Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T09:16:58.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Embedding theorems for LTL and its variants†

Published online by Cambridge University Press:  02 December 2014

NORIHIRO KAMIDE
NORIHIRO KAMIDE*
Affiliation:
Department of Human Information Systems, Faculty of Science and Engineering, Teikyo University, Toyosatodai 1-1, Utsunomiya-shi, Tochigi 320-8551, Japan Email: drnkamide08@kpd.biglobe.ne.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we prove some embedding theorems for LTL (linear-time temporal logic) and its variants: viz. some generalisations, extensions and fragments of LTL. Using these embedding theorems, we give uniform proofs of the completeness, cut-elimination and/or decidability theorems for LTL and its variants. The proposed embedding theorems clarify the relationships between some LTL-variations (for example, LTL, a dynamic topological logic, a fixpoint logic, a spatial logic, Prior's logic, Davies' logic and an NP-complete LTL) and some traditional logics (for example, classical logic, intuitionistic logic and infinitary logic).

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Issue 1 , January 2015 , pp. 83 - 134
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Sections 2, 4, 5, 6.1 and 6.3 in the current paper are based on some refinements of technical parts of the conference presentations Kamide (2009; 2010b; 2010d; 2010e; 2011).

References

Artemov, S. N., Davoren, J. M. and Nerode, A. (1997) Modal logics and topological semantics for hybrid systems. Technical Report MSI 97-05, Cornell University.CrossRefGoogle Scholar
Baratella, S. and Masini, A. (2004) An approach to infinitary temporal proof theory. Archive for Mathematical Logic 43 (8)965–990.Google Scholar
Baratella, S and Masini, A. (2003) A proof-theoretic investigation of a logic of positions. Annals of Pure and Applied Logic 123 135–162.Google Scholar
Biere, A., Cimatti, A., Clarke, E. M., Strichman, O. and Zhu, Y. (2003) Bounded model checking. Advances in Computers 58 118–149.Google Scholar
Clarke, E. M., Grumberg, O. and Peled, D. A. (1999) Model checking, The MIT Press.Google Scholar
Davies, R. (1996) A temporal-logic approach to binding-time analysis. In: Proceedings of the 11th IEEE Annual Symposium on Logic in Computer Science (LICS'96) 184–195.CrossRefGoogle Scholar
Demri, S. and Schnoebelen, Ph. (2002) The complexity of propositional linear temporal logics in simple cases. Information and Computation 174 (1)84–103.Google Scholar
Emerson, E. A. (1990) Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, Formal Models and Semantics (B), Elsevier and MIT Press 995–1072.Google Scholar
Etessami, K., Vardi, M. Y. and Wilke, T. (1997) First-order logic with two variables and unary temporal logic. In: Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science (LICS'97) 228–235.CrossRefGoogle Scholar
Feferman, S. (1968) Lectures on proof theory. In: Proceedings of the summer school in logic. Springer-Verlag Lecture Notes in Mathematics 70 1–107.Google Scholar
Holzmann, G. J. (2006) The SPIN model checker: Primer and reference manual, Addison-Wesley.Google Scholar
Kamide, N. (2006a) Linear and affine logics with temporal, spatial and epistemic operators. Theoretical Computer Science 353 (1–3)165–207.Google Scholar
Kamide, N. (2006b) An equivalence between sequent calculi for linear-time temporal logic. Bulletin of the Section of the Logic 35 (4)187–194.Google Scholar
Kamide, N. (2009) Embedding linear-time temporal logic into infinitary logic: Application to cut-elimination for multi-agent infinitary epistemic linear-time temporal logic. In: Fisher, M., Sadri, F. and Thielscher, M. (eds.) Proceedings of the 9th International Workshop on Computational Logic in Multi-Agent Systems (CLIMA9). Springer-Verlag Lecture Notes in Computer Science 5405 57–76.Google Scholar
Kamide, N. (2010a) Branching-time versus linear-time: A cooperative and feasible approach. In: Proceedings of the 2nd International Conference on Agents and Artificial Intelligence (ICAART 2010) 522–526.Google Scholar
