Abstract
Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally misses tests, which are needed to model standard programming constructs such as conditionals and \(\mathsf {while}\)-loops. It turns out that integrating tests in CKA is subtle, due to their interaction with parallelism. In this paper we provide a solution in the form of Concurrent Kleene Algebra with Observations (CKAO). Our main contribution is a completeness theorem for CKAO. Our result resorts on a more general study of CKA “with hypotheses”, of which CKAO turns out to be an instance: this analysis is of independent interest, as it can be applied to extensions of CKA other than CKAO.
Chapter PDF
Similar content being viewed by others
References
Anderson, C.J., Foster, N., Guha, A., Jeannin, J.B., Kozen, D., Schlesinger, C., Walker, D.: NetKAT: Semantic foundations for networks. In: POPL. pp. 113–126. ACM (2014)
Birkhoff, G., Bartee, T.C.: Modern applied algebra. McGraw-Hill (1970)
Bonchi, F., Pous, D.: Checking NFA equivalence with bisimulations up to congruence. In: POPL. pp. 457–468 (2013)
Brunet, P., Pous, D., Struth, G.: On decidability of concurrent Kleene algebra. In: CONCUR. pp. 28:1–28:15 (2017)
Cohen, E.: Hypotheses in Kleene algebra. Tech. rep., Bellcore (1994)
Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Ltd., London (1971)
Doumane, A., Kuperberg, D., Pous, D., Pradic, P.: Kleene algebra with hypotheses. In: FOSSACS. pp. 207–223 (2019)
Foster, N., Kozen, D., Milano, M., Silva, A., Thompson, L.: A coalgebraic decision procedure for NetKAT. In: POPL. pp. 343–355 (2015)
Gischer, J.L.: The equational theory of pomsets. Theor. Comput. Sci. 61, 199–224 (1988)
Grabowski, J.: On partial languages. Fundam. Inform. 4(2), 427 (1981)
Hoare, T., Möller, B., Struth, G., Wehrman, I.: Concurrent Kleene algebra. In: CONCUR. pp. 399–414 (2009)
Jipsen, P., Moshier, M.A.: Concurrent Kleene algebra with tests and branching automata. J. Log. Algebr. Meth. Program. 85(4), 637–652 (2016)
Kappé, T., Brunet, P., Rot, J., Silva, A., Wagemaker, J., Zanasi, F.: Kleene algebra with observations. In: CONCUR. pp. 41:1–41:16 (2019)
Kappé, T., Brunet, P., Silva, A., Wagemaker, J., Zanasi, F.: Concurrent Kleene algebra with observations: from hypotheses to completeness (2020), arXiv:2002.09682
Kappé, T., Brunet, P., Silva, A., Zanasi, F.: Concurrent Kleene algebra: Free model and completeness. In: ESOP. pp. 856–882 (2018)
Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Inf. Comput. 110(2), 366–390 (1994)
Kozen, D.: Kleene algebra with tests and commutativity conditions. In: TACAS. pp. 14–33 (1996)
Kozen, D.: On the complexity of reasoning in Kleene algebra. Inf. Comput. 179(2), 152–162 (2002)
Kozen, D.: On the coalgebraic theory of Kleene algebra with tests. In: Başkent, C., Moss, L.S., Ramanujam, R. (eds.) Rohit Parikh on Logic, Language and Society, Outstanding Contributions to Logic, vol. 11, pp. 279–298. Springer (2017)
Kozen, D., Mamouras, K.: Kleene algebra with equations. In: ICALP. pp. 280–292 (2014)
Krob, D.: A complete system of B-rational identities. In: ICALP. pp. 60–73 (1990)
Kuratowski, C.: Sur l’opération Ā de l’Analysis Situs. Fundamenta Mathematicae 3(1), 182–199 (1922)
Laurence, M.R., Struth, G.: Completeness theorems for bi-Kleene algebras and series-parallel rational pomset languages. In: RAMiCS. pp. 65–82 (2014)
Laurence, M.R., Struth, G.: Completeness theorems for pomset languages and concurrent Kleene algebras (2017), arXiv:1705.05896
Lodaya, K., Weil, P.: Series-parallel languages and the bounded-width property. Theoretical Computer Science 237(1), 347–380 (2000)
Prisacariu, C.: Synchronous Kleene algebra. The Journal of Logic and Algebraic Programming 79(7), 608 – 635 (2010)
Salomaa, A.: Two complete axiom systems for the algebra of regular events. J. ACM 13(1), 158–169 (1966)
Smolka, S., Foster, N., Hsu, J., Kappé, T., Kozen, D., Silva, A.: Guarded Kleene algebra with tests: verification of uninterpreted programs in nearly linear time. In: POPL. pp. 61:1–61:28 (2020)
Wagemaker, J., Bonsangue, M., Kappé, T., Rot, J., Silva, A.: Completeness and incompleteness of synchronous Kleene algebra. In: MPC (2019)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2020 The Author(s)
About this paper
Cite this paper
Kappé, T., Brunet, P., Silva, A., Wagemaker, J., Zanasi, F. (2020). Concurrent Kleene Algebra with Observations: From Hypotheses to Completeness. In: Goubault-Larrecq, J., König, B. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2020. Lecture Notes in Computer Science(), vol 12077. Springer, Cham. https://doi.org/10.1007/978-3-030-45231-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-45231-5_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-45230-8
Online ISBN: 978-3-030-45231-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
