Skip to main content
SpringerLink
Log in

The logic of events

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

An event space is a set of instantaneous events that vary both in time and specificity. The concept of an event space provides a foundation for a logical – i.e., modular and open – approach to causal reasoning. In this article, we propose intuitively transparent axioms for event spaces. These axioms are constructive in the intuitionistic sense, and hence they can be used directly for causal reasoning in any computational logical framework that accommodates type theory. We also put the axioms in classical form and show that in this form they are adequate for the representation in terms of event trees established by Shafer [40] using stronger axioms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Baral, M. Gelfond and A. Provetti, Representing actions: Laws, observations and hypotheses, Journal of Logic Programming 31 (1997) 201-243.

    Article  MATH  MathSciNet  Google Scholar 

  2. B. Baras et al., The Coq Proof Assistant Manual. Version 6.2.4(January 28, 1999).

  3. N. Belnap, Branching space-time, Synthese 92 (1992) 385-434.

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Bolc and A. Szalas, Time and Logic: A Computational Approach(UCL Press, London, 1995).

    MATH  Google Scholar 

  5. B. Bru, Postface, in [9] (1986).

  6. R.L. Constable et al., Implementing Mathematics with the NuPRL Proof Development System(Prentice-Hall, Englewood Cliffs, NJ, 1986).

    Google Scholar 

  7. T. Coquand and G. Huet, The calculus of constructions, Information and Computation 76 (1988) 95-120.

    Article  MATH  MathSciNet  Google Scholar 

  8. B.A. Davey and H.A. Priestley, Introduction to Lattices and Order(Cambridge University Press, New York, 1990).

    MATH  Google Scholar 

  9. P.S. de Laplace, Essai philosophique sur les probabilit´es(Christian Bourgois, Paris, 1986) (Laplace's Essaiwas first published in 1814, and the fifth and definitive edition was published in 1825).

    Google Scholar 

  10. G. Dowek, A. Felty, H. Herbelin, G. Huet, C. Murthy, C. Parent, C. Paulin-Mohring and B. Werner, The Coq Proof Assistant User's Guide, Version 5.8(INRIA-Rocquencourt and CNRS-ENS, Lyon, 1993). http://pauillac.inria.fr/coq/assis-eng.html.

  11. M. Dummett, Elements of Intuitionism(Oxford University Press, Oxford, 1977).

    MATH  Google Scholar 

  12. E.A. Emerson, Temporal and modal logic, in: Handbook of Theoretical Computer Science, Vol. B, ed. J. Leeuwen (Elsevier, Amsterdam, 1990) pp. 995-1072.

    Google Scholar 

  13. E.A. Emerson and J.Y. Halpern, "Sometimes" and "not never" revisited: On branching versus linear time temporal logic, Journal of the Association for Computing Machinery 33(1) (1986) 151-171.

    MATH  MathSciNet  Google Scholar 

  14. M. Gelfond and V. Lifschitz, Representing actions and change by logic programs, Journal of Logic Programming 17 (1993) 301-323.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Goldblatt, Logics of Time and Computation, 2nd edition, revised and expanded (CSLI Publications, Stanford, 1992).

    Google Scholar 

  16. R.A. Kowalski and M.J. Sergot, A logic-based calculus of events, New Generation Computing 4 (1986) 67-95.

    Article  Google Scholar 

  17. H. Levesque, R. Reiter, Y. Lesperance, F. Lin and R.B. Scherl, GOLOG: A logic programming language for dynamic domains, Journal of Logic Programming (1997).

  18. F. Lin and R. Reiter, State constraints revisited, Journal of Logic and Computation 4 (1994) 566-678.

    MathSciNet  Google Scholar 

  19. F. Lin and R. Reiter, How to progress a database, Artificial Intelligence 92 (1997) 131-167.

    Article  MATH  MathSciNet  Google Scholar 

  20. Z. Luo, Computation and Reasoning: A Type Theory for Computer Science(Clarendon Press, Oxford, 1994).

    MATH  Google Scholar 

  21. Z. Luo and R. Pollack, Lego Proof Development System: User's Manual, LFCS Report ECS-LFCS-92-211, Department of Computer Science, University of Edinburgh (May 7, 1992). http://www.dcs.ed.ac.uk/home/lego.

  22. L. Magnusson and B. Nordstr¨om, The ALF proof editor and its proof engine, in: Types for Proofs and Programs, eds. H. Barendregt and T. Nipkow, Lecture Notes in Computer Science, Vol. 806 (Springer, New York, 1994) pp. 213-237.

    Google Scholar 

  23. T. Martin, Probabilit´es et critique philosophique selon Cournot(Vrin, Paris, 1996).

    Google Scholar 

  24. P. Martin-L¨of, Constructive mathematics and computer programming, in: Proceedings of the Sixth International Congress of Logic, Methodology, and Philosophy of Science, eds. L.J. Cohen, J. Los, H. Pfeiffer and K.-P. Podewski (North-Holland, Amsterdam, 1982) pp. 153-175.

    Google Scholar 

  25. P. Martin-L¨of, Intuitionistic Type Theory(Bibliopolis, Naples, 1984).

    Google Scholar 

  26. J. McCarthy and P. Hayes, Some philosophical problems from the standpoint of artificial intelligence, in: Machine Intelligence, Vol. 4, eds. B. Meltzer and D. Michie (Edinburgh University Press, Edinburgh, 1969) pp. 463-502.

    Google Scholar 

  27. B. Nordstr¨om, K. Petersson and J.M. Smith, Programming in Martin-L¨of's Type Theory(Clarendon Press, Oxford, 1990).

    Google Scholar 

  28. L. Paulson, Isabelle: A Generic Theorem Prover, Lecture Notes in Computer Science, Vol. 828 (Springer, New York, 1994). http://www.cl.cam.ac.uk/Research/HVG/Isabelle/dist/.

