U ,
2) for each element, E, in VS, E I is a subset of U , and
3) for each zone, z, z I is the intersection of the sets
represented by the curves that z is inside minus the union
of the sets represented by the curves that z is outside.
We now proceed to define a notion of truth in an extended
interpretation, which forms the basis of a definition of satisfaction. To define truth in an extended interpretation, the first step
is to define the semantic formula of a unitary diagram. This
captures the constraints, provided by the unitary diagram, on
the relationships between the represented individuals, atomic
concepts, and roles along with the properties of the variables
used in the diagram.
Definition 4. Let d be a unitary concept diagram and let I =
(U, ·I ) be an extended interpretation. The semantic formula
for d given I, denoted f orm(d, I), is the conjunction of the
following conditions:
1) Curves Condition In each underlying Euler diagram the
union of the sets represented by the zones is U .
2) Shading Condition In each underlying Euler diagram,
d, every shaded zone contains only elements represented
by spiders (strictly, spider labels) in d.
3) Spiders’ Location Condition Each spider, s, represents
an element that lies in one of sets represented by the
zones in which its vertices are placed.
2 Again,
our examples did not make use of this helpful piece of syntax.
4) Spiders’ Distinctness Condition Any two distinct spiders, s1 and s2 , in d represent the same element if and
only if they are joined by =.
5) Arrows Condition For each arrow with source s, target t
and label r, the image of rI under the domain restriction
to sI equals tI .
Given an extended interpretation, we can determine whether
the semantic formula for a unitary diagram d is true. This
leads to a definition of satisfaction within an interpretation, I:
d is satisfied by I if the semantic formula is true under all
extensions of I. The extension of the definition of satisfies
to concept diagrams is straightforward, mirroring standard
inductive approaches used in symbolic logics.
IV. I NFERENCE RULES
There are several different kinds of inference rules exhibited
in section 2. We had standard first-order logical rules, such
as modus ponens: from A ⇒ B and A, deduce B; and
specialisation: from ∀xA(x) (or, more simply, from A(x)),
deduce A(t) for some particular t. Secondly, we had rules
which deleted syntax from a diagram: for example, given a
diagram with curves representing A ⊆ B and B ⊆ C, we
can remove the curve B to obtain a diagram with only the
information A ⊆ C. Thirdly, there were those rules which
combine information from two diagrams: given two diagrams
which share a region, we can combine (some of) the spiders
and shading from each diagram into a new diagram with this
region. We will specify rules which are of each type. Rules
are displayed in a two dimensional way, the conclusions and
premisses separated by a horizontal line. For instance, if R was
a two-premiss rule with premisses d1 and d2 and conclusion
d3 , then instances of this rule would be displayed as
d1
d2
d3
R
Step (1) of lemma 1 involved syntax deletion, a commonly
occurring inference step. Removing syntax from a diagram is
a form of information weakening. For instance, any interpretation which is a model for a unitary diagram containing an
arrow, will also be a model for the same diagram without the
arrow. This is because all of the other model conditions will
still be satisfied automatically. However, we must be careful
to delete only information in the positive fragment of the
diagrammatic logic. In other words, if d1 ⇒ d2 , and d1 is
d1 with some syntax deleted, it does not necessarily follow
that d1 ⇒ d2 .
To formalise a syntax deletion rule, we define a sub-diagram
of a unitary diagram:
Definition 5. Let d1 and d2 be unitary diagrams. If d2 can
be obtained from d1 by deleting any items of syntax then d2
is a sub-diagram of d1 .
