Knowledge Representation WASP 28.10.2016
Knowledge Representation for Autonomous Systems
WASP Autonomous Systems Course
Jacek Malec
Dept. of Computer Science, Lund University
October 28, 2016
Jacek Malec, Computer Science, Lund University
1(47)
Knowledge Representation WASP 28.10.2016
IBM Watson example
https://www.youtube.com/watch?v=DywO4zksfXw
VOLUME 56, NUMBER 3/4, MAY/JUL. 2012
Journal of Research
and Development
Including IBM Systems Journal
This Is Watson
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Knowrob: Why is knowledge so important?
if the robot does not know about the task, the environment, or
the robot, then the programmer has to hardcode everything
programming/instructing at an abstract/semantic level
put the bolt into the nut and fasten it
pour water into the glass
...
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Knowrob: Ontology (knowrob.owl)
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Knowrob: A task ontology
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Knowrob: A task ontology
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Knowrob: Knowledge types
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
KnowRob Components
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Knowrob: Procedural attachments
Compute symbolic
knowledge on demand
from data structures that
already exist on the robot
by attaching procedures
to semantic classes and
properties
Re-use existing
information and make
sure abstract knowledge
is grounded
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Knowrob: Inferring storage location
Jacek Malec, Computer Science, Lund University
10(47)
Knowledge Representation WASP 28.10.2016
Knowrob: Summary
declarative knowledge: ontologies
procedural attachment
logical inference
multi-modal representation
Video (13 mins):
https://www.youtube.com/watch?v=4usoE981e7I
Jacek Malec, Computer Science, Lund University
11(47)
Knowledge Representation WASP 28.10.2016
Plan for today
1
Knowledge-based systems
Tacit knowledge
Inferred knowledge
Domain-specific stuff
Changing premises
Uncertainty
Semantic anchoring
2
Architectures
3
Self-awareness
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Tacit knowledge
Facts about:
Jacek Malec, Computer Science, Lund University
13(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Facts about:
objects
Jacek Malec, Computer Science, Lund University
13(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Facts about:
objects
places
Jacek Malec, Computer Science, Lund University
13(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Facts about:
objects
places
times
Jacek Malec, Computer Science, Lund University
13(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Facts about:
objects
places
times
events
processes
behaviours
Jacek Malec, Computer Science, Lund University
13(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Facts about:
objects
places
times
events
processes
behaviours
vehicle dynamics
rigid body interactions
traffic laws
...
Jacek Malec, Computer Science, Lund University
13(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Background knowledge for all this includes:
Jacek Malec, Computer Science, Lund University
14(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Background knowledge for all this includes:
ontologies
Jacek Malec, Computer Science, Lund University
14(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Background knowledge for all this includes:
ontologies
theories
Jacek Malec, Computer Science, Lund University
14(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Background knowledge for all this includes:
ontologies
theories
physics
mereology
...
Jacek Malec, Computer Science, Lund University
14(47)
Knowledge Representation WASP 28.10.2016
Tacit knowledge
Background knowledge for all this includes:
ontologies
theories
physics
mereology
...
Not everything needs to be explicit, nor expressed in one
monolithic formalism
Jacek Malec, Computer Science, Lund University
14(47)
Knowledge Representation WASP 28.10.2016
Inferred knowledge
(or: turning implicit into explicit)
1
logics (language)
2
theorem proving (mechanics)
3
modes of reasoning
Jacek Malec, Computer Science, Lund University
15(47)
Knowledge Representation WASP 28.10.2016
Logics
Every logical language consists of:
1
syntax (correctly formed expressions, well-formed formulae,
wffs)
2
semantics (meaning of expressions)
It is usually associated with a set of algorithms for deriving or
verifying semantic status of wffs.
Jacek Malec, Computer Science, Lund University
16(47)
Knowledge Representation WASP 28.10.2016
Logic: propositional
propositional variables: P, Q, R, S
operators (connectives): ¬, _, ^, !
formulae: P, ¬Q ^ R, ¬(Q _ R), Q ! P
Language, i.e. the set of all well-formed formulae (wff):
{P, Q, ¬P, ¬Q, P ^ Q, P _ Q, . . .}
So: what does P mean?
