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Completeness for Identity-free Kleene Lattices

Amina Doumane, Damien Pous, Michael Wagner
2018 International Conference on Concurrency Theory  
Our proof builds on the completeness 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] .  ... 
doi:10.4230/lipics.concur.2018.18 dblp:conf/concur/DoumaneP18 fatcat:d2un5rbvxvfxlezgavrnchlqd4

Kleene Theorems for Product Systems [chapter]

Kamal Lodaya, Madhavan Mukund, Ramchandra Phawade
2011 Lecture Notes in Computer Science  
We prove 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.  ... 
doi:10.1007/978-3-642-22600-7_19 fatcat:llpsqaw6q5hormddm5g6kbkgnm

Feasibly Reducing KAT Equations to KA Equations [article]

James Worthington
2008 arXiv   pre-print
A 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.  ... 
arXiv:0801.2368v1 fatcat:zdr6bpjhx5gc7lksvbmkd4erni

Synchronous Kleene algebra

Cristian Prisacariu
2010 The Journal of Logic and Algebraic Programming  
The resulting algebraic structure is called synchronous 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.  ... 
doi:10.1016/j.jlap.2010.07.009 fatcat:osi22hgbefcrfleg2vmlbl7dr4

On Probabilistic Kleene Algebras, Automata and Simulations [chapter]

Annabelle McIver, Tahiry M. Rabehaja, Georg Struth
2011 Lecture Notes in Computer Science  
We show that a class of 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.  ... 
doi:10.1007/978-3-642-21070-9_20 fatcat:2x62n4brdbbefku4afxkvd4tvq

Timed regular expressions

Eugene Asarin, Paul Caspi, Oded Maler
2002 Journal of the ACM  
This result is the timed analogue of 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.  ... 
doi:10.1145/506147.506151 fatcat:wy2cjdv5dbbodaums2utyxcoye

A Probabilistic Kleene Theorem [chapter]

Benedikt Bollig, Paul Gastin, Benjamin Monmege, Marc Zeitoun
2012 Lecture Notes in Computer Science  
We provide a 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).  ... 
doi:10.1007/978-3-642-33386-6_31 fatcat:m4lnnln4gngxjmtqnsfs3yblz4

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]

Ramchandra Phawade, Kamal Lodaya
2015 Lecture Notes in Computer Science  
These systems have been related to labelled 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.  ... 
doi:10.1007/978-3-662-48650-4_5 fatcat:3epy2t6fkfgpniv3ybypss6w5m

Probabilistic Concurrent Kleene Algebra

Annabelle McIver, Tahiry Rabehaja, Georg Struth
2013 Electronic Proceedings in Theoretical Computer Science  
We provide an extension of concurrent 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.  ... 
doi:10.4204/eptcs.117.7 fatcat:tnzjyxycqzfxhke2o5ivgwgjbm

On the Coalgebraic Theory of Kleene Algebra with Tests [chapter]

Dexter Kozen
2017 Outstanding Contributions to Logic  
We develop a coalgebraic theory of 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.  ... 
doi:10.1007/978-3-319-47843-2_15 fatcat:z3jeflh7gve75apwm5blrhr4cq

A coalgebraic take on regular and ω-regular behaviours [article]

Tomasz Brengos
2021 arXiv   pre-print
We end the paper with coalgebraic 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).  ... 
arXiv:1902.02601v6 fatcat:n3lads47u5c2rgdcgjneidss6q

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.  ...  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  ... 
arXiv:1301.7153v1 fatcat:443dnyeoi5ff3l3my6w33fo34i

Petri Automata [article]

Paul Brunet, Damien Pous
2017 arXiv   pre-print
We prove a 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.  ... 
arXiv:1702.01804v2 fatcat:yqprcbsxsbh6hhss2abqvwfv24

A coalgebraic take on regular and $\omega$-regular behaviours

Tomasz Brengos
2021 Logical Methods in Computer Science  
We end the paper with coalgebraic 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).  ... 
doi:10.46298/lmcs-17(4:24)2021 fatcat:haemg5ias5c2fcia744klc266i
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