Categorical Methods in Qualitative Reasoning: The Case for Weak Representations

This paper argues for considering qualitative spatial and temporal reasoning in algebraic and category-theoretic terms. A central notion in this context is that of weak representation (WR) of the algebra governing the calculus. WRs are ubiquitous in qualitative reasoning, appearing both as domains of interpretation and as constraints. Defining the category of WRs allows us to express the basic notion of satisfiability (or consistency) in a simple way, and brings clarity to the study of various variants of consistency. The WRs of many popular calculi are of interest in themselves. Moreover, the classification of WRs leads to non-trivial model-theoretic results. The paper provides a not-too-technical introduction to these topics and illustrates it with simple examples.

[1]  Alfred Tarski,et al.  Relational selves as self-affirmational resources , 2008 .

[2]  Jean-François Condotta,et al.  Ultimately periodic qualitative constraint networks for spatial and temporal reasoning , 2005, 17th IEEE International Conference on Tools with Artificial Intelligence (ICTAI'05).

[3]  Gérard Ligozat,et al.  Weak Representations of Interval Algebras , 1990, AAAI.

[4]  Bernhard Nebel,et al.  Qualitative Spatio-Temporal Reasoning with RCC-8 and Allen's Interval Calculus: Computational Complexity , 2002, ECAI.

[5]  Gérard Ligozat,et al.  Reasoning about Cardinal Directions , 1998, J. Vis. Lang. Comput..

[6]  Anthony G. Cohn,et al.  A Spatial Logic based on Regions and Connection , 1992, KR.

[7]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[8]  Jean-François Condotta,et al.  On the consistency problem for the INDU calculus , 2003, 10th International Symposium on Temporal Representation and Reasoning, 2003 and Fourth International Conference on Temporal Logic. Proceedings..

[9]  Joxan Jaffar,et al.  Principles and Practice of Constraint Programming – CP’99 , 1999, Lecture Notes in Computer Science.

[10]  Christian Freksa,et al.  Spatial Information Theory. Cognitive and Computational Foundations of Geographic Information Science , 1999, Lecture Notes in Computer Science.

[11]  Max J. Egenhofer,et al.  Relation algebras over containers and surfaces: An ontological study of a room space , 1999, Spatial Cogn. Comput..

[12]  Jean-François Condotta,et al.  Spatial and temporal reasoning: beyond Allen's calculus , 2004, AI Commun..

[13]  Michael F. Goodchild,et al.  Foundations of Geographic Information Science , 2003 .

[14]  I. Hodkinson,et al.  Relation Algebras by Games , 2002 .

[15]  Gérard Ligozat.  Simple Models for Simple Calculi , 1999, COSIT.

[16]  Gérard Ligozat,et al.  Problems with Local Consistency for Qualitative Calculi , 2004, ECAI.

[17]  Gérard Ligozat,et al.  On Generalized Interval Calculi , 1991, AAAI.

[18]  Chengqi Zhang,et al.  PRICAI 2004: Trends in Artificial Intelligence , 2004, Lecture Notes in Computer Science.

[19]  Matteo Cristani,et al.  The Complexity of Reasoning about Spatial Congruence , 1999, J. Artif. Intell. Res..

[20]  Gérard Ligozat.  When Tables Tell It All: Qualitative Spatial and Temporal Reasoning Based on Linear Orderings , 2001, COSIT.

[21]  Gérard Ligozat,et al.  What Is a Qualitative Calculus? A General Framework , 2004, PRICAI.

[22]  Peter Jonsson,et al.  Towards a Complete Classification of Tractability in Point Algebras for Nonlinear Time , 1999, CP.

[23]  R. Maddux.  Some varieties containing relation algebras , 1982 .

[24]  Peter B. Ladkin,et al.  On binary constraint problems , 1994, JACM.

[25]  Abdul Sattar,et al.  INDU: An Interval and Duration Network , 1999, Australian Joint Conference on Artificial Intelligence.

[26]  Matteo Cristani,et al.  The complexity of constraint satisfaction problems for small relation algebras , 2004, Artif. Intell..

[27]  Jochen Renz,et al.  A Spatial Odyssey of the Interval Algebra: 1. Directed Intervals , 2001, IJCAI.