Jacek Malec, Computer Science, Lund University
17(47)
Knowledge Representation WASP 28.10.2016
Logic: propositional
propositional variables: P, Q, R, S
operators (connectives): ¬, _, ^, !
formulae: P, ¬Q ^ R, ¬(Q _ R), Q ! P
Language, i.e. the set of all well-formed formulae (wff):
{P, Q, ¬P, ¬Q, P ^ Q, P _ Q, . . .}
So: what does P mean?
What does JacekIs59 mean?
Jacek Malec, Computer Science, Lund University
17(47)
Knowledge Representation WASP 28.10.2016
Logic: first order
Predicates (relations, properties):
Jacek Malec, Computer Science, Lund University
18(47)
Knowledge Representation WASP 28.10.2016
Logic: first order
Predicates (relations, properties):
AgeOf , Bald, CapitalOf , YoungerThan, <, =, P, Q, . . .
Constants:
Jacek Malec, Computer Science, Lund University
18(47)
Knowledge Representation WASP 28.10.2016
Logic: first order
Predicates (relations, properties):
AgeOf , Bald, CapitalOf , YoungerThan, <, =, P, Q, . . .
Constants:
Jacek , 59, Stockholm, Sweden, Fredrik , table59, c, d, . . .
Functions:
Jacek Malec, Computer Science, Lund University
18(47)
Knowledge Representation WASP 28.10.2016
Logic: first order
Predicates (relations, properties):
AgeOf , Bald, CapitalOf , YoungerThan, <, =, P, Q, . . .
Constants:
Jacek , 59, Stockholm, Sweden, Fredrik , table59, c, d, . . .
Functions: fatherOf , ageOf , lengthOf , locationOf , . . .
Terms: constants, variables, functions thereof
Atomic sentences: relation over appropriate amount of terms
AgeOf (Jacek , 59), Bald(Jacek ), 8 < x,
YoungerThan(Jacek , fatherOf (Jacek )), P(x, y , z)
locationOf (TJR048) = PDammgården, . . .
Jacek Malec, Computer Science, Lund University
18(47)
Knowledge Representation WASP 28.10.2016
Logic: first order
Predicates (relations, properties):
AgeOf , Bald, CapitalOf , YoungerThan, <, =, P, Q, . . .
Constants:
Jacek , 59, Stockholm, Sweden, Fredrik , table59, c, d, . . .
Functions: fatherOf , ageOf , lengthOf , locationOf , . . .
Terms: constants, variables, functions thereof
Atomic sentences: relation over appropriate amount of terms
AgeOf (Jacek , 59), Bald(Jacek ), 8 < x,
YoungerThan(Jacek , fatherOf (Jacek )), P(x, y , z)
locationOf (TJR048) = PDammgården, . . .
Well-formed formulae: as before plus
8xA and 9xA are wffs if A is a wff
Jacek Malec, Computer Science, Lund University
18(47)
Knowledge Representation WASP 28.10.2016
Logics: semantics
1
Truth value
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Knowledge Representation WASP 28.10.2016
Logics: semantics
1
Truth value
JacekIs59,
Jacek Malec, Computer Science, Lund University
19(47)
Knowledge Representation WASP 28.10.2016
Logics: semantics
1
Truth value
JacekIs59, AgeOf (Jacek , 59),
Jacek Malec, Computer Science, Lund University
19(47)
Knowledge Representation WASP 28.10.2016
Logics: semantics
1
Truth value
JacekIs59, AgeOf (Jacek , 59), ageOf (Jacek ) = 59
Jacek Malec, Computer Science, Lund University
19(47)
Knowledge Representation WASP 28.10.2016
Logics: semantics
1
Truth value
JacekIs59, AgeOf (Jacek , 59), ageOf (Jacek ) = 59
2
Interpretations
3
Models
4
Satisfiability
5
Validity
6
Inconsistency
Jacek Malec, Computer Science, Lund University
19(47)
Knowledge Representation WASP 28.10.2016
Consequences (or: so what?)
1
Entailment (semantics):
T |= ↵
2
Verification procedure: model checking
Inference (syntax):
T `i ↵
Verification procedure: theorem proving
Jacek Malec, Computer Science, Lund University
20(47)
Knowledge Representation WASP 28.10.2016
Consequences (or: so what?)
