## Kleene Coalgebra -- an overview

### BibTeX

@MISC{Silva_kleenecoalgebra,

author = {Alexandra Silva and Marcello Bonsangue and Jan Rutten},

title = {Kleene Coalgebra -- an overview},

year = {}

}

### OpenURL

### Abstract

Coalgebras provide a uniform framework for the study of dynamical systems, including several types of automata. The coalgebraic view on systems has recently been proved relevant by the development of a number of expression calculi which generalize classical results by Kleene, on regular expressions, and by Kozen, on Kleene algebra. This note contains an overview of the motivation and results of the generic framework we developed – Kleene Coalgebra – to uniformly derive the aforementioned calculi. We present an historical overview of work on regular expressions and axiomatizations, as well a discussion of related work. We show applications of the framework to three types of probabilistic systems: simple Segala, stratified and Pnueli-Zuck.

### Citations

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Citation Context ...odel of computation in which non-determinism is allowed [39]. He introduced a language for finitely presented behaviors, which can be seen as a fragment of the calculus of communicating systems (CCS) =-=[38]-=-, and a sound and complete axiomatization with respect to bisimilarity. The paper of Milner served as inspiration for many researchers in the concurrency community. Probabilistic extensions of CCS, to... |

374 |
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(Show Context)
Citation Context ...ideline for the development of basic notions for new models of computation. One of the simplest models of computation is that of a deterministic finite automaton. In his seminal paper in 1956, Kleene =-=[32]-=- described finite deterministic automata (which he called nerve nets), together with a specification language: regular expressions. One of his most important results is the theorem which states that a... |

300 | Universal coalgebra: a theory of systems
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(Show Context)
Citation Context ...ults. More importantly, we should be able to derive new languages and axiomatizations. We combine the work of Kleene with coalgebra, hence the name Kleene coalgebra. The theory of universal coalgebra =-=[45]-=- provides a standard equivalence and a universal domain of behaviors, uniquely based on the type of the system, given by a functor F. It is our main aim to show how the type of the system also allows ... |

277 |
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Citation Context ...ular expressions, which would enable algebraic reasoning. The first answer to this question was given in 1966 by Salomaa [47], who presented two complete axiomatizations. The 1971 monograph of Conway =-=[21]-=- presents an extended overview of results on regular expressions and their axiomatizations. Later, in 1990, Kozen [33] showed that Salomaa’s axiomatization is non-algebraic, in the sense that it is un... |

271 | Probabilistic simulations for probabilistic processes
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(Show Context)
Citation Context ...he concurrency community. Probabilistic extensions of CCS, together with sound and complete axiomatizations (with respect to the appropriate notion of equivalence) have been presented for instance in =-=[22,23,51]-=-. In addition to the models already mentioned, other important models of computation include Mealy machines [37] (automata with input and output), automata on guarded strings [34] and weighted automat... |

212 | Derivatives of regular expressions
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Citation Context ...plication of regular expressions at that time, in the design of digital circuits, allowing a more easy conversion of a natural language specification of problems into a regular expression. Brzozowski =-=[17,18]-=- introduced the notion of derivative of regular expressions, which allowed him to prove Kleene’s theorem for the extended set of regular expressions without having to recur to non-deterministic automa... |

201 | The foundations of Esterel
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Citation Context ...and most beautiful) algorithms to translate regular expressions into automata was devised by Berry and Sethi [8]. This algorithm became the basis of one of the first compilers of the language Esterel =-=[7]-=-, a synchronous programming language, based on regular expressions, dedicated to embedded systems. Esterel is one of the most successful follow ups of Kleene’s work: it is used as a specification lang... |

186 | A completeness theorem for Kleene algebras and the algebra of regular events
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(Show Context)
Citation Context ...aa [47], who presented two complete axiomatizations. The 1971 monograph of Conway [21] presents an extended overview of results on regular expressions and their axiomatizations. Later, in 1990, Kozen =-=[33]-=- showed that Salomaa’s axiomatization is non-algebraic, in the sense that it is unsound under substitution of alphabet symbols by arbitrary regular expressions, and presented an algebraic axiomatizati... |