Definition 6 (Rule: Delete Syntax). Let d1 be a unitary
diagram and let d2 be a sub-diagram of d1 , obtained by
deleting any syntax except that: (i) no occurrences of = are
deleted from d1 , unless at least one of the spiders incident
with a deleted occurrence of = is also deleted, and (ii) no
spider is deleted from d1 that has a node placed in a shaded
zone, unless that shading is also deleted. Then
d1 u
−
d2
implications. We can also use modus ponens to eliminate the
iff connective, ⇔. An instance of this rule can be see here:
animal
animal
Rex
Þ
Another step in the proof of lemma 1 also uses this rule,
namely step (4), where syntax is deleted from the diagram at
the preceding step. This instance is represented as:
animal
animal
person
isPetOf
Rex
Rex
mp
animal
person
isPetOf
Rex
person
isPetOf
Mick
Rex
Returning our focus to the proof of lemma 1, we can use
specialisation and modus ponens to infer line (2) from line
(1) and diagram (d). Thus, we define a derived rule that uses
specialisation and modus ponens, in that the inference made
results from applying first specialisation then modus ponens:
del
person
Mick
Of course, syntax deletion rules can be applied more generally
than just to unitary diagrams. In particular, we also have the
following deletion rules, where d1 and d2 are obtained from
d1 and d2 , respectively, by an instance of the rule u− :
d1 ⇒ d2 ⇒
−
d1 ⇒ d2
d1 ∧ d2
∧−
d1 ∧ d2
d1 ∨ d2
∨−
d1 ∨ d2
For the next rule, we observe, given a diagram which
contains a spider whose label is a free variable, we can
instantiate this variable by replacing all free occurrences of
it with an individual name. All other syntax in the diagram
remains the same. For example, in diagram (d), we can
replace a with Rex:
animal
(l):
animal
Rex
Þ
person
isPetOf
Rex
To present the definition of the specialisation rule we
introduce notation that indicates the substitution of one spider
for another within a diagram: Let d be a concept diagram
containing a free variable, x, and let y be an element of
IN ∪ VE that does not occur anywhere in d. Then each free
occurrence of x can be replaced with y, yielding d0 ; we denote
d0 by d[y/x], read d with y for x. Thus, this replacement
relabels the spiders in d that are labelled with a free occurrence
of x with the label y. Thus, a diagrammatic version of the
standard symbolic logic specialisation rule is:
Definition 7 (Rule: Specialisation). Let d be a diagram where
x is free in d and y ∈ IN ∪ VE is not in d. Then
d
spec
d[y/x]
In most logical inference systems, we have some kind
of device for eliminating implications, usually called modus
ponens. This is a vital rule, since we often need to eliminate
Definition 8 (Rule: Transitive specialisation). Let d1 ⇒ d2
be a diagram where x is free and y ∈ IN ∪ VE is not in
d1 ⇒ d2 . Then
d1 ⇒ d2 d1 [y/x]
tspec
d2 [y/x]
An instance of this rule can be seen here:
animal
animal
a
Þ
person
animal
isPetOf
Rex
a
tspec
animal
person
isPetOf
Rex
There is one final rule that we need to define for the proof of
lemma 1. Step (3) adds a spider labelled Mick to the diagram in
the previous step. In general, suppose we have two diagrams,
d1 and d2 , each containing a curve A. If d2 has a spider, s, in A
and d1 had only spiders in A that we know to represent distinct
elements from s then we can add the information about s to
the curve A in d1 . This can be seen as a form of substitution.
To see more clearly why this is substitution, take the following
example: suppose dA contains an arrow from some spider a to
a derived curve X, which is contained in a curve B. Further,
suppose dB contains an arrow from a to X, and X contains a
spider b. Then, we can combine the information in these two
diagrams by substituting the information from dB into dA : we
would obtain a diagram with a spider a, an arrow from a to
X containing a spider b, contained in B.
Definition 9 (Rule: Copy Spider). Let d1 and d2 be two
diagrams, each containing a curve A, and no shading. Let
s be a spider contained in A in d2 , but not in d1 , and d1
contains no other spider vertices inside A. Then the diagram
d3 is the diagram d1 but with s contained in A.
Instances of this rule are labelled c. In the proof of lemma 1,
we can apply the copy spider rule to introduce Mick to (2),
giving the (3), so here (2) plays the role of d2 and (3) is d3 ;
(i) is d1 . Thus, we now have enough rules to formally prove
lemma 1, shown in figure 1.
Most of the rules needed to derive lemma 2 have already
been defined. However, some new rules are needed. Both
involve adding a curve from one diagram, say d1 , to another,
say d2 , to give d3 . The rule comes in two flavours, depending
on the information available. In the first, we have that d1
contains a curve A, and in d2 , we have that A is contained
in B. Thus, in d3 , we copy B into d1 , so we have that A
is contained in B. For every zone which was outside A, we
create two new zones: one inside B and one outside B. This
is what happens in step (5) in the proof of lemma 2, where d1
is (h), d2 is (b), and d3 is (5) (this is shown below definition
10). Instances of this rule are labelled cc1.
an instance of eliminating a universal quantifier, here we
seek to introduce an existential quantifier: from a specific
individual, we deduce that there exists some individual. Since
specific individuals are represented as named spiders, and
existential quantifiers are represented as unlabelled spiders
(using a slight abuse of notation), this rule has instances of
the form:
Definition 10 (Rule: Copy curve 1). Let d1 and d2 be two
diagrams, each containing a curve A. Let B be a curve
containing A in d2 which does not appear in d1 . Then, the
diagram d3 is d1 , but with B added so that it contains A
and splits all zones outside A into two: one inside B and
one outside B. Any spiders which are in the split zones are
updated in a similar fashion.