1
Entailment (semantics):
T |= ↵
2
Verification procedure: model checking
Inference (syntax):
T `i ↵
Verification procedure: theorem proving
soundness, completeness
Jacek Malec, Computer Science, Lund University
20(47)
Knowledge Representation WASP 28.10.2016
Consequences (or: so what?)
1
Entailment (semantics):
T |= ↵
2
Verification procedure: model checking
Inference (syntax):
T `i ↵
Verification procedure: theorem proving
soundness, completeness
{↵, ↵ ! } |=
{↵, ↵ ! } `
↵, ↵ !
Jacek Malec, Computer Science, Lund University
20(47)
Knowledge Representation WASP 28.10.2016
Logics: modal
1
take a logical language, let ↵ be a wff
2
⇤↵ is a wff
3
⌃↵ is a wff
4
normally ⇤↵ $ ¬⌃¬↵
Intended meaning?
Jacek Malec, Computer Science, Lund University
21(47)
Knowledge Representation WASP 28.10.2016
Logics: modal
1
take a logical language, let ↵ be a wff
2
⇤↵ is a wff
3
⌃↵ is a wff
4
normally ⇤↵ $ ¬⌃¬↵
Intended meaning?
1
⇤↵ means Necessarily ↵
Jacek Malec, Computer Science, Lund University
21(47)
Knowledge Representation WASP 28.10.2016
Logics: modal
1
take a logical language, let ↵ be a wff
2
⇤↵ is a wff
3
⌃↵ is a wff
4
normally ⇤↵ $ ¬⌃¬↵
Intended meaning?
1
⇤↵ means Necessarily ↵
2
⇤↵ means Agent knows ↵
Jacek Malec, Computer Science, Lund University
21(47)
Knowledge Representation WASP 28.10.2016
Logics: modal
1
take a logical language, let ↵ be a wff
2
⇤↵ is a wff
3
⌃↵ is a wff
4
normally ⇤↵ $ ¬⌃¬↵
Intended meaning?
1
⇤↵ means Necessarily ↵
2
⇤↵ means Agent knows ↵
3
⇤↵ means Agent believes ↵
Jacek Malec, Computer Science, Lund University
21(47)
Knowledge Representation WASP 28.10.2016
Logics: modal
1
take a logical language, let ↵ be a wff
2
⇤↵ is a wff
3
⌃↵ is a wff
4
normally ⇤↵ $ ¬⌃¬↵
Intended meaning?
1
⇤↵ means Necessarily ↵
2
⇤↵ means Agent knows ↵
3
⇤↵ means Agent believes ↵
4
⇤↵ means Always in the future ↵
Jacek Malec, Computer Science, Lund University
21(47)
Knowledge Representation WASP 28.10.2016
Logics: modal
1
take a logical language, let ↵ be a wff
2
⇤↵ is a wff
3
⌃↵ is a wff
4
normally ⇤↵ $ ¬⌃¬↵
Intended meaning?
1
⇤↵ means Necessarily ↵
2
⇤↵ means Agent knows ↵
3
⇤↵ means Agent believes ↵
4
⇤↵ means Always in the future ↵
5
G↵ means Always in the future (or: Globally) ↵
Jacek Malec, Computer Science, Lund University
21(47)
Knowledge Representation WASP 28.10.2016
Logics: Kripke semantics
Actually, meaning of modal formulae is defined on graph structures
Nodes: possible worlds
Edges: reachability relation
~p,q,~r
p,q,~r
~p,q,r
p,q,r
p,q,~r
~p,~q,r
p,q,r
p,~q,r
Jacek Malec, Computer Science, Lund University
22(47)
Knowledge Representation WASP 28.10.2016
Logics: temporal
1
2
Globally (always):
Finally (eventually):
3
Next:
4
Until:
⇤
⌃
U
Jacek Malec, Computer Science, Lund University
23(47)
Knowledge Representation WASP 28.10.2016
Logics: temporal
1
2
Globally (always):
⇤
Finally (eventually):
3
Next:
4
Until:
⌃
U
Cf. Richard Murray’s lecture:
(
e
init
^⇤
e
safe
^ ⇤⌃
Jacek Malec, Computer Science, Lund University
e
prog )
!(
s
init
^⇤
s
safe
^ ⇤⌃
s
prog )
23(47)
Knowledge Representation WASP 28.10.2016
Logics: description
Earlier known as semantic networks. Formal version of semantic
web languages (OIL, DAML, OWL).