154 | Reactive, generative, and stratified models of probabilistic processes
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Citation Context ...m is equivalent to the following�� � �� � �� simpler one. � • 1/2 � 1/2 ����� �� • • • 1 a The language of expressions we propose for these systems is a subset of the language originally proposed in =-=[28]-=- (there a parallel composition operator is also considered). More interestingly, there was no axiomatization of the language in [28] and thus the axiomatization we present here is completely new. Pnue... |

122 |
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(Show Context)
Citation Context ...ce and formal models of computation play a central role. In 1981, Milner adapted Kleene and Salomaa’s results to labeled transition systems: a model of computation in which non-determinism is allowed =-=[39]-=-. He introduced a language for finitely presented behaviors, which can be seen as a fragment of the calculus of communicating systems (CCS) [38], and a sound and complete axiomatization with respect t... |

113 |
On the definition of a family of automata
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(Show Context)
Citation Context ...In addition to the models already mentioned, other important models of computation include Mealy machines [37] (automata with input and output), automata on guarded strings [34] and weighted automata =-=[50]-=-. These three models have applications, for instance, in digital circuit design, compiler optimization and image recognition, respectively. The aim of our work is to make use of the coalgebraic view o... |

104 |
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(Show Context)
Citation Context ...ect to the appropriate notion of equivalence) have been presented for instance in [22,23,51]. In addition to the models already mentioned, other important models of computation include Mealy machines =-=[37]-=- (automata with input and output), automata on guarded strings [34] and weighted automata [50]. These three models have applications, for instance, in digital circuit design, compiler optimization and... |

100 |
From regular expressions to deterministic automata
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- 1986
(Show Context)
Citation Context ...en regular expressions started to be widely used for pattern matching. One of the fastest (and most beautiful) algorithms to translate regular expressions into automata was devised by Berry and Sethi =-=[8]-=-. This algorithm became the basis of one of the first compilers of the language Esterel [7], a synchronous programming language, based on regular expressions, dedicated to embedded systems. Esterel is... |

94 |
Regular Expressions and State Graphs for Automata
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- 1960
(Show Context)
Citation Context ...n-algebraic, in the sense that it is unsound under substitution of alphabet symbols by arbitrary regular expressions, and presented an algebraic axiomatization: Kleene algebras. McNaughton and Yamada =-=[36]-=- gave algorithms to build a non-deterministic automaton from a regular expression and back, and introduced a notion of extended regular expression with intersection and complementation operators. This... |

89 | Coalgebraic logic
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(Show Context)
Citation Context ...ss and completeness of the axiomatization we will introduce for the set of generalized regular expressions. In the last few years several proposals for specification languages for coalgebras appeared =-=[40,43,30,29,20,11,12,49,35]-=-. The languages derived in our framework (for details see the thesis of the first author [52] or the joint papers with theother authors [13,15,14,54,10,53]) are similar in spirit to that of Rössiger ... |

78 |
Two complete axiom systems for the algebra of regular events
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(Show Context)
Citation Context ...ld exist a finite, sound and complete, axiomatization of the equivalence of regular expressions, which would enable algebraic reasoning. The first answer to this question was given in 1966 by Salomaa =-=[47]-=-, who presented two complete axiomatizations. The 1971 monograph of Conway [21] presents an extended overview of results on regular expressions and their axiomatizations. Later, in 1990, Kozen [33] sh... |

62 | Automata and coinduction (an exercise in coalgebra
- Rutten
- 1998
(Show Context)
Citation Context ...ession describing the behavior of s1, and an example of a valid axiom. 1.1 Related work The connection between Kleene’s regular expressions, deterministic automata and coalgebra was first explored in =-=[44,46]-=-. Rutten studied the coalgebraic structure of the set of regular expressions, given by Brzozowski derivatives [18], in order to show that the coalgebraic semantics coincides with the standard inductiv... |

