Abstract
This paper is concerned with the application of the resolution theorem proving method to reified logics. The logical systems treated include the branching temporal logics and logics of belief based on K and its extensions. Two important problems concerning the application of the resolution rule to reified systems are identified. The first is the redundancy in the representation of truth functional relationships and the second is the axiomatic reasoning about modal structure. Both cause an unnecessary expansion in the search space. We present solutions to both problems which allow the axioms defining the reified logic to be eliminated from the database during theorem proving hence reducing the search space while retaining completeness. We describe three theorem proving methods which embody our solutions and support our analysis with empirical results.
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References
Aitken, S., Reichgelt, H. and Shadbolt, N. (1992). Planning, knowledge division and agency,Proc. 11th UK Planning Special Interest Group.
Auffray, Y. and Enjalbert, P. (1989), Modal theorem proving: An equational viewpoint,Proc. IJCAI-11 441–445.
Auffray, Y., Enjalbert, P. and Hebrard, J. (1990), Strategies for modal resolution: results and problemsJ. Automated Reasoning 6 1–38.
Boyer, R. S. and Moore, J. S. (1972), The sharing of structure in theorem proving programs, in B. Meltzer and D. Michie (eds.),Machine Intelligence Vol 7, Edinburgh University Press, pp. 101–116
Catach, L. (1991), TABLEAUX: A general theorem prover for modal logics,J. Automated Reasoning 7 489–510.
Eisinger, N., Ohlbach, H. J. and Pracklein, A. (1991), Reduction rules for resolution based systems,Artificial Intelligence 50 141–181.
Farinas del Cerro, L. (1985), Resolution modal logic,Logique et Analyse 28 153–172.
Fitting, M. (1988), First-order modal tableaux,J. Automated Reasoning 4 191–213.
Hughes, G. E. and Cresswell, M. J. (1968),An Introduction to Modal Logic, Methuen, London.
Jackson, P. and Reichgelt, H. (1989), A general proof method for modal predicate logic, in Jackson, P., Reichgelt, H. and van Harmelen, F. (eds.),Logic Based Knowledge Representation, MIT Press.
Konolige, K. (1986),A Deduction Model of Belief, Pitman, London.
Kripke, S. A. (1971), Semantical considerations on modal logic, in Linsky, L. (1971).
Linskey, L. (1971),Reference and Modality, Oxford University Press.
Moore, R. C. (1985), A formal theory of knowledge and action, in Hobbs, R. J. and Moore, R. C. (eds.),Formal Theories of the Commonsense World, Ablex Publishing, New Jersey.
Ohlbach, H. J. (1988), A resolution calculus for modal logics,Proc. 9th Int. Conf. on Automated Deduction, Lecture Notes in Computer Science No. 310, Springer-Verlag, New York, pp. 500–516.
Reichgelt, H. (1989a), Logics for reasoning about knowledge and belief,The Knowledge Engineering Review 4 119–139.
Reichgelt, H. (1989b), A comparison of modal and first order logics of time, in Jackson, P., Reichgelt, H. and van Harmelen, F. (eds.),Logic Based Knowledge Representation, MIT Press.
Rescher, N. and Urquhart, A. (1971),Temporal logic, Springer-Verlag, New York.
Robinson, J. A. (1979),Logic Form and Function, Edinburgh University Press.
Robinson, J. A. (1992), Logic and logic programming,Comm. ACM 35 41–65.
Shoham, Y. (1986), Reified temporal logics: Semantical and ontological considerations,Proc. ECAI-7 390–397.
Staples, J. and Robinson, P. J. (1990), Structure sharing for quantified terms: fundamentals,J. Automated Reasoning 6 115–145.
Wallen, L. A. (1987), Matrix proof methods for modal logics,Proc. IJCAI-10 917–923.
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Much of the research reported in this paper was supported by DTI IED SERC grant No. GR/F 35968, and was carried out whilst Han Reichgelt was at the University of Nottingham.
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Aitken, J.S., Reichgelt, H. & Shadbolt, N. Resolution theorem proving in reified modal logics. J Autom Reasoning 12, 103–129 (1994). https://doi.org/10.1007/BF00881845
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DOI: https://doi.org/10.1007/BF00881845