Hostname: page-component-76d6cb85b7-7262s Total loading time: 0 Render date: 2026-07-10T16:27:46.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Convolution and concurrency

Published online by Cambridge University Press:  23 March 2022

James Cranch ,
Simon Doherty  and
Georg Struth Open the ORCID record for Georg Struth [Opens in a new window]
James Cranch
Affiliation:
University of Sheffield, Sheffield S10 2TN, UK
Simon Doherty
Affiliation:
University of Sheffield, Sheffield S10 2TN, UK
Georg Struth*
Affiliation:
University of Sheffield, Sheffield S10 2TN, UK
*
*Corresponding author. Email: g.struth@sheffield.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras of functions from relational structures with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras, among others. The elements of the quantales can be understood as weights; the case where weights are drawn from the booleans corresponds to languages. We develop a correspondence theory between properties of the relational structures and algebraic properties in the weight and convolution algebras in the sense of modal and substructural logics, or boolean algebras with operators. The resulting correspondence triangles yield in particular general construction principles for models of concurrent quantales and Kleene algebras as convolution algebras from much simpler relational structures, including weighted ones for quantitative applications. As examples, we construct the concurrent quantales and Kleene algebras of weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.

Keywords

Relational monoidsquantalesKleene algebrasconvolution algebrasconcurrency theory

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 8 , September 2021 , pp. 918 - 949
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramsky, S. and Jung, A. (1994). Domain theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science, vol. III, New York, Oxford University Press, 1–168.Google Scholar
Armstrong, A., Struth, G. and Weber, T. (2014). Programming and automating mathematics in the Tarski-Kleene hierarchy. Journal of Logical and Algebraic Methods in Programming 83 (2) 87–102.CrossRefGoogle Scholar
Berstel, J. and Reutenauer, C. (1984) Les séries rationnelles et leurs langagues, Paris, Masson.Google Scholar
Calk, C., Goubault, E., Malbos, P. and Struth, G. (2020). Algebraic coherent confluence and higher-dimensional globular Kleene algebras. CoRR, abs/2006.16129.Google Scholar
Connes, A. (1995). Noncommutative Geometry, San Diego, Academic Press.Google Scholar
Connes, A. and Consani, C. (2010). From monoids to hyperstructures: In search of an absolute arithmetic. In: Dijk, G. and Wakayama, M. (eds.) Casimir Force, Casimir Operators and the Riemann Hypothesis, Berlin New York, de Gruyter, 147–189.Google Scholar
Courcelle, B. and Engelfriet, J. (2012). Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Cranch, J., Doherty, S. and Struth, G. (2020), Relational semigroups and object-free categories. CoRR abs/2001.11895.Google Scholar
de Roever, W. P., de Boer, F. S., Hannemann, U., Hooman, J., Lakhnech, Y., Poel, M. and Zwiers, J. (2001). Concurrency Verification: Introduction to Compositional and Noncompositional Methods, Cambridge, Cambridge University Press.Google Scholar
Dongol, B., Gomes, V. B. F., Hayes, I. J. and Struth, G. (2017). Partial semigroups and convolution algebras. Archive of Formal Proofs.Google Scholar
Dongol, B., Hayes, I. J. and Struth, G. (2016). Convolution as a unifying concept: Applications in separation logic, interval calculi, and concurrency. ACM TOCL 17 (3) 15:1–15:25.CrossRefGoogle Scholar
Dongol, B., Hayes, I. J. and Struth, G. (2021). Convolution algebras: Relational convolution, generalised modalities and incidence algebras. Logical Methods in Computer Science 17 (1) 13:1–13:34.Google Scholar
Droste, M., Kuich, W. and Vogler, H. (eds.) (2009). Handbook of Weighted Automata , Berlin Heidelberg, Springer.Google Scholar
Eckmann, B. and Hilton, P. J. (1962). Group-like structures in general categories I. multiplications and comultiplications. Mathematische Annalen 145 (3) 227–255.CrossRefGoogle Scholar
