Abstract
In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research.
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References
P.J. Hayes, “The naïve physics manifesto,” in Expert Systems in the Micro-Electronic Age, edited by D. Mitchie, Edinburgh University Press, 1979.
J.R. Hobbs and R.C. Moore (eds.), Formal Theories of the Commonsense World, Ablex, 1985.
E. Davis, Representations of Commonsense Knowledge, Morgan Kaufmann: San Mateo, 1990.
A. Requicha, “Representations for rigid solids: Theory, methods and systems,” ACM Comput. Surv., vol. 12, pp. 437–464, 1980.
J.F. Allen, “An interval-based representation of temporal knowledge,” in Proceedings 7th IJCAI, 1981, pp. 221–226.
R.H. Güting, M.H. Böhlen, M. Erwig, C.C. Jenssen, N.A. Lorentzos, M. Schneider, and M. Vazirgiannis, “A foundation for representing and querying moving objects,” ACM Transactions on Database Systems, vol. 25, 2000.
J. McCarthy and P. Hayes, “Some philosophical problems from the standpoint of artificial intelligence,” in Machine Intelligenge, edited by B. Meltzer and D. Mitchie, vol. 4, Edinburgh University Press, pp. 463–502, 1969.
B. Clarke, “A calculus of individuals based on ‘connection’,” Notre Dame Journal of Formal Logic, vol. 23, no. 3, pp. 204–218, 1981.
J.F. Allen, “Towards a general theory of action and time,” Arti-ficial Intelligence, vol. 23, no. 2, pp. 123–154, 1984.
P.J. Hayes, “The second naive physics manifesto,” in Formal Theories of the Commonsense World, edited by J.R. Hobbs and B. Moore, Ablex, pp. 1–36, 1985.
R. Kowalski and M. Sergot, “A logic-based calculus of events,” New Generation Computing, vol. 4, pp. 67–95, 1986.
D.A. Randell, A.G. Cohn, and Z. Cui, “Naive topology: Modelling the force pump,” in Advances in Qualitative Physics, edited by P. Struss and B. Faltings, MIT Press, pp. 177–192, 1992.
A. Sistla and E. Clarke, “The complexity of propositional linear temporal logics,” Journal of the Association for Computing Machinery, vol. 32, pp. 733–749, 1985.
J. Kamp, “Tense logic and the theory of linear order,” Ph.D. Thesis, Michigan State University, 1968.
Z. Manna and A. Pnueli, The Temporal Logic of Reactive and Concurrent Systems, Springer-Verlag: Berlin, 1992.
C. Stirling, “Modal and temporal logics,” in Handbook of Logic in Computer Science, vol. 2, edited by S. Abramsky, D. Gabbay, and T. Maibaum, Clarendon Press, pp. 478–551, 1992.
M. Stone, “Topological representations of distributive lattices and Brouwerian logics,” Casopis pro péstovani matematiky a fysiky, vol. 67, pp. 1–25, 1937/1938.
A. Tarski, “Der Aussagenkalkül und die Topologie [Sentential Calculus and Topology],” Fundamenta Mathematicae, vol. 31, pp. 103–134, 1938. English translation in A. Tarski, Logic, Semantics, Metamathematics, Oxford Clarendon Press, 1956.
T.-C. Tang, “Algebraic postulates and a geometric interpretation for the Lewis calculus of strict implication,” Bulletin of the American Mathematical Society, vol. 44, pp. 737–744, 1938.
J. McKinsey and A. Tarski, “Some theorems about the sentential calculi of Lewis and Heyting,” Journal of Symbolic Logic, vol. 13, pp. 1–15, 1948.
A. Chagrov and M. Zakharyaschev, Modal Logic, Clarendon Press: Oxford, 1997.
B. Bennett, “Modal logics for qualitative spatial reasoning,” Bulletin of the Interest Group in Pure and Applied Logic (IGPL), vol. 4, no. 1, pp. 23–45, 1996.WWWaddress ftp://ftp.mpisb. mpg.de/pub/igpl/Journal/V4-1/index.html.