55 | A Kleene Theorem for Timed Automata
- Asarin, Caspi, et al.
- 1997
(Show Context)
Citation Context ... of weighted automata) coincides with the set of rational formal power series. For timed automata, there were several proposals, including the papers by Bouyer and Petit [16], Asarin, Caspi and Maler =-=[4]-=-, and Asarin and Dima [5]. Recently, the results of Bouyer and Petit as well as those of Schützenberger have been extended to the class of weighted timed automata by Droste and Quaas [24]. Furthermore... |

50 |
Monadic computation and iterative algebraic theories
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(Show Context)
Citation Context ...cursive specifications and therefore formalize potentially infinite computations. This type of computations were studied also in the context of iterative theories, which have been introduced by Elgot =-=[25]-=-. The main example of an iterative theory is the theory of regular trees, that is trees which have finitely many distinct subtrees. Adámek, Milius and Velebil have presented Elgot’s work from a coalge... |

50 | Behavioural differential equations: a coinductive calculus of streams, automata, and power series, Theoret
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(Show Context)
Citation Context ...ession describing the behavior of s1, and an example of a valid axiom. 1.1 Related work The connection between Kleene’s regular expressions, deterministic automata and coalgebra was first explored in =-=[44,46]-=-. Rutten studied the coalgebraic structure of the set of regular expressions, given by Brzozowski derivatives [18], in order to show that the coalgebraic semantics coincides with the standard inductiv... |

48 | O.: Timed regular expressions
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(Show Context)
Citation Context ...incides with the set of rational formal power series. For timed automata, there were several proposals, including the papers by Bouyer and Petit [16], Asarin, Caspi and Maler [4], and Asarin and Dima =-=[5]-=-. Recently, the results of Bouyer and Petit as well as those of Schützenberger have been extended to the class of weighted timed automata by Droste and Quaas [24]. Furthermore, Kozen has extended Klee... |

33 | Coalgebras and modal logic
- Rößiger
(Show Context)
Citation Context ...ss and completeness of the axiomatization we will introduce for the set of generalized regular expressions. In the last few years several proposals for specification languages for coalgebras appeared =-=[40,43,30,29,20,11,12,49,35]-=-. The languages derived in our framework (for details see the thesis of the first author [52] or the joint papers with theother authors [13,15,14,54,10,53]) are similar in spirit to that of Rössiger ... |

31 |
Duality for logics of transition systems
- Bonsangue, Kurz
- 2005
(Show Context)
Citation Context ...ss and completeness of the axiomatization we will introduce for the set of generalized regular expressions. In the last few years several proposals for specification languages for coalgebras appeared =-=[40,43,30,29,20,11,12,49,35]-=-. The languages derived in our framework (for details see the thesis of the first author [52] or the joint papers with theother authors [13,15,14,54,10,53]) are similar in spirit to that of Rössiger ... |

28 | Iteration theories: the equational logic of iterative processes
- Bloom, Ésik
- 1993
(Show Context)
Citation Context ..., modulo the axioms is closely related to the work above: it is an initial iterative algebra. This also shows the connection of our work with the work by Bloom and Ésik on iterative algebras/theories =-=[9]-=-. Kleene’s theorem has been extended in various ways. Büchi [19] extended it to infinite words and ω-automata, introducing an ω operator on languages. Ochmanski [41] introduced a concurrent version of... |

26 |
Presenting functors by operations and equations
- Bonsangue, Kurz
- 2006
(Show Context)
Citation Context |

24 |
Regular behaviour of concurrent systems
- Ochma'nski
- 1985
(Show Context)
Citation Context ...Ésik on iterative algebras/theories [9]. Kleene’s theorem has been extended in various ways. Büchi [19] extended it to infinite words and ω-automata, introducing an ω operator on languages. Ochmanski =-=[41]-=- introduced a concurrent version of the Kleene star operator, which lead him to define a notion of co-rational languages, obtained as the rational ones by simply replacing the star by the concurrent i... |

23 | Axiomatizations for probabilistic finite-state behaviors
- Deng, Palamidessi
- 2007
(Show Context)
Citation Context ...he concurrency community. Probabilistic extensions of CCS, together with sound and complete axiomatizations (with respect to the appropriate notion of equivalence) have been presented for instance in =-=[22,23,51]-=-. In addition to the models already mentioned, other important models of computation include Mealy machines [37] (automata with input and output), automata on guarded strings [34] and weighted automat... |