Effective reasoning:
inheritance via SubsetOf (SubClass) and MemberOf (isA) links
intersection paths
special meaning of some links (e.g. cardinality constraints)
classification, consistency, subsumption
Jacek Malec, Computer Science, Lund University
24(47)
Knowledge Representation WASP 28.10.2016
Representation: ontologies
Lots of robot-related ontologies:
knowrob, IEEE CORA (Standard 1872-2015), intelligent systems
ontology (2005, NIST), ...
Jacek Malec, Computer Science, Lund University
25(47)
Knowledge Representation WASP 28.10.2016
Modes of reasoning: Deduction
RedLightAt(intersection1)
thus
8(x)RedLightAt(x) !
StopBefore(x)
StopBefore(intersection1)
General Pattern:
1
prior facts
2
domain knowledge
3
observations
Jacek Malec, Computer Science, Lund University
26(47)
Knowledge Representation WASP 28.10.2016
Modes of reasoning: Deduction
RedLightAt(intersection1)
thus
8(x)RedLightAt(x) !
StopBefore(x)
StopBefore(intersection1)
General Pattern:
1
prior facts
2
domain knowledge
3
observations
4
conclusions
Sound.
Jacek Malec, Computer Science, Lund University
26(47)
Knowledge Representation WASP 28.10.2016
Modes of reasoning: Deduction
RedLightAt(intersection1)
thus
8(x)RedLightAt(x) !
StopBefore(x)
StopBefore(intersection1)
General Pattern:
1
prior facts
2
domain knowledge
3
observations
4
conclusions
Sound. But note:
Birds fly. Tweety is a penguin. Penguins are birds.
Jacek Malec, Computer Science, Lund University
26(47)
Knowledge Representation WASP 28.10.2016
Modes of reasoning: Induction
OnDesk (monitor 1) ^ Monitor (monitor 1),
OnDesk (monitor 2) ^ Monitor (monitor 2),
OnDesk (monitor 3) ^ Monitor (monitor 3),
OnDesk (monitor 4) ^ Monitor (monitor 4),
OnDesk (monitor 5) ^ Monitor (monitor 5)
thus
8(x)Monitor (x) ! OnDesk (x)
General pattern:
1
Observe
2
Generalize
Fallible. Constructs hypotheses, not true facts. However, most of
our practical reasoning, in particular learning, is of this kind.
Jacek Malec, Computer Science, Lund University
27(47)
Knowledge Representation WASP 28.10.2016
Modes of reasoning: Abduction
General pattern:
1
prior facts
2
domain knowledge
3
observations
Jacek Malec, Computer Science, Lund University
28(47)
Knowledge Representation WASP 28.10.2016
Modes of reasoning: Abduction
General pattern:
1
prior facts
2
domain knowledge
3
observations
4
explain the observation
Given a theory T and observations O
E is an explanation of O given T if
E [ T |= O and E [ T is consistent.
Usually we are interested in most plausible E, sometimes minimal
E, most elegant E, ...
Probablilistic abduction: lecture yesterday afternoon.
Jacek Malec, Computer Science, Lund University
28(47)
Knowledge Representation WASP 28.10.2016
What do we want to represent?
objects
places
times
events
processes
behaviours
vehicle dynamics
rigid body interactions
traffic laws
...
Jacek Malec, Computer Science, Lund University
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Knowledge Representation WASP 28.10.2016
Qualitative spatial reasoning
Jacek Malec, Computer Science, Lund University
30(47)
Knowledge Representation WASP 28.10.2016
Qualitative spatial reasoning
Jacek Malec, Computer Science, Lund University
31(47)
Knowledge Representation WASP 28.10.2016
Qualitative spatial reasoning
RCC8: region connection calculus
Given e.g.,
contains(A, B) ^ covers(B, C) we can conclude contains(A, C)
⇤(meet(A, B) !
