## Journal of Symbolic Logic

### Logical consecutions in discrete linear temporal logic

V. V. Rybakov

#### Abstract

We investigate logical consequence in temporal logics in terms of logical consecutions, i.e., inference rules. First, we discuss the question: what does it mean for a logical consecution to be ’correct’ in a propositional logic. We consider both valid and admissible consecutions in linear temporal logics and discuss the distinction between these two notions. The linear temporal logic LDTL, consisting of all formulas valid in the frame 〈 𝒵, ≤, ≥ 〉 of all integer numbers, is the prime object of our investigation. We describe consecutions admissible in LDTL in a semantic way—via consecutions valid in special temporal Kripke/Hintikka models. Then we state that any temporal inference rule has a reduced normal form which is given in terms of uniform formulas of temporal degree 1. Using these facts and enhanced semantic techniques we construct an algorithm, which recognizes consecutions admissible in LDTL. Also, we note that using the same technique it follows that the linear temporal logic ℒ(𝒩) of all natural numbers is also decidable w.r.t. inference rules. So, we prove that both logics LDTL and ℒ(𝒩) are decidable w.r.t. admissible consecutions. In particular, as a consequence, they both are decidable (known fact), and the given deciding algorithms are explicit.

#### Article information

Source
J. Symbolic Logic, Volume 70, Issue 4 (2005), 1137-1149.

Dates
First available in Project Euclid: 18 October 2005

https://projecteuclid.org/euclid.jsl/1129642119

Digital Object Identifier
doi:10.2178/jsl/1129642119

Mathematical Reviews number (MathSciNet)
MR2194241

Zentralblatt MATH identifier
1110.03010

#### Citation

Rybakov, V. V. Logical consecutions in discrete linear temporal logic. J. Symbolic Logic 70 (2005), no. 4, 1137--1149. doi:10.2178/jsl/1129642119. https://projecteuclid.org/euclid.jsl/1129642119

#### References

• R. Bloem, K. Ravi, and F. Somenzi Efficient decision procedures for model checking of linear time logic properties, Conference on Computer Aided Verification (CAV) (Terento, Italy), LNCS, vol. 1633, Springer-Verlag,1999.
• A. Chagrov and M. Zakharyaschev Modal logic, Oxford Logic Guides, vol. 35, Clarendon Press, Oxford,1997.
• A. I. Citkin On admissible rules of intuitionistic propositional logic, Math. USSR Sbornik, vol. 31 (1977), no. 2, pp. 279--288.
• E. Clarke, O. Grumberg, and K. P. Hamaguchi Another look at LTL model checking, Conference on Computer Aided Verification (CAV) (Stanford, California), LNCS, vol. 818, Springer-Verlag,1994.
• H. B. Curry Foundations of mathematical logic, Dover Publications, New-York,1977.
• H. Friedman One hundred and two problems in mathematical logic, Journal of Symbolic Logic, vol. 40 (1975), no. 3, pp. 113--130.
• D. Gabbay Model theory for tense logics, U.S. Air Forse Office of Science Research, contract no. F61052-68-C-0036, report no.1, April 1969.
• D. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev Many-dimensional modal logics: Theory and applications, Elsevier Science Pub Co.,2003.
• S. Ghilardi Unification in intuitionistic logic, Journal of Symbolic Logic, vol. 64 (1999), no. 2, pp. 859--880.
• R. Goldblatt Logics of time and computation, second ed., CSLI Lecture Notes, vol. 7,1992.
• R. Harrop Concerning formulas of the types $A\to B\vee C$, $A\to \exists x B(x)$ in intuitionistic formal system, Journal of Symbolic Logic, vol. 25 (1960), pp. 27--32.
• R. Iemhoff A(nother) characterization of intuitionistic propositional logic, Annals of Pure and Applied Logic, vol. 113 (2001), no. 1--3, pp. 161--173.
• P. Lorenzen Einführung in die operative Logik und Mathematik, Springer-Verlag,1955.
• M. Marx and Y. Venema Multi-dimensional modal logics, Kluwer Academic Publishers,1997.
• G. E. Mints Derivability of admissible rules, Journal of Soviet Mathematics, vol. 6 (1976), no. 4, pp. 417--421.
• A. Pnueli The temporal logic of programs, Proceedings of the 18th Annual Symposium on Foundations of Computer Science, IEEE,1997, pp. 46--57.
• A. Prior Time and modaliy, Oxford,1957.
• V. V. Rybakov A criterion for admissibility of rules in the modal system $S4$ and the intuitionistic logic, Algebra and Logic, vol. 23 (1984), no. 5, pp. 369--384, English Translation.
• V. V. Rybakov, V. R. Kiyatkin, and T. Oner On finite model property for admissible rules, Mathematical Logic Quarterly, vol. 45 (1999), no. 4, pp. 505--520.
• V. V. Rybakov, M. Terziler, and V. V. Rimazki Basis in semi-reduced form for the admissible rules of the intuitionistic logic IPC, Mathematical Logic Quarterly, vol. 46 (2000), no. 2, pp. 207--218.
• K. Segerberg Modal logics with linear alternative relations, Theoria, vol. 36 (1970), pp. 301--322.
• S. K. Thomason Semantic analysis of tense logic, Journal of Symbolic Logic, vol. 37 (1972), no. 1.
• J. van Benthem The logic of time, Synthese Libraryšpace*\fill, vol. 156, Reidel,1983.