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.
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Notes
- 1.
This idea is confirmed in some Representation Theorems of Euclidean continuous functions.
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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.
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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
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