B. Bennett, “Determining consistency of topological relations,” Constraints, vol. 3, no. 2/3, pp. 213–225, 1998.
S. Demri, “Efficient strategies for automated reasoning in modal logics,” in Logics in Artificial Intelligence, edited by C. MacNish, D. Pearce, and L. Pereira, vol. 833 of Lecture Notes in Artificial Intelligence, Proceedings of the Workshop JELIA'94, 1994.
F. Giunchiglia and R. Sebastiani, “Building decision procedures for modal logics from propositional decision procedures-the case study of modal K,” in Proceedings of the 13th International Conference on Automated Deduction, CADE-13, edited by M. McRobbie and J. Slaney, vol. 1104 of LNCS. Berlin, pp. 583–597, 1996.
I.R. Horrocks, “Using an expressive description logic: FaCT or fiction?,” in Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98), edited by A.G. Cohn, L. Schubert, and S. Shapiro, pp. 636–647, 1998.
C. Dixon, M. Fisher, and M. Wooldridge, “Resolution for temporal logics of knowledge,” Journal of Logic and Computation, vol. 8, no. 3, 1998.
A. Voronkov, “KK: A theorem prover for K,” in Proceedings of CADE, 1999, pp. 383–387.
H. De Nivelle, R.A. Schmidt, and U. Hustadt, “Resolution-based methods for modal logics,” Logic Journal of the IGPL, to appear.
M. Kracht and F. Wolter, “Properties of independently axiomatizable bimodal logics,” Journal of Symbolic Logic, vol. 56, pp. 1469–1485, 1991.
E. Spaan, “Complexity of modal logics,” Ph.D. Thesis, Department of Mathematics and Computer Science, University of Amsterdam, 1993.
M. Marx and Y. Venema, Multi-Dimensional Modal Logic, Applied Logic, Kluwer, 1997.
D.M. Gabbay, Fibring Logics, Oxford University Press: Oxford, 1998.
F. Wolter, The Decision Problem for Combined Modal Logics, Habilitationsschrift, Leipzig University, 2000.
D. Gabbay, A.Kurucz, F.Wolter, and M. Zakharyaschev, “Many dimensional modal logics: Theory and applications,” Available from http://www.dcs.kcl.ac.uk/staff/mz, 2000.
K. Fine and G. Schurz, “Transfer theorems for multimodal logics,” in Logic and Reality: Essays on the Legacy of Arthur Prior, edited by J. Copeland, Oxford University Press: Oxford, pp. 169–213, 1996.
K. Segerberg, “Two-dimensional modal logic,” Journal of Philosophical Logic, vol. 2, pp. 77–96, 1973.
V. Shehtman, “Two-dimensional modal logics,” Mathematical Notices of the USSR Academy of Sciences, vol. 23, pp. 417–424, 1978.
D. Gabbay and V. Shehtman, “Products of modal logics, part 1,” Journal of the IGPL, vol. 6, pp. 73–146, 1998.
M. Marx, “Complexity of products of modal logics,” Journal of Logic and Computation, vol. 9, pp. 221–238, 1999.
R. Fagin, J. Halpern, Y. Moses, and M. Vardi, Reasoning About Knowledge, MIT Press: Cambridge, MA, 1995.
M. Reynolds, “A decidable temporal logic of parallelism,” Notre Dame Journal of Formal Logic, vol. 38, pp. 419–436, 1997.
F. Baader and H. Ohlbach, “A multi-dimensional terminological knowledge representation language,” Journal of Applied Non-Classical Logic, vol. 5, pp. 153–197, 1995.
F. Wolter and M. Zakharyaschev, “Satisfiability problem in description logics with modal operators,” in Proceedings of the sixth Conference on Principles of Knowledge Representation and Reasoning, Trento, Italy, pp. 512–523, 1998.
F. Wolter and M. Zakharyaschev, “Modal description logics: Modalizing roles,” Fundamenta Informaticae, vol. 39, pp. 411–438, 1999.
F. Wolter and M. Zakharyaschev, “Temporalizing description logics,” in Frontiers of Combining Systems 2, edited by M. de Rijke and D. Gabbay, Research Studies Press: Baldock, Hertfordshire, England, pp. 377–401, 2000.
