Qualitative Spatial-Temporal Reasoning

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Qualitative Spatial-Temporal Reasoning

• Jason J. Li
• Advanced Topics in A.I.
• The Australian National University

2
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.

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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)

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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

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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

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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

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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

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Trinity of a Qualitative Calculus

• Algebra of relations
• Domain
• Weak-Representation

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Algebra of Relations

• Formally, its 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.

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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 Bs relation to A
• Intersection/union
• Defined set-theoretically
• Complement
• A is not related to B by Rel_A, then what is the
  relation?

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Example Point Algebra

• Points along a line
• Composition of relations
• lt lt
• lt, lt lt
• lt,gt lt lt,,gt
• lt, gt,

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Example RCC8
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Domain

• The set of spatial-temporal objects we wish to
  reason
• Example
• 2D Generic Regions
• Points in time

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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

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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

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Network 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

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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

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Example of Networks
Czech
NTPP
U
EC
EU
Russia
U
DC
TPP
Greece
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Path-Consistency

• Any two variable assignment can be extended to
  three variables assignment
• Forall 1 lt i, j, k lt n
• Rij Rij n Rik Rkj

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Example of Path-Consistency
Czech
NTPP
U
EC
EU
Russia
U
DC
TPP
Greece
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Example of Path-Consistency
Conv(NTPP) NTPPi
Czech
NTPP
DC
EC NTPPi DC
EC
EU
Russia
DC
U
TPP
Greece
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Example of Path-Consistency
Conv(DC) DC
Czech
NTPP
DC
DC DC U
EC
EU
Russia
U
DC
TPP
Greece
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Example of Path-Consistency
TPP NTPPi DC,EC,PO,TPPi, NTPPi
Conv(NTPP) NTPPi
Czech
NTPP
DC
EC
EU
Russia
DC
TPP
Greece
DC,EC,PO,TPPi,NTPPi
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Example of Path-Consistency

• From the information given, we were able to
  eliminate some possibilities of the relation
  between Czech and Greece

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Consistency

• A network is consistent iff
• There is an instantiation in the domain such that
  all constraints are satisfied.

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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

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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!

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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?

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Important Problems in Qualitative
Spatial-Temporal Reasoning

• Computational Complexity
• What is the complexity for deciding consistency?
• P? NP? NP-Hard? P-SPACE? EXP-SPACE?

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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.

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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 (PAPA), INDU (IA Size), etc
• BIG Calculus containing all information?
• Meta-reasoning to switch calculi?

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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?

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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

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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

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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)

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Transforming Problem

• Boolean satisfiability problem has
• Variables
• Literals
• Constraints
• Transform each component to spatial networks

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Transforming Problem

• Show deciding consistency is same as deciding
  consistency for SAT problem, and vice versa
• Program written to do this automatically (Renz
  Li, KR2008)

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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.

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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, 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