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Equivalence checking for weak bi-Kleene algebra [article]

Tobias Kappé and Paul Brunet and Bas Luttik and Alexandra Silva and Fabio Zanasi
2021 arXiv   pre-print
Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a  ...  We characterize a fragment of pomset automata that admits a decision procedure for language equivalence.  ...  We would like to thank the anonymous reviewers for their insightful comments, which helped to substantially improve the initial version of this paper.  ... 
arXiv:1807.02102v5 fatcat:pgnylodopvddplmfzjfqrdgjt4

Convolution and Concurrency [article]

James Cranch and Simon Doherty and Georg Struth
2020 arXiv   pre-print
concurrent quantales or Kleene algebras Q.  ...  We show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras Q^X of functions from structures X with two ternary relations that satisfy relational interchange laws into  ...  Acknowledgement: The authors would like to thank Tony Hoare for discussions on models of concurrent Kleene algebras, and to Brijesh Dongol and Ian Hayes for their collaboration on convolution algebras  ... 
arXiv:2002.02321v1 fatcat:dyjzozvmhzd75cmus2v6hhpwxa

Concurrent Kleene algebra with tests and branching automata

Peter Jipsen, M. Andrew Moshier
2016 Journal of Logical and Algebraic Methods in Programming  
Highlights • Guarded strings are generalized to guarded series-parallel strings giving a concrete language model for concurrent Kleene algebra with tests that satisfies the weak exchange law. • We define  ...  Abstract We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Möller  ...  In general it can be difficult to check this equivalence in the free algebra, but for KA and KAT the semantic models of all rational languages and all guarded rational languages give a concrete representation  ... 
doi:10.1016/j.jlamp.2015.12.005 fatcat:nhu7vxfcijcwbhvixcyulxnfrq

Concurrent Dynamic Algebra [article]

Hitoshi Furusawa, Georg Struth
2014 arXiv   pre-print
The modal operators of concurrent dynamic algebra are obtained from abstract axioms for domain and antidomain operators; the Kleene star is modelled as a least fixpoint.  ...  We reconstruct Peleg's concurrent dynamic logic in the context of modal Kleene algebras.  ...  at Kyushu University for a pleasant short stay and financial support.  ... 
arXiv:1407.5819v1 fatcat:bfbftlau55czhd6kg6zmqkxiym

Weak Concurrent Kleene Algebra with Application to Algebraic Verification [article]

Annabelle McIver and Tahiry Rabehaja and Georg Struth
2013 arXiv   pre-print
The algebra is proved sound with respect to a model of automata modulo a variant of rooted η-simulation equivalence.  ...  We propose a generalisation of concurrent Kleene algebra Hoa09 that can take account of probabilistic effects in the presence of concurrency.  ...  (Subsection 3.2) forms a weak concurrent Kleene algebra.  ... 
arXiv:1301.7153v1 fatcat:443dnyeoi5ff3l3my6w33fo34i

PBZ*-Lattices: Structure Theory and Subvarieties [article]

Roberto Giuntini, Claudia Mureşan, Francesco Paoli
2019 arXiv   pre-print
These lattices with additional structure originate in the foundations of quantum mechanics and can be viewed as a common generalisation of orthomodular lattices and Kleene algebras expanded by an extra  ...  The class of bounded involution lattices is a variety, here denoted by BI. The involution of a pseudo-Kleene algebra is called Kleene complement.  ...  The variety of pseudo-Kleene algebras, for which see e.g. [10] , is denoted by PKA. Distributive pseudo-Kleene algebras are variously called Kleene lattices or Kleene algebras in the literature.  ... 
arXiv:1804.08136v3 fatcat:v4xz2xyi7bf7rlftjcknijg52q

Layered graph traversals and Hamiltonian path problems — An algebraic approach [chapter]

Thomas Brunn, Bernhard Möller, Martin Russling
1998 Lecture Notes in Computer Science  
In the rst case, we are even able to abstract to the very general setting of Kleene algebras.  ...  Using an algebra of paths we present abstract algebraic derivations for two problem classes concerning graphs, viz. layer oriented traversal and computing sets of Hamiltonian paths.  ...  The pointwise extension of join yields the following result for unary relations S und T: S 1 T = S \ T : (3) 3 Kleene Algebras and Closures Kleene Algebras A Kleene algebra is a quintuple (S; ; ; 0;1  ... 
doi:10.1007/bfb0054287 fatcat:bxgkpwzkf5bitiodnkhpdhnleu

