Combining Temporal Logic Systems
Marcelo Finger and Dov Gabbay
Source: Notre Dame J. Formal Logic Volume 37, Number 2
(1996), 204-232.
Abstract
This paper investigates modular combinations of temporal logic systems. Four combination methods are described and studied with respect to the transfer of logical properties from the component one-dimensional temporal logics to the resulting combined two-dimensional temporal logic. Three basic logical properties are analyzed, namely soundness, completeness, and decidability. Each combination method comprises three submethods that combine the languages, the inference systems, and the semantics of two one-dimensional temporal logic systems, generating families of two-dimensional temporal languages with varying expressivity and varying degrees of transfer of logical properties. The temporalization method and the independent combination method are shown to transfer all three basic logical properties. The method of full join of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several cases. So a weaker method of restricted join is proposed and shown to transfer all three basic logical properties.
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Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1040046087
Mathematical Reviews number (MathSciNet): MR1403818
Digital Object Identifier: doi:10.1305/ndjfl/1040046087
Zentralblatt MATH identifier: 0857.03008
References
[1] Aqvist, L., ``A conjectured axiomatization of two-dimensional Reichenbachian tense logic,'' Journal of Philosophical Logic, vol. 8 (1979), pp. 1--45.
Mathematical Reviews (MathSciNet): MR80d:03014
Zentralblatt MATH: 0407.03024
[2] Büchi, J. R., ``On a decision method in restricted second order arithmetic,'' pp. 1--11 in Logic, Methodology, and Philosophy of Science: Proceedings of the 1960 International Congress, Stanford University Press, Stanford, 1962.
Mathematical Reviews (MathSciNet): MR32:1116
[3] Burgess, J. P., and Y. Gurevich, ``The decision problem for linear logic,'' Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 566--582.
Zentralblatt MATH: 0573.03004
Mathematical Reviews (MathSciNet): MR783592
Project Euclid: euclid.ndjfl/1093870820
Digital Object Identifier: doi:10.1305/ndjfl/1093870820
[4] Burgess, J. P., ``Axioms for tense logic I: `Since' and `Until','' Notre Dame Journal of Formal Logic, vol. 23 (1982), pp. 367--374.
Mathematical Reviews (MathSciNet): MR84j:03031a
Zentralblatt MATH: 0452.03021
Project Euclid: euclid.ndjfl/1093870149
Digital Object Identifier: doi:10.1305/ndjfl/1093870149
[5] Burgess, J. P., ``Basic tense logic,'' pp. 89--133 in Handbook of Philosophical Logic, volume II, edited by D. Gabbay and F. Guenthner, Reidel, Dordrecht, 1984.
Mathematical Reviews (MathSciNet): MR844597
Zentralblatt MATH: 0875.03046
[6] Finger, M., and D. M. Gabbay. ``Adding a Temporal Dimension to a Logic System,'' Journal of Logic, Language and Information, vol. 1 (1992), pp. 203--233.
Mathematical Reviews (MathSciNet): MR95h:03037
Zentralblatt MATH: 0798.03031
Digital Object Identifier: doi:10.1007/BF00156915
[7] Finger, M., ``Handling database updates in two-dimensional temporal logic,'' Journal of Applied Non-Classical Logic, vol. 2 (1992), pp. 201--224.
Mathematical Reviews (MathSciNet): MR94m:68032
Zentralblatt MATH: 0798.68048
[8] Finger, M., Changing the Past: Database Applications of Two-dimensional Temporal Logics, Ph.D. Thesis, Imperial College, Department of Computing, February, 1994.
[9] Gabbay, D. M., ``An irreflexivity lemma,'' pp. 67--89 in Aspects of Philosophical Logic, edited by U. Monnich, Reidel, Dordrecht, 1981.
Mathematical Reviews (MathSciNet): MR83f:03015
Zentralblatt MATH: 0519.03008
[10] Gabbay, D. M., ``Expressive functional completeness in tense logic,'' pp. 91--117 in Aspects of Philosophical Logic, edited by U. Monnich, Reidel, Dordrecht, 1981.
