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Completeness for Identity-free Kleene Lattices

2018
*
International Conference on Concurrency Theory
*

Our proof builds on the completeness

doi:10.4230/lipics.concur.2018.18
dblp:conf/concur/DoumaneP18
fatcat:d2un5rbvxvfxlezgavrnchlqd4
*theorem**for**Kleene*algebra, and on a novel*automata*construction that makes it possible to extract axiomatic proofs using a*Kleene*-like algorithm. ... We provide a finite set of axioms*for*identity-*free**Kleene*lattices, which we prove sound and complete*for*the equational theory of their relational models. ... Second, this is the fragment*for*which the*Kleene**theorem**for*Petri*automata*is proved the most naturally [11] . ...##
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Kleene Theorems for Product Systems
[chapter]

2011
*
Lecture Notes in Computer Science
*

We prove

doi:10.1007/978-3-642-22600-7_19
fatcat:llpsqaw6q5hormddm5g6kbkgnm
*Kleene**theorems**for*two subclasses of labelled product systems which are inspired from well-studied subclasses of 1bounded Petri nets. ...*For*product*free**choice*systems with a restriction of structural cyclicity, that is, the initial global state is a feedback vertex set, going from systems to expressions is still polynomial time; in the ... Given a renaming ρ defining a*distributed**alphabet*C, a labelled product system of*automata*N [ρ] is a product system N*over*C. ...##
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Feasibly Reducing KAT Equations to KA Equations
[article]

2008
*
arXiv
*
pre-print

A

arXiv:0801.2368v1
fatcat:zdr6bpjhx5gc7lksvbmkd4erni
*Kleene*algebra with tests (KAT) is a*Kleene*algebra with an embedded Boolean subalgebra. ...*Kleene*algebra (KA) is the algebra of regular events. Familiar examples of*Kleene*algebras include regular sets, relational algebras, and trace algebras. ... In fact, the set of regular languages*over*an*alphabet*Σ is the*free**Kleene*algebra on Σ. ...##
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Synchronous Kleene algebra

2010
*
The Journal of Logic and Algebraic Programming
*

The resulting algebraic structure is called synchronous

doi:10.1016/j.jlap.2010.07.009
fatcat:osi22hgbefcrfleg2vmlbl7dr4
*Kleene*algebra. Models are given in terms of sets of synchronous strings and finite*automata*accepting synchronous strings. ... The extension of synchronous*Kleene*algebra with Boolean tests is presented together with models on sets of guarded synchronous strings and the associated*automata*on guarded synchronous strings. ... Thanks go also to Martin Steffen, Olaf Owe, and Gerardo Schneider*for*their useful comments. ...##
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On Probabilistic Kleene Algebras, Automata and Simulations
[chapter]

2011
*
Lecture Notes in Computer Science
*

We show that a class of

doi:10.1007/978-3-642-21070-9_20
fatcat:2x62n4brdbbefku4afxkvd4tvq
*automata*modulo simulation equivalence forms a model of probabilistic*Kleene*algebra. ... Hence an identity is valid in continuous probabilistic*Kleene*algebras if and only if the associated*automata*are simulation equivalent. ... We would like to thank Ernie Cohen and Hitoshi Furusawa*for*helpful discussions, and to Jeff Sanders*for*valuable comments. ...##
###
Timed regular expressions

2002
*
Journal of the ACM
*

This result is the timed analogue of

doi:10.1145/506147.506151
fatcat:wy2cjdv5dbbodaums2utyxcoye
*Kleene**Theorem*and, similarly to that result, the hard part in the proof is the translation from*automata*to expressions. ... In this paper we define timed regular expressions, a formalism*for*specifying discrete behaviors augmented with timing information, and prove that its expressive power is equivalent to the timed*automata*... equations*over*this*alphabet*of transitions is as simple as*for*untimed*automata*. ...##
###
A Probabilistic Kleene Theorem
[chapter]

2012
*
Lecture Notes in Computer Science
*

We provide a

doi:10.1007/978-3-642-33386-6_31
fatcat:m4lnnln4gngxjmtqnsfs3yblz4
*Kleene**Theorem**for*(Rabin) probabilistic*automata**over*finite words. Probabilistic*automata*generalize deterministic finite*automata*and assign to a word an acceptance probability. ... We provide probabilistic expressions with probabilistic*choice*, guarded*choice*, concatenation, and a star operator. ... This allows them to derive algebraic expressions and a corresponding*Kleene**Theorem*, as well as full axiomatizations*for*many of those (and even*for*weighted*automata**over*arbitrary semirings). ...##
###
Page 3869 of Mathematical Reviews Vol. , Issue 83i
[page]