    MATH  Google Scholar 

  29. J.A. Pinto, Temporal reasoning in the situation calculus. Ph.D. Dissertation, Department of Computer Science, University of Toronto, Toronto, Ontario (1994). Available as report KRR-TR-94-1.

  30. J.A. Pinto, Occurrences and narratives as constraints in the branching structure of the Situation calculus, Journal of Logic and Computation 9 (1998) 777-808.

    Article  MathSciNet  Google Scholar 

  31. F. Pirri and R. Reiter, Some contributions to the metatheory of the situation calculus, Journal of the ACM 45 (1999) 261-325.

    MathSciNet  Google Scholar 

  32. A. Ranta, Type Theoretical Grammar(Oxford University Press, Oxford, 1994).

    MATH  Google Scholar 

  33. R. Reiter, The frame problem in the situation calculus: A simple solution (sometimes) and a completeness result for goal regression, in: Artificial Intelligence and Mathematical Theory of Computation: Papers in Honor of John McCarthy(Academic Press, San Diego, 1991) pp. 359-380.

    Google Scholar 

  34. R. Reiter, Proving properties of states in the situation calculus, Artificial Intelligence 64 (1993) 337-351.

    Article  MATH  MathSciNet  Google Scholar 

  35. R. Reiter, Knowledge in Action: Logical Foundations for Describing and Implementing Dynamical Systems, Book draft, 2000.

  36. J.A. Robinson, A machine-oriented logic based on the resolution principle, Journal of the ACM 12 (1965) 23-41.

    Article  MATH  Google Scholar 

  37. E. Sandewall, Features and Fluents: The Representation of Knowledge about Dynamical Systems, Vol. 1 (Clarendon Press, Oxford, 1994).

    MATH  Google Scholar 

  38. R. Scherl and G. Shafer, A logic of action, causality, and the temporal relations of events, in: Proceedings of the Fifth International Workshop on Temporal Representation and Reasoning (Time-98), eds. L. Khatib and R. Morris (IEEE Computer Society, Los Alamitos, CA, 1998) pp. 89-96.

    Google Scholar 

  39. G. Shafer, The Art of Causal Conjecture(MIT Press, Cambridge, MA, 1996).

    MATH  Google Scholar 

  40. G. Shafer, Mathematical foundations for probability and causality, in: Mathematical Aspects of Artificial Intelligence, ed. F. Hoffman, Symposia in Applied Mathematics, Vol. 55 (American Mathematical Society, Providence, RI, 1998) pp. 207-270.

    Google Scholar 

  41. G. Shafer, Causal logic, in: Proceedings of ECAI98 (13th European Conference on Artificial Intelligence), ed. H. Prade (Wiley, Chichester, 1998) pp. 711-719.

    Google Scholar 

  42. M. Shanahan, Solving the Frame Problem: A Mathematical Investigation of the Common Sense Law of Inertia(MIT Press, Cambridge, MA, 1997).

    Google Scholar 

  43. Y. Shoham, Reasoning about Change: Time and Causation from the Standpoint of Artificial Intelligence(MIT Press, Cambridge, MA, 1988).

    Google Scholar 

  44. L. Stein and L. Morgenstern, Motivated action theory: a formal theory of causal reasoning, Artificial Intelligence 71 (1994) 1-42.

    Article  MATH  MathSciNet  Google Scholar 

  45. G. Sz´asz, Introduction to Lattice Theory, 3rd revised and enlarged edition (Academic Press, New York, 1963).

    Google Scholar 

  46. A. Troelstra and D. van Dalen, Constructivism in Mathematics, Vol. 2 (North-Holland, Amsterdam, 1988).

    Google Scholar 

  47. D. van Dalen, Intuitionistic logic, in: Handbook of Philosophical Logic, Vol. III, eds. D. Gabbay and F. Guenther (Reidel, 1986) pp. 225-339.

  48. D. van Dalen, Logic and Structure(Springer, 1997).

  49. L. Vila, A survey on temporal reasoning in artificial intelligence, Artificial Intelligence Communications 7(1) (1994) 4-28.

    MathSciNet  Google Scholar 

  50. J. von Plato, The axioms of constructive geometry, Annals of Pure and Applied Logic 76 (1995) 169-200.

    Article  MATH  MathSciNet  Google Scholar 

  51. J. von Plato, Organization and development of a constructive axiomatization, in: Types for Proofs and Programs, eds. S. Berardi and M. Coppo, Lecture Notes in Computer Science, Vol. 1158 (Springer, New York, 1996) pp. 288-296.

    Google Scholar 

  52. J. von Plato, Positive partial order, lattices, and Heyting algebras, unpublished (1997).

Download references

Author information

Authors and Affiliations

  1. Department of Accounting and Information Systems, Faculty of Management, Rutgers University, 180 University Avenue, Newark, NJ, 07102‐1897, USA, also

    Glenn Shafer

  2. Department of Computer Science, Royal Holloway College, University of London, UK Email

    Glenn Shafer

  3. Department of Accounting and Information Systems, Faculty of Management, Rutgers University, Janice H. Levin Building, 94 Rockafeller Road, Piscataway, NJ, 08854‐8054, USA Email

    Peter R. Gillett

  4. Department of Computer and Information Science, New Jersey Institute of Technology, Newark, NJ, 07102‐1982, USA Email

    Richard Scherl

Authors
  1. Glenn Shafer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter R. Gillett
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Richard Scherl
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shafer, G., Gillett, P.R. & Scherl, R. The logic of events. Annals of Mathematics and Artificial Intelligence 28, 315–389 (2000). https://doi.org/10.1023/A:1018964524717

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018964524717

Keywords

Search

Navigation