Completeness and Incompleteness of Synchronous Kleene Algebra

Wagemaker, Jana; Bonsangue, Marcello M.; Kappé, Tobias; Rot, Jurriaan; Silva, Alexandra

doi:10.1007/978-3-030-33636-3_14

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…included an inference rule with a side condition that prevented it from being algebraic in the sense that the validity of an equation is not preserved when substituting letters for arbitrary regular expressions. Nevertheless, this inspired axiomatizations of several variations and extensions of Kleene algebra [48,44,43], as well as Milner's axiomatization of the algebra of star behaviours [34]. The side condition introduced by Salomaa is often called the empty word property, an early version of a concept from process theory called guardedness 9 that is also fundamental to the theory of iteration [7].…”

Section: Related Workmentioning

confidence: 99%

A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests

Schmid¹,

Kappé²,

Silva³

2023

Preprint

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Guarded Kleene Algebra with Tests (GKAT) is a fragment of Kleene Algebra with Tests (KAT) that was recently introduced to reason efficiently about imperative programs. In contrast to KAT, GKAT does not have an algebraic axiomatization, but relies on an analogue of Salomaa's axiomatization of Kleene Algebra. In this paper, we present an algebraic axiomatization and prove two completeness results for a large fragment of GKAT in the form of skip-free programs.5 A probabilistic semantics in terms of sub-Markov kernels is also possible [44].

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Section: Related Workmentioning

confidence: 99%

A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests

Schmid¹,

Kappé²,

Silva³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…included an inference rule with a side condition that prevented it from being algebraic in the sense that the validity of an equation is not preserved when substituting letters for arbitrary regular expressions. Nevertheless, this inspired axiomatizations of several variations and extensions of Kleene algebra [46,42,41], as well as Milner's axiomatization of the algebra of star behaviours [32]. The side condition introduced by Salomaa is often called the empty word property, an early version of a concept from process theory called guardedness 9 that is also fundamental to the theory of iteration [6].…”

Section: Related Workmentioning

confidence: 99%

A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests

Schmid

Kappé

Silva

2023

Lecture Notes in Computer Science

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Guarded Kleene Algebra with Tests (GKAT) is a fragment of Kleene Algebra with Tests (KAT) that was recently introduced to reason efficiently about imperative programs. In contrast to KAT, GKAT does not have an algebraic axiomatization, but relies on an analogue of Salomaa’s axiomatization of Kleene Algebra. In this paper, we present an algebraic axiomatization and prove two completeness results for a large fragment of GKAT consisting of skip-free programs.

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Concurrent Kleene Algebra with Observations: From Hypotheses to Completeness

Kappé

Brunet

Silva

et al. 2020

Lecture Notes in Computer Science

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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 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. PreliminariesWe recall basic definitions on pomset languages, used in the semantics of CKA, which generalise languages to allow letters in words to be partially ordered. We fix a (possibly infinite) alphabet Σ. When defining sets parametrised by Σ, say S(Σ), if Σ is clear from the context we use S to refer to S(Σ). Pomsets [9,10] are labelled posets, up to isomorphism. Posets and PomsetsDefinition 2.1 (Labellet poset). A labelled poset over Σ is a tuple u = S, ≤, λ , where S is a finite set (the carrier of u), ≤ u is a partial order on S (the order of u), and λ : S → Σ is a function (the labelling of u).

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Completeness and Incompleteness of Synchronous Kleene Algebra

Cited by 3 publications

References 23 publications

A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests

A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests

A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests

Concurrent Kleene Algebra with Observations: From Hypotheses to Completeness

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