1983
*
Mathematical Reviews
*

Let L be a

*distributive*lattice with zero and unit element. An L-fuzzy event*over*X is a mapping from X* into L where X* is the*free*monoid generated by X. (X is a finite*alphabet*.) ... The regular L-fuzzy events*over*X are characterized by a*Kleene**theorem*in strict analogy to the usual case. In a more general setting the*theorem*was proved by E. S. ...##
###
Kleene Theorems for Synchronous Products with Matching
[chapter]

2015
*
Lecture Notes in Computer Science
*

These systems have been related to labelled

doi:10.1007/978-3-662-48650-4_5
fatcat:3epy2t6fkfgpniv3ybypss6w5m
*free**choice*nets. 1 A preliminary version of this paper appeared at the 8th PNSE workshop in Tunis [PL14] . ... We thank the organizers*for*all the help provided*for*the presentation. ... We would also like to thank Jörg Desel*for*his patience as editor of the ToPNOC special issue. ...##
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Probabilistic Concurrent Kleene Algebra

2013
*
Electronic Proceedings in Theoretical Computer Science
*

We provide an extension of concurrent

doi:10.4204/eptcs.117.7
fatcat:tnzjyxycqzfxhke2o5ivgwgjbm
*Kleene*algebras to account*for*probabilistic properties. ... The algebra yields a unified framework containing nondeterminism, concurrency and probability and is sound with respect to the set of probabilistic*automata*modulo probabilistic simulation. ... Generally, a transition is composed of a state and a*distribution**over*Σ × P but restriction to*automata*with simple transitions suffices*for*our results in this paper. ...##
###
On the Coalgebraic Theory of Kleene Algebra with Tests
[chapter]

2017
*
Outstanding Contributions to Logic
*

We develop a coalgebraic theory of

doi:10.1007/978-3-319-47843-2_15
fatcat:z3jeflh7gve75apwm5blrhr4cq
*Kleene*algebra with tests (KAT) along the lines of Rutten (1998)*for**Kleene*algebra (KA) and Chen and Pucella (2003)*for*a limited version of KAT, resolving some technical ... Our treatment includes a simple definition of the Brzozowski derivative*for*KAT expressions and an*automata*-theoretic interpretation involving*automata*on guarded strings. ... The completeness result of [9] says that the algebra of regular sets of strings*over*a finite*alphabet*Σ is the*free**Kleene*algebra on generators Σ. ...##
###
A coalgebraic take on regular and ω-regular behaviours
[article]

2021
*
arXiv
*
pre-print

We end the paper with coalgebraic

arXiv:1902.02601v6
fatcat:n3lads47u5c2rgdcgjneidss6q
*Kleene*-type*theorems**for*(ω-) regular input. The framework is instantiated on non-deterministic (Büchi)*automata*, tree*automata*and probabilistic*automata*. ... The first part of the paper is devoted to presenting a construction of a monad suitable*for*modelling (in)finite behaviour. ...*Theorem*5.10 (*Kleene**theorem**for*regular behaviour). ...##
###
Weak Concurrent Kleene Algebra with Application to Algebraic Verification
[article]

2013
*
arXiv
*
pre-print

The algebra is proved sound with respect to a model of

arXiv:1301.7153v1
fatcat:443dnyeoi5ff3l3my6w33fo34i
*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. ... In a*Kleene*algebra, the semiring structure supports two*distributivity*laws: Left*distributivity*: xy + xz = x(y + z),(4) Right*distributivity*: (x + y)z = xz + yz. (5) Equation (4) the*choice*between ...##
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Petri Automata
[article]

2017
*
arXiv
*
pre-print

We prove a

arXiv:1702.01804v2
fatcat:yqprcbsxsbh6hhss2abqvwfv24
*Kleene**theorem**for*this*automata*model: the sets of graphs recognisable by Petri*automata*are precisely the sets of graphs definable through the extended regular expressions we consider. ... Petri*automata*allow us to obtain decidability of identity-*free*relational*Kleene*lattices, i.e., the equational theory generated by binary relations on the signature of regular expressions with intersection ... We thus restrict to Petri*automata**over*X and to simple expressions. At the algebraic level, this means we consider identity-*free**Kleene*lattices. ...##
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A coalgebraic take on regular and $\omega$-regular behaviours

2021
*
Logical Methods in Computer Science
*

We end the paper with coalgebraic

doi:10.46298/lmcs-17(4:24)2021
fatcat:haemg5ias5c2fcia744klc266i
*Kleene*-type*theorems**for*($\omega$-) regular input. The framework is instantiated on non-deterministic (B\"uchi)*automata*, tree*automata*and probabilistic*automata*. ... The first part of the paper is devoted to presenting a construction of a monad suitable*for*modelling (in)finite behaviour. ...*Theorem*5.10 (*Kleene**theorem**for*regular behaviour). ...
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