Post on 31-Jan-2016
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Qualitative Spatial-Temporal Qualitative Spatial-Temporal ReasoningReasoning
Jason J. LiAdvanced Topics in A.I.
The Australian National University
Spatial-Temporal ReasoningSpatial-Temporal Reasoning
• Space is ubiquitous in intelligent systems
– We wish to reason, make predictions, and plan for events in space
– Modelling space is similar to modelling time.
• Space is ubiquitous in intelligent systems
– We wish to reason, make predictions, and plan for events in space
– Modelling space is similar to modelling time.
Quantitative ApproachesQuantitative Approaches
• Spatial-temporal configurations can be described by specifying coordinates:
– At 10am object A is at position (1,0,1), at 11am it is at (1,2,2)
– From 9am to 11am, object B is at (1,2,2)– At 11am object C is at (13,10,12), and at 1pm it
is at (12,11,12)
• Spatial-temporal configurations can be described by specifying coordinates:
– At 10am object A is at position (1,0,1), at 11am it is at (1,2,2)
– From 9am to 11am, object B is at (1,2,2)– At 11am object C is at (13,10,12), and at 1pm it
is at (12,11,12)
A Qualitative PerspectiveA Qualitative Perspective
• Often, a qualitative description is more adequate
– Object A collided with object B, then object C appeared
– Object C was not near the collision between A and B when it took place
• Often, a qualitative description is more adequate
– Object A collided with object B, then object C appeared
– Object C was not near the collision between A and B when it took place
Qualitative RepresentationsQualitative Representations
• Uses a finite vocabulary– A finite set of relations
• Efficient when precise information is not available or not necessary
• Handles well with uncertainty– Uncertainty represented by disjunction of
relations
• Uses a finite vocabulary– A finite set of relations
• Efficient when precise information is not available or not necessary
• Handles well with uncertainty– Uncertainty represented by disjunction of
relations
Qualitative vs. FuzzyQualitative vs. Fuzzy
• Fuzzy representations take approximations of real values
• Qualitative representations make only as much distinctions as necessary
– This ensures the soundness of composition
• Fuzzy representations take approximations of real values
• Qualitative representations make only as much distinctions as necessary
– This ensures the soundness of composition
Qualitative Spatial-Temporal ReasoningQualitative Spatial-Temporal Reasoning
• Represent space and time in a qualitative manner
• Reasoning using a constraint calculus with infinite domains
– Space and time is continuous
• Represent space and time in a qualitative manner
• Reasoning using a constraint calculus with infinite domains
– Space and time is continuous
Trinity of a Qualitative CalculusTrinity of a Qualitative Calculus
• Algebra of relations
• Domain
• Weak-Representation
• Algebra of relations
• Domain
• Weak-Representation
Algebra of RelationsAlgebra of Relations
• Formally, it’s called Nonassociatve Algebra– Relation Algebra is a subset of such algebras
that its composition is associative– It prescribes the constraints between elements
in the domain by the relationship between them.
• Formally, it’s called Nonassociatve Algebra– Relation Algebra is a subset of such algebras
that its composition is associative– It prescribes the constraints between elements
in the domain by the relationship between them.
Algebra of RelationsAlgebra of Relations
• It usually has these operations:– Composition:
• If A is related to B, B is related to C, what is A to C
– Converse:• If A is related to B, what is B’s relation to A
– Intersection/union: • Defined set-theoretically
– Complement:• A is not related to B by Rel_A, then what is the relation?
• It usually has these operations:– Composition:
• If A is related to B, B is related to C, what is A to C
– Converse:• If A is related to B, what is B’s relation to A
– Intersection/union: • Defined set-theoretically
– Complement:• A is not related to B by Rel_A, then what is the relation?
