Abstract
In this paper we show the adequacy of tense logic with unary operators for dealing with finite trees. We prove that models on finite trees can be characterized by tense formulas, and describe an effective method to find an axiomatization of the theory of a given finite tree in tense logic. The strength of the characterization is shown by proving that adding the binary operators "Until" and "Since" to the language does not result in a better description than that given by unary tense logic; although the greater expressive power of "Until" and "Since" can be exploited by using the semantics of e-frames instead of traditional Kripke semantics.
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References
Baker, K. A., ‘Finite equational bases for finite algebras in a congruence-distributive equational class’, Advances in Mathematics 24: 207–243, 1977.
Bellissima, F., ‘Atoms of tense algebras’, Algebra Universalis 28(1): 52–78, 1991.
Bellissima, F., and A. Bucalo, ‘A distinguishable model theorem for the minimal US-tense logic’, Notre Dame Journal of Formal Logic 36(4): 585–594, 1995.
Bellissima, F., and S. Cittadini, ‘Duality and completeness for US-logics’, Rapp. Mat. 319, Università di Siena, Dipartimento di Matematica, 1997, submitted.
Bellissima, F., and S. Cittadini, ‘Minimal axiomatization in modal logic’, Mathematical Logic Quarterly 43(1):92–102, 1997.
Bellissima, F., and S. Cittadini, ‘Minimal p-morphic images, axiomatizations and coverings in the modal logic K4’, 1997, to appear in Studia Logica.
van Benthem, J. F. A. K., ‘Temporal logic’, in D. Gabbay, C. Hogger, and J. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, vol. III, Oxford University Press, Oxford, 1990.
Kamp, J. A. W., Tense Logic and the Theory of Linear Order, PhD thesis, University of California at Los Angeles, 1968.
Kracht, M., ‘Even more about the lattice of tense logics’, Archive for Mathematical Logic 31(4):243–257, 1992.
Kracht, M., and F. Wolter, ‘Properties of independently axiomatizable bimodal logics’, The Journal of Symbolic Logic 56(4):1469–1485, 1991.
Laroussinie, F., and Ph. Schnoebelen, ‘A hierarchy of temporal logics with past’. Theorelical Computer Science 148(2):303–324, 1995.
Makinson, D. C., ‘Some embedding theorems for modal logics’, Notre Dame Journal of Formal Logic 12:252–254, 1971.
Rautenberg, W., ‘More about the lattice of tense logics’, Bull. Sect. of Logic 8:21–29, 1979.
Segerberg, K., ‘Modal logics with linear alternative relation’, Theoria 36:301–322, 1970.
Segerberg, K., An Essay in Classical Modal Logic, Filosofiska Studier, Uppsala, 1971.
Wolter, F., Lattices of Modal Logics. PhD thesis, FU Berlin, 1993.
Wolter, F., ‘Tense logic without tense operators’, Mathematical Logic Quarterly 42(2):145–171, 1996.
Wolter, F., ‘The structure of lattices of subframe logics’, Annals of Pure and Applied Logic 86(1):47–100, 1997.
Xu, M., ‘On some U,S-tense logics’ Journal of Philosophical Logic 17:181–202, 1988.
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Bellissima, F., Cittadini, S. Finite Trees in Tense Logic. Studia Logica 62, 121–140 (1999). https://doi.org/10.1023/A:1026456817552
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DOI: https://doi.org/10.1023/A:1026456817552