PBZ*-Lattices: Structure Theory and Subvarieties

Roberto Giuntini, Claudia Mureşan, Francesco Paoli
2020 Reports on Mathematical Logic  
These lattices with additional structure originate in the foundations of quantum mechanics and can be viewed as a common generalisation of orthomodular lattices and Kleene algebras expanded by an extra  ...  Finally, we thank an anonymous referee for her/his precious comments.  ...  The class of bounded involution lattices is a variety, here denoted by BI. The involution of a pseudo-Kleene algebra is called Kleene complement.  ... 
doi:10.4467/20842589rm.20.001.12433 fatcat:lei2p5utx5dnxisbyqmuogrq4i

Commutative integral bounded residuated lattices with an added involution

Roberto Cignoli, Francesc Esteva
2009 Annals of Pure and Applied Logic  
Since these conditions are rather restrictive (for instance, on a symmetric Heyting algebra ε is an interior operator if and only the equation (x → 0) ∨ ((x → 0) → 0) = 1 is satisfied) we consider when  ...  We give necessary and sufficient conditions for ε being an interior operator.  ...  (ii) A satisfies the Kleene equation (2.16). (iii) A is normal. Hence it follows from Lemmas 1.5 and 1.6 that (i) and (ii) are equivalent.  ... 
doi:10.1016/j.apal.2009.05.008 fatcat:i4yxfu54sfgltgorrokwtsfok4

Concurrent Kleene Algebra and its Foundations

Tony Hoare, Bernhard Möller, Georg Struth, Ian Wehrman
2011 The Journal of Logic and Algebraic Programming  
A Concurrent Kleene Algebra offers two composition operators, related by a weak version of an exchange law: when applied in a trace model of program semantics, one of them stands for sequential execution  ...  On this basis, we develop a series of richer algebras; the richest validates a proof calculus for programs similar to that of a Jones style rely/guarantee calculus.  ...  Acknowledgements We are grateful for valuable comments by J. Desharnais  ... 
doi:10.1016/j.jlap.2011.04.005 fatcat:7zx4n42mojaqhaktidvrdzl6cq

On the Fine-Structure of Regular Algebra

Simon Foster, Georg Struth
2014 Journal of automated reasoning  
We use Isabelle/HOL for a detailed systematic study of the regular algebra axioms given by Boffa, Conway, Kozen and Salomaa.  ...  Regular algebra is the algebra of regular expressions as induced by regular language identity.  ...  We would like to thank Christian Urban and Tobias Nipkow for information on the implementation of regular languages in Isabelle, and the anonymous referees for comments and suggestions that helped us to  ... 
doi:10.1007/s10817-014-9318-9 fatcat:tt7toa7wwbdghgymeqowvzxkdu

Monoids with tests and the algebra of possibly non-halting programs [article]

Marcel Jackson, Tim Stokes
2014 arXiv   pre-print
composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions.  ...  All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigroup.  ...  Within weak comparison monoids, quasi-equational law (32) for disagreeable monoids is equivalent to the following equational laws: T (X) is closed under weak comparison, where the operation is called  ... 
arXiv:1408.4498v1 fatcat:vf2a5y2vufferblustxp3cwtwi

Monoids with tests and the algebra of possibly non-halting programs

Marcel Jackson, Tim Stokes
2015 Journal of Logical and Algebraic Methods in Programming  
composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions.  ...  All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigroup.  ...  Within weak comparison monoids, quasi-equational law (32) for disagreeable monoids is equivalent to the following equational laws: T (X) is closed under weak comparison, where the operation is called  ... 
doi:10.1016/j.jlamp.2014.08.007 fatcat:6g36upllqnhihjjxt5l3hbomxi

Concurrent Kleene Algebra with Tests [chapter]

Peter Jipsen
2014 Lecture Notes in Computer Science  
Concurrent Kleene algebras were introduced by Hoare, Möller, Struth and Wehrman in [HMSW09,HMSW09a,HMSW11] as idempotent bisemirings that satisfy a concurrency inequation and have a Kleene-star for both  ...  Concurrent Kleene algebras with tests (CKAT) combine these concepts and give a relatively simple algebraic model for reasoning about operational semantics of concurrent programs.  ...  The bi-Kleene algebra of series-rational gsp-languages 1 , denoted by C Σ,T , is the subalgebra generated by {G(t) : t ∈ T } ∪ {G(p) : p ∈ Σ}.  ... 
doi:10.1007/978-3-319-06251-8_3 fatcat:tnekgiex4jfsbpeesa52q2vt7a

Convolution, Separation and Concurrency [article]

Brijesh Dongol and Ian J. Hayes and Georg Struth
2014 arXiv   pre-print
A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales.  ...  separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for  ...  over X, for instance a Kleene algebra with constants from X.  ... 
arXiv:1410.4235v1 fatcat:lk4kbkkrkzdq3p4ng677vabn44
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