Example – Point AlgebraExample – Point Algebra
• Points along a line• Composition of
relations– {<} ; {=} = {<}– {<,=} ; {<} = {<}– {<,>} ; {<} = {<,=,>} – {<,=} ; {>,=} = {=}
• Points along a line• Composition of
relations– {<} ; {=} = {<}– {<,=} ; {<} = {<}– {<,>} ; {<} = {<,=,>} – {<,=} ; {>,=} = {=}
Example – RCC8Example – RCC8
DomainDomain
• The set of spatial-temporal objects we wish to reason
• Example:– 2D Generic Regions– Points in time
• The set of spatial-temporal objects we wish to reason
• Example:– 2D Generic Regions– Points in time
Weak-RepresentationWeak-Representation
• How the algebra is mapped to the domain (JEPD)
– Jointly Exhaustive: everything is related to everything else
– Pairwise Disjoint: any two entities in the domain is related by an atomic relation
• How the algebra is mapped to the domain (JEPD)
– Jointly Exhaustive: everything is related to everything else
– Pairwise Disjoint: any two entities in the domain is related by an atomic relation
Mapping of Point AlgebraMapping of Point Algebra
• Domain: Real values– Between any two value there is a value– We say the weak representation is a
representation– Any consistent network can be consistently
extended• Domain: Discrete values (whole numbers)
– Weak representation not representation
• Domain: Real values– Between any two value there is a value– We say the weak representation is a
representation– Any consistent network can be consistently
extended• Domain: Discrete values (whole numbers)
– Weak representation not representation
Network of RelationsNetwork of Relations
• Always complete graphs (JEPD)• Set of vertices (VN) and label of edges (LN)• Vertice VN(i) denotes the ith spatial-temporal variable• Label LN(i,j) denote the possible relations between
the two variables VN(i), VN(j) • A network M is a subnetwork of another network N iff
all nodes and labels of M are in N
• Always complete graphs (JEPD)• Set of vertices (VN) and label of edges (LN)• Vertice VN(i) denotes the ith spatial-temporal variable• Label LN(i,j) denote the possible relations between
the two variables VN(i), VN(j) • A network M is a subnetwork of another network N iff
all nodes and labels of M are in N
Example of NetworksExample of Networks
• Greece is part of EU and on its boarder
• Czech Republic is part of EU and not on its boarder
• Russia is externally connected to EU and disconnected to Greece
• Greece is part of EU and on its boarder
• Czech Republic is part of EU and not on its boarder
• Russia is externally connected to EU and disconnected to Greece
Example of NetworksExample of Networks
Greece
EU Russia
Czech
TPP
NTPP
EC
DC
U
U
Path-ConsistencyPath-Consistency
• Any two variable assignment can be extended to three variables assignment
• Forall 1 <= i, j, k <= n– Rij = Rij ∩ Rik ; Rkj
• Any two variable assignment can be extended to three variables assignment
• Forall 1 <= i, j, k <= n– Rij = Rij ∩ Rik ; Rkj
Example of Path-ConsistencyExample of Path-Consistency
Greece
EU Russia
Czech
TPP
NTPP
EC
DC
U
U
Example of Path-ConsistencyExample of Path-Consistency
Greece
EU Russia
Czech
TPP
NTPP
EC
DC
DC
U
EC ; NTPPi = DC
Conv(NTPP) = NTPPi
Example of Path-ConsistencyExample of Path-Consistency
Greece
EU Russia
Czech
TPP
NTPP
EC
DC
DC
U
DC ; DC = U
Conv(DC) = DC
Example of Path-ConsistencyExample of Path-Consistency
Greece
EU Russia
Czech
TPP
NTPP
EC
DC
DC
DC,EC,PO,TPPi,NTPPi
TPP ; NTPPi = {DC,EC,PO,TPPi, NTPPi}
Conv(NTPP) = NTPPi
Example of Path-ConsistencyExample of Path-Consistency
• From the information given, we were able to eliminate some possibilities of the relation between Czech and Greece
• From the information given, we were able to eliminate some possibilities of the relation between Czech and Greece
ConsistencyConsistency
• A network is consistent iff– There is an instantiation in the domain
such that all constraints are satisfied.