Y. Venema, “Expressiveness and completeness of an interval tense logic,” Notre Dame Journal of Formal Logic, vol. 31, pp. 529–547, 1990.
H. Ono and A. Nakamura, “On the size of refutation Kripke models for some linear modal and tense logics,” Studia Logica, vol. 39, pp. 325–333, 1980.
M. Marx and M. Reynolds, “Undecidability of compass logic,” Journal of Logic and Computation, vol. 9, no. 6, pp. 897–941, 1999.
M. Reynolds and M. Zakharyaschev, “On the products of linear modal logics,” Journal of Logic and Computation, vol. 11, pp. 909–931, 2001.
J. Halpern and Y. Shoham, “A propositional modal logic of time intervals,” in Proceedings, Symposium on Logic in Computer Science, Boston, 1986.
R. Maddux, “The equational theory of CA 3 is undecidable,” Journal of Symbolic Logic, vol. 45, pp. 311–316, 1980.
R. Hirsch, I. Hodkinson, and A. Kurucz, “Every logic between K 3 and S53 is undecidable and non-finitely axiomatizable,” Manuscript; available at http://www.doc.ic. ac.uk/~kuag/k3undec.ps, 2000.
D. Randell, Z. Cui, and A.G. Cohn, “A spatial logic based on regions and connection,” in Proc. 3rd Int. Conf. on Knowledge Representation and Reasoning, San Mateo, 1992, pp. 165–176.
M. Egenhofer and R. Franzosa, “Point-set topological spatial relations,” International Journal of Geographical Information Systems, vol. 5, no. 2, pp. 161–174, 1991.
N.M. Gotts, “An axiomatic approach to topology for spatial information systems,” Technical report, Report 96.25, School of Computer Studies, University of Leeds, 1996.
B. Bennett, “Logical representations for automated reasoning about spatial relationships,” Ph.D. Thesis, School of Computer Studies, The University of Leeds, 1997. Abstract and postscript at http://www.scs.leeds.ac.uk/brandon/thesis.html.
A. Grzegorczyk, “Undecidability of some topological theories,” Fundamenta Mathematicae, vol. 38, pp. 137–152, 1951.
N.M. Gotts, “Using the RCC formalism to describe the topology of spherical regions,” Technical Report, Report 96.24, School of Computer Studies, University of Leeds, 1996.
C. Dornheim, “Undecidability of plane polygonal mereotopology,” in Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98), edited by A.G. Cohn, L. Schubert, and S. Shapiro, 1998, pp. 342–353.
M. Knauff, R. Rauh, and J. Renz, “A cognitive assessment of topological spatial relations: Results from an empirical investigation,,” in Proceedings of the 3rd International Conference on Spatial Information Theory (COSIT'97), 1997.
B. Bennett, “Spatial reasoning with propositional logics,” in Principles of Knowledge Representation and Reasoning: Proceedings of the 4th International Conference (KR94), edited by J. Doyle, E. Sandewall, and P. Torasso, San Francisco, CA, 1994.
V. Goranko and S. Passy, “Using the universal modality: Gains and questions,” Journal of Logic and Computation, vol. 2, pp. 5–30, 1992.
J. Renz, “A canonical model of the region connection calculus,” in Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98), edited by A.G. Cohn, L. Schubert, and S. Shapiro, pp. 330–341, 1998.
F. Wolter and M. Zakharyaschev, “Spatial reasoning in RCC-8 with Boolean region terms,” in Proceedings of ECAI 2000, Berlin, 2000, pp. 244–248.
J. Renz and B. Nebel, “On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus,” in Proceedings of IJCAI-97, 1997.
J. Renz and B. Nebel, “On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus,” Artificial Intelligence, vol. 108, no. 1/2, pp. 69–123, 1999.
P. Muller, “A qualitative theory of motion based on spatiotemporal primitives,” in Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98), edited by A.G. Cohn, L. Schubert, and S. Shapiro, pp. 131–141, 1998.
