Abstract
We introduce a temporal logic TAL and prove that it has several nice features. The formalism is a two-dimensional modal system in the sense that formulas of the language are evaluated at pairs of time points. Many known formalisms with a two-dimensional flavor can be expressed in TAL, which can be seen as the temporal version of square arrow logic.
We first pin down the expressive power of TAL to the three-variable fragment of first-order logic; we prove that this induces an expressive completeness result of ‘flat’ TAL with respect to monadic first order logic (over the class of linear flows of time).
Then we treat axiomatic aspects: our main result is a completeness proof for the set of formulas that are ‘flatly’ valid in well-ordered flows of time and in the flow of time of the natural numbers.
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References
H. Andréka, J.D. Monk & I. Németi (eds.), Algebraic Logic; Proceedings of the 1988 Budapest Conference on Algebraic Logic, North-Holland, Amsterdam, 1991.
K. Doets, “Monadic Π 11 -theories of Π 11 -properties”, Notre Dame Journal of Formal Logic, 30 (1989) 224–240.
D.M. Gabbay, “An irreflexivity lemma with applications to axiomatizations of conditions on linear frames”, in:, pp. 67–89.
D.M. Gabbay, “Expressive functional completeness in tense logic”, in:, pp. 91–117.
D.M. Gabbay, I.M. Hodkinson & M. Reynolds, Temporal Logic: Mathematical Foundations and Computational Aspects, Oxford University Press, Oxford, 1992, to be published.
D.M. Gabbay & I.M. Hodkinson, “An axiomatization of the temporal logic with Since and Until over the real numbers”, Journal of Logic and Computation, 1 (1990) 229–259.
J.Y. Halpern & Y. Shoham “A propositional modal logic of time intervals”, Proceedings of the First IEEE Symposium on Logic in Computer Science, Cambridge, Massachusettes, 279–292.
D. Harel, D. Kozen & R. Parikh, “Process Logic: expressiveness, decidability, completeness, Journal of Computer and System Sciences, 26 (1983) 222–243.
N. Immerman and D. Kozen, “Definability with bounded number of bound variables”, in:, pp. 236–244.
J.A.W. Kamp, Tense Logic and the Theory of Linear Order, PhD Dissertation, University of California, Los Angeles, 1968.
Proceedings of the Symposium on Logic in Computer Science 1987, Ithaka, New York, Computer Society Press, Washington, 1987.
R. Maddux, “A Sequent Calculus for Relation Algebras”, Annals of Pure and Applied Logic, 25 (1983) 73–101.
R.D. Maddux, “Finitary Algebraic Logic”, Zeitschrift für Mathematisch Logik und Grundlagen der Mathematik, 35 (1989) 321–332, 39 (1993) 566–569.
M. Masuch, L. Pólós & M. Marx (eds.), Arrow Logic, Oxford University Press, to appear.
M. Marx, Sz. Mikulás, I. Németi & I. Sain, “Investigations in arrow logic”, to appear in [14].
U. Mönnich, ed., Aspects of Philosophical Logic, Reidel, Dordrecht, 1981.
I. Németi, “Algebraizations of quantifier logics: an introductory overview”, Studia Logica 50 (1991) 485–570.
E. Orłowska, “Relational interpretation of modal logics”, in:, 443–471.
M. Reynolds, “An axiomatization for Since and Until over the reals without the IRR rule”, Studia Logica, to appear.
M. de Rijke (ed.), Diamonds and Defaults, Kluwer Academic Publishers, Dordrecht, 1993.
A. Tarski and S. Givant, A formalization of set theory without variables, AMS Colloquium Publications 41, 1987.
Y. Venema, “Relational Games”, in, 695–718.
Y. Venema, Many-Dimensional Modal Logic, Doctoral Dissertation, University of Amsterdam, 1992, submitted to Oxford University Press.
Y. Venema, “Completeness by completeness: Since and Until”, in:, 349–359.
Y. Venema, “Derivation rules as anti-axioms in modal logic”, Journal of Symbolic Logic, 58 (1993) 1003–1054.
Y. Venema, “A crash course in arrow logic”, Logic Group Preprint Series 107, February 1994, Department of Philosophy, Utrecht University; to appear in [14].
L. Åqvist, “A conjectured axiomatization of two-dimensional Reichenbachian tense logic”, Journal of Philosophical Logic, 8 (1982) 315–333.
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© 1994 Springer-Verlag Berlin Heidelberg
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Venema, Y. (1994). Completeness through flatness in two-dimensional temporal logic. In: Gabbay, D.M., Ohlbach, H.J. (eds) Temporal Logic. ICTL 1994. Lecture Notes in Computer Science, vol 827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0013986
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DOI: https://doi.org/10.1007/BFb0013986
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