• A network is consistent iff– There is an instantiation in the domain
such that all constraints are satisfied.
ConsistencyConsistency
• A nice property of a calculus, would be that path-consistency entails consistency for CSPs with only atomic constraints.
– If all the transitive constraints are satisfied, then it can be realized.
• RCC8, Point Algebra all have this property• But many do not…
• A nice property of a calculus, would be that path-consistency entails consistency for CSPs with only atomic constraints.
– If all the transitive constraints are satisfied, then it can be realized.
• RCC8, Point Algebra all have this property• But many do not…
Path-Consistency and ConsistencyPath-Consistency and Consistency
• Path-consistency is different to (general) consistency
– Consider 5 circular disks– All externally connected to
each other– This is PC, but not Consistent!
• Path-consistency is different to (general) consistency
– Consider 5 circular disks– All externally connected to
each other– This is PC, but not Consistent!
Important Problems in Qualitative Spatial-Temporal Reasoning
Important Problems in Qualitative Spatial-Temporal Reasoning
• A very nice property of a qualitative calculus is that if path-consistency entails consistency
– If the network is path-consistent, then you can get an instantiation in the domain
– Usually, it requires a manual proof – Any way to do it automatically?
• A very nice property of a qualitative calculus is that if path-consistency entails consistency
– If the network is path-consistent, then you can get an instantiation in the domain
– Usually, it requires a manual proof – Any way to do it automatically?
Important Problems in Qualitative Spatial-Temporal Reasoning
Important Problems in Qualitative Spatial-Temporal Reasoning
• Computational Complexity– What is the complexity for deciding
consistency?• P? NP? NP-Hard? P-SPACE? EXP-SPACE?
• Computational Complexity– What is the complexity for deciding
consistency?• P? NP? NP-Hard? P-SPACE? EXP-SPACE?
Important Problems in Qualitative Spatial-Temporal Reasoning
Important Problems in Qualitative Spatial-Temporal Reasoning
• Unified theory of spatial-temporal reasoning– Many spatial-temporal calculi have been
proposed• Point Algebra, Interval Algebra, RCC8, OPRA, STAR,
etc.
– How do we combine efficient reasoning calculi for more expressive queries.
• Unified theory of spatial-temporal reasoning– Many spatial-temporal calculi have been
proposed• Point Algebra, Interval Algebra, RCC8, OPRA, STAR,
etc.
– How do we combine efficient reasoning calculi for more expressive queries.
Important Problems in Qualitative Spatial-Temporal Reasoning
Important Problems in Qualitative Spatial-Temporal Reasoning
• Unified theory of spatial-temporal reasoning
– Some approaches combines two calculi to form a new calculi, with mixed results• IA (PA+PA), INDU (IA + Size), etc• BIG Calculus containing all information?• Meta-reasoning to switch calculi?
• Unified theory of spatial-temporal reasoning
– Some approaches combines two calculi to form a new calculi, with mixed results• IA (PA+PA), INDU (IA + Size), etc• BIG Calculus containing all information?• Meta-reasoning to switch calculi?
Important Problems in Qualitative Spatial-Temporal Reasoning
Important Problems in Qualitative Spatial-Temporal Reasoning
• Qualitative representations may have different levels of granularity
– How coarse/fine you want to define the relations• Do you care PP vs. TPP?
– What resolution do you want your representation?
– What level of information do you want to use?
• Qualitative representations may have different levels of granularity
– How coarse/fine you want to define the relations• Do you care PP vs. TPP?
– What resolution do you want your representation?
– What level of information do you want to use?