(meet(A, B) _ disjoint(A, B) _ overlap(A, B)))
Jacek Malec, Computer Science, Lund University
32(47)
Knowledge Representation WASP 28.10.2016
Juggling example (Apt)
Jacek Malec, Computer Science, Lund University
33(47)
Object Models
Geometry, size, material, …
Bounding box
Width/height/depth
Origin
X/Y/Z-axis
Mass
Material properties
Polygon triangulation
Grasp positions
Deep geometry representation
…
5
OntoBREP - An Ontology for CAD Data © fortiss GmbH
Ljubljana, 2065-03-23
Boundary Representation (BREP) of objects
Basic BREP Structure
Topological Entities
7
OntoBREP - An Ontology for CAD Data © fortiss GmbH
Geometric Entities
Ljubljana, 2065-03-23
OntoBREP
Semantic Description Language
• Using the Web Ontology Language (OWL)
• Taxonomy of topological and geometric entities
• Properties, i.e. topological relations and geometric parameters
Example: cylinder
8
OntoBREP - An Ontology for CAD Data © fortiss GmbH
Ljubljana, 2065-03-23
Geometric Interrelation Constraints
Semantic description transparent to end-user
9
OntoBREP - An Ontology for CAD Data © fortiss GmbH
Ljubljana, 2065-03-23
Example Applications
OntoBREP use-cases
Perzylo 2015
RSS WS
• Parameterization of semantic task descriptions
– Geometric constraints as assembly parameters
– Constraints are solved based on perception of
involved objects
– Generates target poses
Somani 2015
ROBIO
• Object recognition and pose estimation
– Improves primitive shape based recognition
– Underspecified object poses can be described,
e.g. for symmetrical objects
Somani 2015
IROS
• Constrained-based robot control
– Task constraints translate to constraints on the
executing robot‘s pose
– Robot controller may exploit nullspace information
10
OntoBREP - An Ontology for CAD Data © fortiss GmbH
Ljubljana, 2065-03-23
Knowledge Representation WASP 28.10.2016
Interval calculus (Allen 1983)
Jacek Malec, Computer Science, Lund University
34(47)
Knowledge Representation WASP 28.10.2016
Invalidating conclusions
Tweety is a bird.
So it flies.
Jacek Malec, Computer Science, Lund University
35(47)
Knowledge Representation WASP 28.10.2016
Invalidating conclusions
Tweety is a bird.
So it flies.
But Tweety is a penguin.
So it doesn’t fly.
Jacek Malec, Computer Science, Lund University
35(47)
Knowledge Representation WASP 28.10.2016
Invalidating conclusions
Tweety is a bird.
So it flies.
But Tweety is a penguin.
So it doesn’t fly.
Non-monotonic reasoning.
Truth-maintenance systems.
Default reasoning. Circumscription. Closed World Assumption.
Negation as failure. . . .
Jacek Malec, Computer Science, Lund University
35(47)
Knowledge Representation WASP 28.10.2016
Uncertainty
Every perception is associated with uncertainty. Account for that.
(Yesterday lectures. Perception module.)
Approaches:
probabilistic representations
fuzzy approaches
multi-valued logics
Transformations between representations as needed.
Jacek Malec, Computer Science, Lund University
36(47)
Knowledge Representation WASP 28.10.2016
Back to KnowRob
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37(47)
Knowledge Representation WASP 28.10.2016
KnowRob lessons
Beetz and Tenorth, AIJ, 2016:
1
No fixed levels of abstraction, no layers, no “black boxes”;
2
A knowledge base should reuse data structures of the robot’s
control program;
3
Symbolic knowledge bases are useful, but not sufficient;
4
Robots need multiple inference methods;
5
Evaluating a robot knowledge base is difficult.
