Abstract
The wide number of languages and programming paradigms, as well as the heterogeneity of ‘programs’ and ‘executions’ require new generalisations of propositional dynamic logic. The dynamisation method, introduced in [20], contributed on this direction with a systematic parametric way to construct Many-valued Dynamic Logics able to handle systems where the uncertainty is a prime concern. The instantiation of this method with the Łukasiewicz arithmetic lattice over [0, 1], that we derive here, supports a general setting to design and to (fuzzy-) reason about systems with uncertainty degrees in their transitions.
For the verification of real systems, however, there are no de facto methods to accommodate exact truth degrees or weights. Instead, the traditional approach within scientific community is to use different kinds of approximation techniques.
Following this line, the current paper presents a framework where the representation values are given by means of intervals. Technically this is achieved by considering an ‘interval version’ of the Kleene algebra based on the [0, 1] Łukasiewicz lattice. We also discuss the ‘intervalisation’ of \(\L \) action lattice (in the lines reported in [28]) and how this class of algebras behaves as an (interval) semantics of many-valued dynamic logic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This idea is confirmed in some Representation Theorems of Euclidean continuous functions.
References
Baltag, A., Smets, S.: Quantum logic as a dynamic logic. Synthese 179(2), 285–306 (2011)
Bedregal, B., Santiago, R.: Some continuity notions for interval functions and representation. Comput. Appl. Math. 32(3), 435–446 (2013)
Bedregal, B.C., Takahashi, A.: Interval valued versions of t-conorms, fuzzy negations and fuzzy implications. In: 2006 IEEE International Conference on Fuzzy Systems , pp. 1981–1987 (2006)
Bedregal, B.R.C., Santiago, R.H.N.: Interval representations, Łukasiewicz implicators and Smets-Magrez axioms. Inf. Sci. 221, 192–200 (2013)
Bedregal, B.R.C., Takahashi, A.: The best interval representations of t-norms and automorphisms. Fuzzy Sets Syst. 157(24), 3220–3230 (2006)
Bou, F., Esteva, F., Godo, L., Rodríguez, R.O.: On the minimum many-valued modal logic over a finite residuated lattice. J. Log. Comput. 21(5), 739–790 (2011)
Bustince, H., Barrenechea, E., Pagola, M., Fernandez, J., Xu, Z., Bedregal, B., Montero, J., Hagras, H., Herrera, F., Baets, B.D.: A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. 24(1), 179–194 (2016)
Cignoli, R., d’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic. Springer, Netherlands (1999)
Conway, J.H.: Regular Algebra and Finite Machines. Printed in GB by William Clowes & Sons Ltd, London (1971)
Fitting, M.: Many-valued modal logics. Fundam. Inform. 15(3–4), 235–254 (1991)
Fitting, M.: Many-valued model logics II. Fundam. Inform. 17(1–2), 55–73 (1992)
Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge (2000)
Hickey, T., Ju, Q., Van Emden, M.H.: Interval arithmetic: from principles to implementation. J. ACM 48(5), 1038–1068 (2001)
Hughes, J., Esterline, A.C., Kimiaghalam, B.: Means-end relations and a measure of efficacy. J. Log. Lang. Inf. 15(1–2), 83–108 (2006)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms, 1st edn. Springer, Berlin (2000)
Kozen, D.: On action algebras (manuscript). In: Logic and Flow of Information, Amsterdam (1991)
Kozen, D.: A probabilistic PDL. J. Comput. Syst. Sci. 30(2), 162–178 (1985)
Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Inf. Comput. 110(2), 366–390 (1994)
Liau, C.-J.: Many-valued dynamic logic for qualitative decision theory. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS (LNAI), vol. 1711, pp. 294–303. Springer, Heidelberg (1999). doi:10.1007/978-3-540-48061-7_36
Madeira, A., Neves, R., Martins, M.A.: An exercise on the generation of many-valued dynamic logics. J. Log. Algebraic Methods Program. 85(5), 1011–1037 (2016)
Madeira, A., Neves, R., Martins, M.A., Barbosa, L.S.: A dynamic logic for every season. In: Braga, C., Martí-Oliet, N. (eds.) SBMF 2014. LNCS, vol. 8941, pp. 130–145. Springer, Heidelberg (2015). doi:10.1007/978-3-319-15075-8_9
Moore, R.E.: Interval arithmetic and automatic error analysis in digital computing. Ph.D. dissertation, Department of Mathematics, Stanford University, Stanford, CA, USA, November 1962. (Also published as Applied Mathematics and Statistics Laboratories Technical report No. 25)
Moore, R.E., Yang, C.T.: Interval analysis I. Technical document LMSD-285875, Lockheed Missiles and Space Division, Sunnyvale, CA, USA (1959)
Mundici, D.: Advanced Łukasiewicz Calculus and MV-Algebras. Trends in Logic. Springer, Netherlands (2011)
Platzer, A.: Logical Analysis of Hybrid Systems - Proving Theorems for Complex Dynamics. Springer, Berlin (2010)
Pratt, V.R.: Semantical considerations on floyd-hoare logic. In: 17th Annual Symposium on Foundations of Computer Science, Houston, Texas, USA, 25–27 October 1976, pp. 109–121. IEEE Computer Society (1976)
Pratt, V.: Action logic and pure induction. In: Eijck, J. (ed.) JELIA 1990. LNCS, vol. 478, pp. 97–120. Springer, Heidelberg (1991). doi:10.1007/BFb0018436
Santiago, R.H.N., Bedregal, B.R.C., Acióly, B.M.: Formal aspects of correctness and optimality of interval computations. Formal Aspects Comput. 18(2), 231–243 (2006)
Sunaga, T.: Theory of an interval algebra and its application to numerical analysis [reprint of Res. Assoc. Appl. Geom. Mem. 2, 29–46 (1958)]. Japan J. Ind. Appl. Math. 26(2–3), 125–143 (2009)
Xu, Y., Ruan, D., Qin, K., Liu, J.: Lattice-Valued Logic: An Alternative Approach to Treat Fuzziness and Incomparability. Studies in Fuzziness and Soft Computing. Springer, Berlin (2012)
Acknowledgements
R. Santiago and B. Bedregal are supported by Marie Curie project PIRSES-GA-2012-318986 GetFun funded by EU-FP7 and by the Brazilian National Council for Scientific and Technological Development (CNPq, Portuguese: Conselho Nacional de Desenvolvimento Científico e Tecnológico) under the Projects 304597/2015-5 and 307681/2012-2. This work is also financed by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 and by National Funds through the Portuguese funding agency, FCT-Fundação para a Ciência e a Tecnologia within project POCI-01-0145-FEDER- 016692. A. Madeira and M. Martins are also supported by the FCT BPD individual grant SFRH/BPD/103004/2014 and UID/MAT/04106/2013 at CIDMA, respectively.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Santiago, R.H.N., Bedregal, B., Madeira, A., Martins, M.A. (2016). On Interval Dynamic Logic. In: Ribeiro, L., Lecomte, T. (eds) Formal Methods: Foundations and Applications. SBMF 2016. Lecture Notes in Computer Science(), vol 10090. Springer, Cham. https://doi.org/10.1007/978-3-319-49815-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-49815-7_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49814-0
Online ISBN: 978-3-319-49815-7
eBook Packages: Computer ScienceComputer Science (R0)