Important Problems in Qualitative Spatial-Temporal Reasoning
Important Problems in Qualitative Spatial-Temporal Reasoning
• Spatial Planning– Most automated planning problems ignore
spatial aspects of the problem– Most real-life applications uses an ad-hoc
representation for reasoning– How do we use make use of efficient reasoning
algorithms to better plan for spatial-change
• Spatial Planning– Most automated planning problems ignore
spatial aspects of the problem– Most real-life applications uses an ad-hoc
representation for reasoning– How do we use make use of efficient reasoning
algorithms to better plan for spatial-change
Solving ComplexitySolving Complexity
• If path-consistency decide consistency, the problem is polynomial
• If not, then some complexity proof is required
– Transform the problem to one of the known problems
• If path-consistency decide consistency, the problem is polynomial
• If not, then some complexity proof is required
– Transform the problem to one of the known problems
Solving ComplexitySolving Complexity
• Show NP-Hardness, you need to show 1-1 transformation for a subset of the problems to a known NP-Complete Problem
– Deciding consistency for some spatial-temporal networks
– Deciding the Boolean satisfiability problem (3-SAT)
• Show NP-Hardness, you need to show 1-1 transformation for a subset of the problems to a known NP-Complete Problem
– Deciding consistency for some spatial-temporal networks
– Deciding the Boolean satisfiability problem (3-SAT)
Transforming ProblemTransforming Problem
• Boolean satisfiability problem has
– Variables– Literals – Constraints
• Transform each component to spatial networks
• Boolean satisfiability problem has
– Variables– Literals – Constraints
• Transform each component to spatial networks
Transforming ProblemTransforming Problem
– Show deciding consistency is same as deciding consistency for SAT problem, and vice versa
– Program written to do this automatically (Renz & Li, KR’2008)
– Show deciding consistency is same as deciding consistency for SAT problem, and vice versa
– Program written to do this automatically (Renz & Li, KR’2008)
SummarySummary
• Qualitative Spatial-Temporal Reasoning uses constraint networks of infinite domains
• It reasons with relations between entities, and make only as few distinctions as necessary
• It is useful for imprecise / uncertain information• Many open questions / problems in the field.
• Qualitative Spatial-Temporal Reasoning uses constraint networks of infinite domains
• It reasons with relations between entities, and make only as few distinctions as necessary
• It is useful for imprecise / uncertain information• Many open questions / problems in the field.
Further ReadingFurther Reading
• A. G. Cohn and J. Renz, Qualitative Spatial Representation and Reasoning, in: F. van Hermelen, V. Lifschitz, B. Porter, eds., Handbook of Knowledge Representation, Elsevier, 551-596, 2008.
• J. J. Li, T. Kowalski, J. Renz, and S. Li, Combining Binary Constraint Networks in Qualitative Reasoning, Proceedings of the 18th European Conference on Artificial Intelligence (ECAI'08), Patras, Greece, July 2008, 515-519.
• G. Ligozat, J. Renz, What is a Qualitative Calculus? A General Framework, 8th Pacific Rim International Conference on Artificial Intelligence (PRICAI'04), Auckland, New Zealand, August 2004, 53-64
• J. Renz, Qualitative Spatial Reasoning with Topological Information, LNCS 2293, Springer-Verlag, Berlin, 2002.
• The above can all be accessed at http://www.jochenrenz.info
• A. G. Cohn and J. Renz, Qualitative Spatial Representation and Reasoning, in: F. van Hermelen, V. Lifschitz, B. Porter, eds., Handbook of Knowledge Representation, Elsevier, 551-596, 2008.
• J. J. Li, T. Kowalski, J. Renz, and S. Li, Combining Binary Constraint Networks in Qualitative Reasoning, Proceedings of the 18th European Conference on Artificial Intelligence (ECAI'08), Patras, Greece, July 2008, 515-519.
• G. Ligozat, J. Renz, What is a Qualitative Calculus? A General Framework, 8th Pacific Rim International Conference on Artificial Intelligence (PRICAI'04), Auckland, New Zealand, August 2004, 53-64
• J. Renz, Qualitative Spatial Reasoning with Topological Information, LNCS 2293, Springer-Verlag, Berlin, 2002.
• The above can all be accessed at http://www.jochenrenz.info