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)


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.


  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). 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). 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). 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). 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