Ésik, Z. (2002). Axiomatizing the subsumption and subword preorders on finite and infinite partial words. Theoretical Computer Science 273 (1–2) 225–248.CrossRefGoogle Scholar
Fahrenberg, U., Johansen, C., Struth, G. and Ziemiański, K. (2021a). Languages of higher-dimensional automata. Mathematical Structures in Computer Science 31 (5) 575–613.CrossRefGoogle Scholar
Fahrenberg, U., Johansen, C., Struth, G. and Ziemiański, K. (2021b). lr-Multisemigroups and modal convolution algebras. CoRR, abs/2105.00188.Google Scholar
Galmiche, D. and Larchey-Wendling, D. (2006). Expressivity properties of boolean BI through relational models. In: Arun-Kumar, S. and Garg, N. (eds.) FSTTCS 2006, LNCS, vol. 4337, Berlin Heidelberg, Springer, 357–368.Google Scholar
Gischer, J. L. (1988). The equational theory of pomsets. Theoretical Computer Science 61 199–224.CrossRefGoogle Scholar
Goguen, J. A. (1967). L-fuzzy sets. Journal of Mathematical Analysis and Applications 18 145–174.CrossRefGoogle Scholar
Goldblatt, R. (1989). Varieties of complex algebras. Annals of Pure and Applied Logic 44 173–242.CrossRefGoogle Scholar
Grabowski, J. (1981). On partial languages. Fundamentae Informaticae 4 427–498.CrossRefGoogle Scholar
Harding, J., Walker, C. and Walker, E. (2018). The convolution algebra. Algebra Universalis 79 (2) 33.CrossRefGoogle Scholar
Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Physik 33 879–893.CrossRefGoogle Scholar
Hoare, T., Möller, B., Struth, G. and Wehrman, I. (2011). Concurrent Kleene algebra and its foundations. Journal of Logic and Algebraic Programming 80 (6) 266–296.CrossRefGoogle Scholar
Jónsson, B. and Tarski, A. (1951). Boolean algebras with operators. Part I. American Journal of Mathematics 73 (4) 891–939.CrossRefGoogle Scholar
Kozen, D. (1994). A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110 (2) 366–390.CrossRefGoogle Scholar
Krasner, M. (1983). A class of hyperrings and hyperfields. International Journal of Mathematics and Mathematical Sciences 6 (2) 207–311.CrossRefGoogle Scholar
Kudryavtseva, G. and Mazorchuk, V. (2015). On multisemigroups. Portugaliae Mathematica 71 (1) 47–80.CrossRefGoogle Scholar
Kuich, W. and Salomaa, A. (1986). Semirings, Automata, Languages , Berlin Heidelberg, Springer.Google Scholar
Lambek, J. (1958). The mathematics of sentence structure. Americal Mathematical Monthly 65 154–170.CrossRefGoogle Scholar
Lamport, L. (1978). Time, clocks, and the ordering of events in a distributed system. Communication of ACM 21 (7) 558–565.CrossRefGoogle Scholar
Lang, S. (2003). Algebra, 3rd ed., New York, Springer.Google Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician, 2nd ed., New York, Springer.Google Scholar
Marty, F. (1934). Sur une généralisation de la notion de groupe. In: Comptes Rendus du 8ème Congrès des Mathématiciens Scandinaves, Lund, Håkon Ohlssons Boktryckeri, 45–49.Google Scholar
Moszkowski, B. C. and Manna, Z. (1983). Reasoning in interval temporal logic. In: Clark, E. M. and Kozen, D. (eds.) Logic of Programs, LNCS, vol. 164, Berlin Heidelberg, Springer, 371–382.Google Scholar
O’Hearn, P. W., Reynolds, J. C. and Yang, H. (2001). Local reasoning about programs that alter data structures. In: Fribourg, L. (ed.) CSL 2001, LNCS, vol. 2142, Berlin Heidelberg, Springer, 1–19.Google Scholar
Rosenthal, K. L. (1990). Quantales and Their Applications, Harlow, Essex, Longman Scientific $\&$ Technical.Google Scholar
Rota, G.-C. (1964). On the foundations of combinatorial theory I: Theory of Möbius functions. Zeitschrift für Wahrscheinli-chkeitstheorie und verwandte Gebiete 2 (4) 340–368.CrossRefGoogle Scholar
Sakarovitch, J. (2003). Éléments de Théorie des Automates, Paris, Vuibert.Google Scholar
Vogler, W. (1992). Modular Construction and Partial Order Semantics of Petri Nets , LNCS, vol. 625, Berlin Heidelberg, Springer.Google Scholar
von Wright, J. (2002). From Kleene algebra to refinement algebra. In: Boiten, E. A. and Möller, B. (eds.) MPC 2002, LNCS, vol. 2386, Berlin Heidelberg, Springer, 233–262.Google Scholar