Abstract
Establishing local consistency is one of the main algorithmic techniques in temporal and spatial reasoning. In this area, one of the central questions for the various proposed temporal and spatial constraint languages is whether local consistency implies global consistency. Showing that a constraint language Γ has this “local-to-global” property implies polynomial-time tractability of the constraint language, and has further pleasant algorithmic consequences.
In the present paper, we study the “local-to-global” property by making use of a recently established connection of this property with universal algebra. Specifically, the connection shows that this property is equivalent to the presence of a so-called quasi near-unanimity polymorphism of the constraint language. We obtain new algorithmic results and give very concise proofs of previously known theorems. Our results concern well-known and heavily studied formalisms such as the point algebra and its extensions, Allen’s interval algebra, and the spatial reasoning language RCC-5.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abian, A.: Categoricity of denumerable atomless boolean rings. Studia Logica 30(1), 63–67 (1972)
Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Reading (1995)
Allen, J.F.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832–843 (1983)
Bodirsky, M., Chen, H.: Oligomorphic clones. Algebra Universalis (to appear, 2007)
Bodirsky, M., Chen, H.: Quantified equality constraints. In: Proceedings of LICS 2007 (to appear, 2007)
Bodirsky, M., Dalmau, V.: Datalog and constraint satisfaction with infinite templates. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 646–659. Springer, Heidelberg (2006) (A journal version is available from the webpage of the first author, 2006)
Bennett, B.: Spatial reasoning with propositional logics. In: International Conference on Knowledge Representation and Reasoning, Morgan Kaufmann, San Francisco (1994)
Bulatov, A., Jeavons, P., Krokhin, A.: The complexity of constraint satisfaction: An algebraic approach (a survey paper). In: Structural Theory of Automata, Semigroups and Universal Algebra (Montreal, 2003). NATO Science Series II: Mathematics, Physics, Chemistry, pp. 181–213 (2005)
Bulatov, A., Krokhin, A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34, 720–742 (2005)
Bodirsky, M., Nešetřil, J.: Constraint satisfaction with countable homogeneous templates. Journal of Logic and Computation 16(3), 359–373 (2006)
Bodirsky, M.: The core of a countably categorical structure. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 100–110. Springer, Heidelberg (2005)
Baker, K.A., Pixley, A.F.: Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z. 143, 165–174 (1974)
Chen, H.: The computational complexity of quantified constraint satisfaction. Ph.D. thesis, Cornell University (August 2004)
Cohen, D., Jeavons, P.: The complexity of constraint languages. Appears in: Handbook of Constraint Programming (2006)
Duentsch, I.: Relation algebras and their application in temporal and spatial reasoning. Artificial Intelligence Review 23, 315–357 (2005)
Dechter, R., van Beek, P.: Local and global relational consistency. TCS 173(1), 283–308 (1997)
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1999)
Evans, D.: Examples of aleph-zero categorical structures. Automorphisms of first-order structures, 33–72 (1994)
Fisher, M., Gabbay, D., Vila, L.: Handbook of Temporal Reasoning in Artificial Intelligence. Elsevier, Amsterdam (2005)
Fraïssé, R.: Theory of Relations. North-Holland, Amsterdam (1986)
Freuder, E.C.: A sufficient condition for backtrack-free search. Journal of the ACM 29(1), 24–32 (1982)
Feder, T., Vardi, M.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing 28, 57–104 (1999)
Hirsch, R.: Relation algebras of intervals. Artificial Intelligence Journal 83, 1–29 (1996)
Hodges, W.: A shorter model theory. Cambridge University Press, Cambridge (1997)
Jeavons, P., Cohen, D., Cooper, M.: Constraints, consistency and closure. AI 101(1-2), 251–265 (1998)
Jonsson, P., Drakengren, T.: A complete classification of tractability in RCC-5. J. Artif. Intell. Res. 6, 211–221 (1997)
Koubarakis, M.: From local to global consistency in temporal constraint networks. Theor. Comput. Sci. 173(1), 89–112 (1997)
Koubarakis, M.: Tractable disjunctions of linear constraints: Basic results and applications to temporal reasoning. Theoretical Computer Science 266, 311–339 (2001)
Kautz, H., van Beek, P., Vilain, M.: Constraint propagation algorithms: A revised report. Qualitative Reasoning about Physical Systems, 373–381 (1990)
Ladkin, P.B., Maddux, R.D.: On binary constraint problems. Journal of the Association for Computing Machinery 41(3), 435–469 (1994)
Mackworth, A.K.: Consistency in networks of relations. AI 8, 99–118 (1977)
Randell, D.A., Cui, Z., Cohn, A.G.: A spatial logic based on regions and connection. In: Conference on Principles on Knowledge Representation and Reasoning (KR 1992), pp. 165–176 (1992)
Renz, J., Nebel, B.: On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artif. Intell. 108(1-2), 69–123 (1999)
Renz, J., Nebel, B.: Qualitative spatial reasoning using constraint calculi. Handbook of Spatial Logics (to appear, 2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodirsky, M., Chen, H. (2007). Qualitative Temporal and Spatial Reasoning Revisited. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-74915-8_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74914-1
Online ISBN: 978-3-540-74915-8
eBook Packages: Computer ScienceComputer Science (R0)