Jacek Malec, Computer Science, Lund University
38(47)
Knowledge Representation WASP 28.10.2016
Architectures of knowledge-based systems
AIMA agents (cf. Fredrik’s introductory lecture)
1
Logical agents - declarative, compositional
2
Rule-based systems - compositionality on the rule level
3
Layered systems (distribution of concerns)
4
Blackboards - compositionality of reasoners (knowledge
sources) (KnowRob, our SIARAS system)
5
Stream-oriented reasoning - Heintz@LiU
Jacek Malec, Computer Science, Lund University
39(47)
Knowledge Representation WASP 28.10.2016
KnowRob as a blackboard
Jacek Malec, Computer Science, Lund University
40(47)
Knowledge Representation WASP 28.10.2016
Self-awareness: Autoepistemic logic
1
Distribution axiom K:
(K ↵ ^ K (↵ ! )) ! K
2
3
Knowledge axiom T:
Positive introspection 4:
K↵ ! ↵
K ↵ ! KK ↵
4
Negative introspection 5:
¬K ↵ ! K ¬K ↵
Jacek Malec, Computer Science, Lund University
41(47)
Knowledge Representation WASP 28.10.2016
Self-awareness: motivation
1
true autonomy requires self-awareness
2
autoepistemic logic captures just one aspect: awareness of
own knowledge
3
resource limitations: anytime algorithms, active logic
4
interaction: distributed knowledge
5
interaction: shared knowledge
6
explanation of own behaviour (trust)
Jacek Malec, Computer Science, Lund University
42(47)
Knowledge Representation WASP 28.10.2016
Assignment
Represent an autonomous car (an autonomous service robot, an
autonomous aerial vehicle) at any given time point taking into
account a number of factors:
Position (what is the reference frame? Multiple ones? Relative
or absolute? Map(s)? GPS? Local? Visual?) and pose;
Dynamic physical state (what are the necessary variables and
why?);
State of the environment;
Knowledge about the future state of myself and the world (or
prescription how to get it);
Mental state. (What can I do? What should I do? What will I
do? Why? What I cannot do and why?)
The purpose is to prepare for building an autonomous car capable
to navigate in city traffic.
Jacek Malec, Computer Science, Lund University
43(47)
Knowledge Representation WASP 28.10.2016
Lab exercise
Implement a simple (e.g. rule-based) reasoner that will control a
simulated autonomous vehicle through an intersection with traffic
lights, meeting traffic, multiple lanes, and pedestrians crossing the
street.
Jacek Malec, Computer Science, Lund University
44(47)
Knowledge Representation WASP 28.10.2016
Thanks
Thanks go to
Michael Beetz @ UBremen (KnowRob slides)
Alexander Perzylo @ fortiss (BREP slides)
Jacek Malec, Computer Science, Lund University
45(47)
Knowledge Representation WASP 28.10.2016
References 1
https://www.youtube.com/watch?v=ymUFadN_MO4 (How Watson
learns)
DOI: 10.1147/JRD.2012.2186519, Automatic knowledge extraction
from documents, J. Fan, A. Kalyanpur, D. C. Gondek, D. A.
Ferrucci, IBM J. RES. DEV. VOL. 56 NO. 3/4 PAPER 5, 2012
YAGO2: A Spatially and Temporally Enhanced Knowledge Base
from Wikipedia, Johannes Hoffart, Fabian M. Suchanek, Klaus
Berberich, Gerhard Weikum, Artificial Intelligence Journal, vol.
194, pp. 28-61, 2013
Representations for robot knowledge in the KnowRob framework,
Moritz Tenorth, Michael Beetz, Artificial Intelligence Journal, in
press, available on the journal site
Logics for Artificial Intelligence, Raymond Turner, Ellis Horwood,
1984
Jacek Malec, Computer Science, Lund University
46(47)
Knowledge Representation WASP 28.10.2016
References 2
Logic In Action, Johan van Benthem, http://www.logicinaction.org,
2012
Rete: A Fast Algorithm for the Many Pattern/ Many Object Pattern
Match Problem, Charles L. Forgy, Artificial Intelligence Journal,
vol.19 (1982), pp. 17-37.
https://arxiv.org/pdf/1201.4089.pdf, A Description Logic Primer,
Markus Kroetzsch, Frantisek Simancik, Ian Horrocks
Qualitative Spatial Representation and Reasoning, Anthony G
Cohn and Jochen Renz, Handbook of Knowledge Representation,
pp. 551-596, Elsevier, 2008
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