21 | Modular construction of modal logics
- Cîrstea, Pattinson
- 2004
(Show Context)
Citation Context |

19 |
Free iterative theories: a coalgebraic view
- Adámek, Milius, et al.
(Show Context)
Citation Context ... of an iterative theory is the theory of regular trees, that is trees which have finitely many distinct subtrees. Adámek, Milius and Velebil have presented Elgot’s work from a coalgebraic perspective =-=[1,2]-=-, simplified some of his original proofs, and generalized the notion of free iterative theory to any finitary endofunctor of every locally presentable category. The language associated with each funct... |

17 | Automata on guarded strings and applications
- Kozen
- 2001
(Show Context)
Citation Context ...or instance in [22,23,51]. In addition to the models already mentioned, other important models of computation include Mealy machines [37] (automata with input and output), automata on guarded strings =-=[34]-=- and weighted automata [50]. These three models have applications, for instance, in digital circuit design, compiler optimization and image recognition, respectively. The aim of our work is to make us... |

16 | Compositional reasoning for probabilistic finite-state behaviors
- Deng, Palamidessi, et al.
- 2005
(Show Context)
Citation Context ...he concurrency community. Probabilistic extensions of CCS, together with sound and complete axiomatizations (with respect to the appropriate notion of equivalence) have been presented for instance in =-=[22,23,51]-=-. In addition to the models already mentioned, other important models of computation include Mealy machines [37] (automata with input and output), automata on guarded strings [34] and weighted automat... |

16 | Probabilistic verification by tableaux - Pnueli, Zuck - 1986 |

16 | Modular algorithms for heterogeneous modal logics
- Schröder, Pattinson
- 2007
(Show Context)
Citation Context |

13 | Kleene coalgebra
- Silva
- 2010
(Show Context)
Citation Context ...st few years several proposals for specification languages for coalgebras appeared [40,43,30,29,20,11,12,49,35]. The languages derived in our framework (for details see the thesis of the first author =-=[52]-=- or the joint papers with theother authors [13,15,14,54,10,53]) are similar in spirit to that of Rössiger [43], Jacobs [30], Pattinson and Schröder [49] in that we use the ingredients of a functor fo... |

12 | A Kleene/Büchi-like theorem for clock languages
- Bouyer, Petit
(Show Context)
Citation Context ...(corresponding to the behavior of weighted automata) coincides with the set of rational formal power series. For timed automata, there were several proposals, including the papers by Bouyer and Petit =-=[16]-=-, Asarin, Caspi and Maler [4], and Asarin and Dima [5]. Recently, the results of Bouyer and Petit as well as those of Schützenberger have been extended to the class of weighted timed automata by Drost... |

12 |
A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages
- Jacobs
(Show Context)
Citation Context ...ng equalities by coinduction. The coinductive proofs turned out to be, in many cases, more concise and intuitive than the alternative algebraic proof using the axioms of Kleene algebra. Later, Jacobs =-=[31]-=- presented a bialgebraic review on deterministic automata and regular expressions, which allowed him to present an alternative coalgebraic proof of Kozen’s result on the completeness of Kleene algebra... |

12 | Coalgebraic automata theory: basic results
- Kupke, Venema
(Show Context)
Citation Context |

10 | A.: Coalgebraic logic and synthesis of Mealy machines
- Bonsangue, Rutten, et al.
- 2008
(Show Context)
Citation Context ... languages for coalgebras appeared [40,43,30,29,20,11,12,49,35]. The languages derived in our framework (for details see the thesis of the first author [52] or the joint papers with theother authors =-=[13,15,14,54,10,53]-=-) are similar in spirit to that of Rössiger [43], Jacobs [30], Pattinson and Schröder [49] in that we use the ingredients of a functor for typing expressions. They differ from the logics presented in ... |

