Advertisement

On the Construction of Multi-valued Concurrent Dynamic Logics

  • Leandro GomesEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12005)

Abstract

Dynamic logic is a powerful framework for reasoning about imperative programs. An extension with a concurrent operator, called concurrent propositional dynamic logic (CPDL) [20], was introduced to formalise programs running in parallel. In a different direction, other authors proposed a systematic method for generating multi-valued propositional dynamic logics to reason about weighted programs [15]. This paper presents the first step of combining these two frameworks to introduce uncertainty in concurrent computations. In the proposed framework, a weight is assigned to each branch of the parallel execution, resulting in a (possible) asymmetric parallelism, inherent to the fuzzy programming paradigm [2, 23]. By adopting such an approach, a family of logics is obtained, called multi-valued concurrent propositional dynamic logics (\(\mathcal {GCDL}(\mathbf {A})\)), parametric on an action lattice \(\mathbf {A}\) specifying a notion of “weight” assigned to program execution. Additionally, the validity of some axioms of CPDL is discussed in the new family of generated logics.

References

  1. 1.
    Baltag, A., Smets, S.: The dynamic turn in quantum logic. Synthese 186(3), 753–773 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cingolani, P., Alcalá-Fdez, J.: jFuzzylogic: a Java library to design fuzzy logic controllers according to the standard for fuzzy control programming. Int. J. Comput. Intell. Syst. 6(sup1), 61–75 (2013)CrossRefGoogle Scholar
  3. 3.
    Conway, J.: Regular Algebra and Finite Machines. Dover Publications, New York (1971)zbMATHGoogle Scholar
  4. 4.
    den Hartog, J., de Vink, E.P.: Verifying probabilistic programs using a Hoare like logic. Int. J. Found. Comput. Sci. 13(3), 315–340 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    den Hartog, J.I.: Probabilistic extensions of semantical models. Ph.D. thesis, Vrije Universiteit, Vrije (2002)Google Scholar
  6. 6.
    Furusawa, H., Kawahara, Y., Struth, G., Tsumagari, N.: Kleisli, Parikh and Peleg compositions and liftings for multirelations. J. Log. Algebraic Methods Program. 90, 84–101 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Furusawa, H., Struth, G.: Taming multirelations. ACM Trans. Comput. Log. 17(4), 28:1–28:34 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gomes, L.: On the construction of multi-valued concurrent dynamic logic. CoRR abs/1911.00462 (2019)Google Scholar
  9. 9.
    Gomes, L., Madeira, A., Barbosa, L.S.: Generalising KAT to verify weighted computations. CoRR abs/1911.01146 (2019)CrossRefGoogle Scholar
  10. 10.
    Hoare, C.A.R.T., Möller, B., Struth, G., Wehrman, I.: Concurrent Kleene algebra. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 399–414. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-04081-8_27CrossRefGoogle Scholar
  11. 11.
    Jipsen, P., Moshier, M.A.: Concurrent Kleene algebra with tests and branching automata. J. Log. Algebraic Method Program. 85(4), 637–652 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kozen, D.: A probabilistic PDL. J. Comput. Syst. Sci. 30(2), 162–178 (1985)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kozen, D.: On action algebras. In: Logic and the Flow of Information, Amsterdam (1993)Google Scholar
  14. 14.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Inf. Comput. 110, 366–390 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Madeira, A., Neves, R., Martins, M.A.: An exercise on the generation of many-valued dynamic logics. JLAMP 1, 1–29 (2016)zbMATHGoogle Scholar
  16. 16.
    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, Cham (2015).  https://doi.org/10.1007/978-3-319-15075-8_9CrossRefGoogle Scholar
  17. 17.
    Martin, C.E., Curtis, S.A., Rewitzky, I.: Modelling angelic and demonic nondeterminism with multirelations. Sci. Comput. Program. 65(2), 140–158 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    McIver, A., Rabehaja, T., Struth, G.: An event structure model for probabilistic concurrent Kleene algebra. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 653–667. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-45221-5_43CrossRefzbMATHGoogle Scholar
  19. 19.
    Parikh, R.: Propositional game logic. In: 24th Annual Symposium on Foundations of Computer Science, pp. 195–200. IEEE Computer Society (1983)Google Scholar
  20. 20.
    Peleg, D.: Concurrent dynamic logic. J. ACM 34(2), 450–479 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Platzer, A.: Logical Analysis of Hybrid Systems - Proving Theorems for Complex Dynamics. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-14509-4CrossRefzbMATHGoogle Scholar
  22. 22.
    Qiao, R., Wu, J., Wang, Y., Gao, X.: Operational semantics of probabilistic Kleene algebra with tests. In: Proceedings - IEEE Symposium on Computers and Communications, pp. 706–713 (2008)Google Scholar
  23. 23.
    Vetterlein, T., Mandl, H., Adlassnig, K.: Fuzzy arden syntax: a fuzzy programming language for medicine. Artif. Intell. Med. 49(1), 1–10 (2010)CrossRefGoogle Scholar
  24. 24.
    Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.HASLab INESC TECUniv. MinhoBragaPortugal

Personalised recommendations