Kamide, N. (2010b) Reasoning about bounded time domain: An alternative to NP-complete fragments of LTL. In: Proceedings of the 2nd International Conference on Agents and Artificial Intelligence (ICAART 2010) 536–539.Google Scholar
Kamide, N. (2010c) Notes on an extension of Davies' logic for binding-time analysis. Far East Journal of Applied Mathematics 44 (1)37–57.Google Scholar
Kamide, N. (2010d) Completeness for generalized first-order LTL. In: Dillmann, R., Beyerer, J., Hanebeck, U. D. and Schultz, T. (eds.) Proceedings of the 33rd Annual German Conference on Artificial Intelligence (KI 2010). Springer-Verlag Lecture Notes in Computer Science 6359 246–254.Google Scholar
Kamide, N. (2010e) A sequent calculus for 3-dimensional space. In: da Rocha Costa, A. C., Vicari, R. M. and Tonidandel, F. (eds.) Advances in Artificial Intelligence – SBIA 2010. Proceedings of the 20th Brazilian Symposium on Artificial Intelligence. Springer-Verlag Lecture Notes in Computer Science 6404 263–272.Google Scholar
Kamide, N. (2010f) A proof system for temporal reasoning with sequential information. In: da Rocha Costa, A. C., Vicari, R. M. and Tonidandel, F. (eds.) Advances in Artificial Intelligence – SBIA 2010. Proceedings of the 20th Brazilian Symposium on Artificial Intelligence. Springer-Verlag Lecture Notes in Computer Science 6404 283–292.Google Scholar
Kamide, N. (2011) Cut-elimination and completeness in dynamic topological and linear-time temporal logics. Logique et Analyse 54 (215)1–16.Google Scholar
Kamide, N. and Wansing, H. (2010) Combining linear-time temporal logic with constructiveness and paraconsistency. Journal of Applied Logic 8 33–61.Google Scholar
Kaneiwa, K. and Kamide, N. (2010) Sequence-indexed linear-time temporal logic: Proof system and application. Applied Artificial Intelligence 24 (10)896–913.Google Scholar
Kaneko, M. and Nagashima, T. (1997) Game logic and its applications II. Studia Logica 58 273–303.Google Scholar
Kawai, H. (1987) Sequential calculus for a first order infinitary temporal logic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 33 423–432.CrossRefGoogle Scholar
Kojima, K. and Igarashi, A. (2008) On constructive linear-time temporal logic. In: Proceedings of Intuitionistic Modal Logics and Applications Workshop (IMLA'08), Pittsburgh.Google Scholar
Konev, B., Kontchakov, R., Wolter, F. and Zakharyaschev, M. (2006) On dynamic topological and metric logics. Studia Logica 84 129–160.Google Scholar
Kremer, P. and Mints, G. (2005) Dynamic topological logic. Annals of Pure and Applied Logic 131 133–158.Google Scholar
Kröger, F. (1977) LAR: a logic of algorithmic reasoning. Acta Informatica 8 243–266.Google Scholar
Lorenzen, P. (1951) Algebraische und logitishe Untersuchungen über freie Verbände. Journal of Symbolic Logic 16 81–106.Google Scholar
Mints, G. (2006) Cut elimination for S4C: a case study. Studia Logica 82 121–132.CrossRefGoogle Scholar
Muscholl, A. and Walukiewicz, I. (2005) An NP-complete fragment of LTL. International Journal of Foundations of Computer Science 16 (4)743–753.Google Scholar
Novikov, P. S. (1961) Inconsistencies of certain logical calculi. In: Infinistic Methods: Proceedings of the Symposium on Foundations of Mathematics, Pergamon Press 71–74.Google Scholar
Pnueli, A. (1977) The temporal logic of programs. In: Proceedings of the 18th IEEE Symposium on Foundations of Computer Science 46–57.CrossRefGoogle Scholar
Prior, A. N. (1957) Time and modality, Clarendon Press.Google Scholar
Sistla, A. P. and Clarke, E. M. (1985) The complexity of propositional linear temporal logics. Journal of the ACM 32 (3)733–749.CrossRefGoogle Scholar
Tanaka, Y. (1999) Representations of algebras and Kripke completeness of infinitary and predicate logics, Doctoral thesis, Japan Advanced Institute of Science and Technology.Google Scholar
Tanaka, Y. (2001) Cut-elimination theorems for some infinitary modal logics. Mathematical Logic Quarterly 47 (3)327–339.Google Scholar
Takeuti, G. (1985) Proof theory, North-Holland.Google Scholar
Walukiewicz, I. (1998) Difficult configurations – On the complexity of LTrL. In: Larsen, K. G., Skyum, S. and Winskel, G. (eds.) Springer-Verlag Lecture Notes in Computer Science 1443 140–151.Google Scholar