10 | A Kleene Theorem for Polynomial Coalgebras
- Bonsangue, Rutten, et al.
- 2009
(Show Context)
Citation Context ... languages for coalgebras appeared [40,43,30,29,20,11,12,49,35]. The languages derived in our framework (for details see the thesis of the first author [52] or the joint papers with theother authors =-=[13,15,14,54,10,53]-=-) are similar in spirit to that of Rössiger [43], Jacobs [30], Pattinson and Schröder [49] in that we use the ingredients of a functor for typing expressions. They differ from the logics presented in ... |

10 |
An Extension of Kleene’s and Ochmanski’s Theorems to Infinite Traces
- Gastin, Petit, et al.
- 1994
(Show Context)
Citation Context ...ly replacing the star by the concurrent iteration. He then generalized Kleene’s theorem showing that the recognizable trace languages are exactly the co-rational languages. Gastin, Petit and Zielonka =-=[26,27]-=- extended Ochmanski’s results to infinite trace languages. For weighted automata, Schützenberger [50] has shown that the set of recognizable formal power series (corresponding to the behavior of weigh... |

7 | Deriving syntax and axioms for quantitative regular behaviours
- Bonchi, Bonsangue, et al.
- 2009
(Show Context)
Citation Context ... languages for coalgebras appeared [40,43,30,29,20,11,12,49,35]. The languages derived in our framework (for details see the thesis of the first author [52] or the joint papers with theother authors =-=[13,15,14,54,10,53]-=-) are similar in spirit to that of Rössiger [43], Jacobs [30], Pattinson and Schröder [49] in that we use the ingredients of a functor for typing expressions. They differ from the logics presented in ... |

7 | An algebra for Kripke polynomial coalgebras
- Bonsangue, Rutten, et al.
- 2009
(Show Context)
Citation Context |

7 | Equational logic of polynomial coalgebras
- Goldblatt
(Show Context)
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6 |
A survey of regular expressions and their applications
- Brzozowski
- 1962
(Show Context)
Citation Context ...plication of regular expressions at that time, in the design of digital circuits, allowing a more easy conversion of a natural language specification of problems into a regular expression. Brzozowski =-=[17,18]-=- introduced the notion of derivative of regular expressions, which allowed him to prove Kleene’s theorem for the extended set of regular expressions without having to recur to non-deterministic automa... |

4 |
Iterative algebras at work
- ADÁMEK, MILIUS, et al.
- 2006
(Show Context)
Citation Context ... of an iterative theory is the theory of regular trees, that is trees which have finitely many distinct subtrees. Adámek, Milius and Velebil have presented Elgot’s work from a coalgebraic perspective =-=[1,2]-=-, simplified some of his original proofs, and generalized the notion of free iterative theory to any finitary endofunctor of every locally presentable category. The language associated with each funct... |

4 |
A Kleene theorem for infinite trace languages
- Gastin, Petit, et al.
- 1991
(Show Context)
Citation Context ...ly replacing the star by the concurrent iteration. He then generalized Kleene’s theorem showing that the recognizable trace languages are exactly the co-rational languages. Gastin, Petit and Zielonka =-=[26,27]-=- extended Ochmanski’s results to infinite trace languages. For weighted automata, Schützenberger [50] has shown that the set of recognizable formal power series (corresponding to the behavior of weigh... |

1 |
On a decision method in restricted second order arithmetic
- Buechi
- 1962
(Show Context)
Citation Context ... an initial iterative algebra. This also shows the connection of our work with the work by Bloom and Ésik on iterative algebras/theories [9]. Kleene’s theorem has been extended in various ways. Büchi =-=[19]-=- extended it to infinite words and ω-automata, introducing an ω operator on languages. Ochmanski [41] introduced a concurrent version of the Kleene star operator, which lead him to define a notion of ... |

1 |
A Kleene-Schützenberger theorem for weighted timed automata
- Droste, Quaas
(Show Context)
Citation Context ...spi and Maler [4], and Asarin and Dima [5]. Recently, the results of Bouyer and Petit as well as those of Schützenberger have been extended to the class of weighted timed automata by Droste and Quaas =-=[24]-=-. Furthermore, Kozen has extended Kleene’s language with Boolean tests as a finitary representation of regular sets of guarded strings and proved an analogue of Kleene’s theorem for automata on guarde... |