Zentralblatt MATH: 0523.03017
[11] Gabbay, D. M., Fibred semantics and the weaving of logics --- Part 1: Modal and intuitionistic logics, Part 2: Fibring non-monotonic logics, Part 3: How to make your logic fuzzy. Lectures given at Logic Colloquim 1992, Vezprém, Hungary 1992. A version of the notes is published as a Technical Report by the University of Stuttgart, Sonderforschungereich 340, Azenbergstr 12, 70174, Stuttgart, Germany, No. 36, 1993. Part 1 is forthcoming in The Journal of Symbolic Logic. Part 2 is forthcoming in Logic Colloquium 92.
Mathematical Reviews (MathSciNet): MR99j:03009
Mathematical Reviews (MathSciNet): MR99j:03010
Mathematical Reviews (MathSciNet): MR2002b:03053
Zentralblatt MATH: 0872.03007
Zentralblatt MATH: 0856.03021
Zentralblatt MATH: 0947.03034
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2275807
[12] Gabbay, D. M., Labelled Deductive Systems: Principles and Applications. Volume 1: Basic Principles, forthcoming from Oxford University Press.
Mathematical Reviews (MathSciNet): MR1430570
Zentralblatt MATH: 0858.03004
[13] Gabbay, D. M., I. M. Hodkinson, and M. A. Reynolds, Temporal Logic: Mathematical Foundations and Computational Aspects, Oxford University Press, Oxford, 1994.
Mathematical Reviews (MathSciNet): MR95h:03040
Zentralblatt MATH: 0921.03023
[14] Halpern, J. Y., and Y. Shoham, ``A propositional modal logic of time intervals,'' pp. 279--292 in Proceedings of the Symposium on Logics in Computer Science, LICS86, Washington, 1986.
[15] Kamp, H., ``Formal properties of now,'' Theoria, vol. 35 (1971), pp. 227--273.
Mathematical Reviews (MathSciNet): MR49:2265
Zentralblatt MATH: 0269.02008
[16] Kracht, M., and F. Wolter, ``Properties of independently axiomatizable bimodal logics,'' The Journal of Symbolic Logic, vol. 56 (1991), pp. 1469--1485.
Mathematical Reviews (MathSciNet): MR93c:03021
Zentralblatt MATH: 0743.03013
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2275487
[17] Rabin, M. O., ``Decidability of second order theories and automata on infinite trees,'' pp. 595--629 in Transactions of the American Mathematical Society, vol. 141, American Mathematical Society, Providence, 1969.
Mathematical Reviews (MathSciNet): MR40:30
Zentralblatt MATH: 0221.02031
Digital Object Identifier: doi:10.2307/1995086
[18] Reynolds, M. A., ``An axiomatisation for until and since over the reals without the IRR rule,'' Studia Logica, vol. 51 (1992), pp. 165--194.
Mathematical Reviews (MathSciNet): MR93m:03033
Zentralblatt MATH: 0785.03006
Digital Object Identifier: doi:10.1007/BF00370112
[19] Segerberg, K., ``Two-dimensional modal logic,'' Journal of Philosophical Logic, vol. 2 (1973), pp. 77--96.
Mathematical Reviews (MathSciNet): MR54:12488
Zentralblatt MATH: 0259.02013
Digital Object Identifier: doi:10.1007/BF02115610
[20] Spaan, E., Complexity of Modal Logics, Ph.D. Thesis, Free University of Amsterdam, 1993.
[21] Thomason, S. K., ``Independent propositional modal logics,'' Studia Logica, vol. 39 (1980), pp. 143--144.
Mathematical Reviews (MathSciNet): MR81m:03026
Zentralblatt MATH: 0457.03017
Digital Object Identifier: doi:10.1007/BF00370317
[22] Venema, Y., ``Expressiveness and completeness of an interval tense logic,'' Notre Dame Journal of Formal Logic, vol. 31 (1990), pp. 529--547.
Mathematical Reviews (MathSciNet): MR92a:03029
Zentralblatt MATH: 0725.03006
Project Euclid: euclid.ndjfl/1093635589
Digital Object Identifier: doi:10.1305/ndjfl/1093635589
[23] Xu, M., ``On some $U,S$-tense logics,'' Journal of Philosophical Logic, vol. 17 (1988), pp. 181--202.
Mathematical Reviews (MathSciNet): MR89d:03017
Zentralblatt MATH: 0648.03010
Digital Object Identifier: doi:10.1007/BF00247911
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