All other steps in the proof of lemma 2 are by rules which
have already been defined: (6) is delete syntax, (7) is copy
curve 2, (9) is delete syntax, (10) is copy curve 2, (11) is
transitive specialisation, (12) is copy spider and (13) is delete
syntax. The proof of theorem 1 is by rules which have already
been defined for lemmas 1 and 2.
In order to evaluate the results in this section, we have
proved that the inference rules are all sound.
whiteThing
vehicle
drives
Mick
ABC1
male
vehicle
∃I
drives
Mick
Theorem 1. The inference rules are sound.
vehicle
van
male
van
ABC1
cc1
whiteThing
van
vehicle
ABC1
In the second flavour of the rule, the information we have
is that d1 contains a spider, s, and in d2 the curve A contains
s. Thus, in d3 we have A contains s: we add the curve A to
d1 . New zones are created inside A and outside A: the curve
A splits all zones in d1 . Instances of this rule are labelled cc2.
We see this rule in step (7) in lemma 2, where d1 is (g), d2
is (6), s is ABC1, and d3 is (7) (shown below definition 11).
Definition 11 (Rule: Copy curve 2). Let d1 and d2 be two
diagrams, each containing a spider s. Further, let there exist
a curve A such that s is in A in d2 . Then, the diagram d3 is
the diagram d1 , but has s in the curve A, and A splits every
zone in d1 . The other spiders are updated as in definition 10.
male
vehicle
drives
ABC1
ABC1
Mick
cc2
male
vehicle
drives
Mick
ABC1
Consider step (8) from lemma 2, which turns ABC1 into
an unlabelled dot. This step in the proof represents a standard
logical rule, akin to specialisation. Whereas specialisation was
Proof: (Sketch) We show that the copy spider rule (definition 9) is sound. We need to show that any interpretation
for the premisses is also an interpretation for the conclusion.
Let I be an interpretation for the premisses. In particular, I
satisfies d1 , with curve A being interpreted as AI . The curve
A also exists in d2 , and moreover the spider s is contained in
A in d2 . Thus, by the spiders’ location condition, this means
that sI ⊆ AI . The diagram d3 is d1 with the spider s added
to the curve A. Since I satisfies d1 , and further I is such that
sI ⊆ AI , we have that I satisfies d3 , as required. The rest of
the rules may be proved sound by similar reasoning.
V. C ONCLUSION
We have presented concept diagrams, that have been designed for formally specifying ontologies, via a case study
style approach. As part of this case study, we demonstrated
how concept diagrams can be used to prove theorems about
the ontology. Since concept diagrams are similar to constraint
diagrams, in that the atomic pieces of syntax used by constraint
diagrams are also used by concept diagrams, we argued that
concept diagrams overcome many of the problems of constraint diagrams whilst retaining their intuitive appeal. Leading
on from this, we have designed a set of sound inference
rules. In some places, the rules are quite restrictive. It would
be possible to generalise some of the rules, however they
would then become less intuitive. In the first instance, we have
favoured intuitiveness over generality.
Future work includes extending the inference rule set, to get
closer to completeness; ideally, we will have large complete
fragments of the notation. In addition, we plan to establish the
expressiveness of concept diagrams, with respect to symbolic
animal
isPetOf
Mick
Rex
del
animal
animal
a
Þ
animal
person
isPetOf
Rex
a
tspec
animal
animal
person
isPetOf
isPetOf
Rex
Rex
Mick
c
animal
person
isPetOf
Mick
Rex
del
person
Mick
Fig. 1.
A proof of lemma 1
logics such as predicate logic and description logic. Concept
diagrams include one and two place predicates and the ability
to quantify over both elements and subsets of the universe,
with the latter making them second-order. We conjecture that
concept diagrams are as expressive as second-order predicate
logic with one and two place predicates that can also quantify
over elements and sets. If this conjecture is proved then we will
have established that concept diagrams are highly expressive
relative to other diagrammatic logics.
ACKNOWLEDGEMENT: This research is supported by EPSRC grants Defining Regular Languages with Diagrams
EP/H012311 and Sketching Euler Diagrams EP/H048480.
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