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