F. Wolter and M. Zakharyaschev, “Spatio-temporal representation and reasoning based on RCC-8,” in Proceedings of the Seventh Conference on Principles of Knowledge Representation and Reasoning, Breckenridge, USA, 2000, pp. 3–14.
D. Gabbay, I. Hodkinson, and M. Reynolds, Temporal Logic: Mathematical Foundations and Computational Aspects, vol. I, Oxford University Press: Oxford, 1994.
I. Hodkinson, F. Woter, and M. Zakaryaschev, “Decidable fragments of first-order temporal logics,” Annals of Pure and Applied Logic, vol. 106, pp. 85–134, 2000.
P. Blackburn, “Fine grained theories of time,” in Semantics of Time, Space and Movement: Working papers of the 4th International Workshop TSM-92, Château de Bonas, edited by M. Aurnague, A. Borillo, M. Borillo, and M. Bras, pp. 327–348, 1992.
M. Vilain, H. Kautz, and P. van Beek, “Constraint propagation algorithms for temporal reasoning: A revised report,” in Proceedings AAAI-86, 1986. Philadelphia. Revised version in Weld and De Kleer (1990).
I. Pratt, “First-order qualitative spatial representation languages with convexity,” Spatial Cognition and Computation, vol. 1, pp. 181–204, 1999.
E. Davis, N. Gotts, and A.G. Cohn, “Constraint networks of topological relations and convexity,” Constraints, vol. 4, no. 3, pp. 241–280, 1999.
P. Balbiani, L. Fariñas del Cerro, T. Tinchev, and D. Vakarelov, “Modal logics for incidence geometries,” Journal of Logic and Computations, vol. 7, no. 1, pp. 59–78, 1997.
J. Kratochvíl, “String graphs II: Recognizing string graphs is NP-Hard,” Journal of Combinatorial Theory, Series B, vol. 52, pp. 67–78, 1991.
J. Kratochvíl and J. Matoušek, “String graphs requiring exponential representations,” Journal of Combinatorial Theory, Series B, vol. 53, pp. 1–4, 1991.
H. Barringer, R. Kuiper, and A. Pnueli, “A really abstract concurrent model and its fully abstract semantics,” in Proceedings of the 13th ACMSymposium on Principles of Programming Languages (POPL), 1986, pp. 173–183.
A.P. Galton, “Space, time and movement,” in Spatial and Temporal Reasoning, edited by O. Stock, Kluwer: Dordrecht, Ch. 10, pp. 321–352, 1997.
A. Galton, “Transitions in continuous time, with an application to qualitative changes in spatial relations,” in Advances in Temporal Logic (Proceedings of the Second International Conference on Temporal Logic, ICTL'97), edited by H.B. et al., 2000, pp. 279–297.
E. Emerson and J. Halpern, “Sometimes’ and ‘not never’ revisited: On branching time versus linear time temporal logics,” Journal of the ACM, vol. 33, no. 1, pp. 151–178, 1986.
U. Hustadt, C. Dixon, R.A. Schmidt, and M. Fisher, “Normal forms and proofs in combined modal and temporal logics,” in Proceedings of the Third International Workshop on Frontiers of Combining Systems (FroCoS'2000), vol. 1794 of LNAI, 2000, pp. 73–87.
M. Marx, S. Mikulas, and S. Schlobach, “Tableau calculus for local cubic modal logics and its implementation,” Journal of the IGPL, vol. 7, no. 6, pp. 755–778, 1999.
A.G. Cohn, B. Bennett, J. Gooday, and N. Gotts, “RCC: A calculus for region based qualitative spatial reasoning,” GeoInformatica, vol. 1, pp. 275–316, 1997.
K. Kuratowski, Topologie, vol. I of Monografie Matematyczne, PWN Polish Scientific Publishers: Warsaw, 1933.
D.S. Weld and J. De Kleer (eds.), Readings in Qualitative Reasoning About Physical Systems. Morgan Kaufman: San Mateo, CA, 1990.
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Bennett, B., Cohn, A.G., Wolter, F. et al. Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning. Applied Intelligence 17, 239–251 (2002). https://doi.org/10.1023/A:1020083231504
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DOI: https://doi.org/10.1023/A:1020083231504