Qualitative Spatial- Temporal Reasoning Jason J. Li Advanced Topics in A.I. The Australian National University Jason J. Li Advanced Topics in A.I. The Australian National University

Spatial-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 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 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 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. 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 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 Calculus Algebra of relations Domain Weak-Representation Algebra of relations Domain Weak-Representation

Algebra 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 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 Algebra Points along a line Composition of relations –{<} ; {=} = {<} –{<,=} ; {<} = {<} –{ } ; { } –{,=} = {=} Points along a line Composition of relations –{<} ; {=} = {<} –{<,=} ; {<} = {<} –{ } ; { } –{,=} = {=}

Example – RCC8

Domain 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-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 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 Relations Always complete graphs (JEPD) Set of vertices (V N ) and label of edges (L N ) Vertice V N (i) denotes the i th spatial-temporal variable Label L N (i,j) denote the possible relations between the two variables V N (i), V N (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 (V N ) and label of edges (L N ) Vertice V N (i) denotes the i th spatial-temporal variable Label L N (i,j) denote the possible relations between the two variables V N (i), V N (j) A network M is a subnetwork of another network N iff all nodes and labels of M are in N

Example 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 Networks Greece EURussia Czech TPP NTPP EC DC U U

Path-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-Consistency Greece EURussia Czech TPP NTPP EC DC U U

Example of Path-Consistency Greece EURussia Czech TPP NTPP EC DC U EC ; NTPPi = DC Conv(NTPP) = NTPPi

Example of Path-Consistency Greece EURussia Czech TPP NTPP EC DC U DC ; DC = U Conv(DC) = DC

Example of Path-Consistency Greece EURussia Czech TPP NTPP EC DC DC,EC,PO,TPPi,NTPPi TPP ; NTPPi = {DC,EC,PO,TPPi, NTPPi} Conv(NTPP) = NTPPi

Example of Path-Consistency From the information given, we were able to eliminate some possibilities of the relation between Czech and Greece

Consistency 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.

Consistency 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 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 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 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 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 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 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 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 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 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 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 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)

Summary 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 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, , 2008.A. G. Cohn 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, J. J. LiT. KowalskiS. LiECAI'08 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, 53-64G. LigozatPRICAI'04 J. Renz, Qualitative Spatial Reasoning with Topological Information, LNCS 2293, Springer-Verlag, Berlin, LNCS 2293 The above can all be accessed at 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, , 2008.A. G. Cohn 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, J. J. LiT. KowalskiS. LiECAI'08 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, 53-64G. LigozatPRICAI'04 J. Renz, Qualitative Spatial Reasoning with Topological Information, LNCS 2293, Springer-Verlag, Berlin, LNCS 2293 The above can all be accessed at