## A Kleene theorem for polynomial coalgebras (2009)

Venue: | In Foundations of Software Science and Computational Structures, 12th International Conference, FOSSACS 2009, volume 5504 of LNCS |

Citations: | 11 - 3 self |

### BibTeX

@INPROCEEDINGS{Bonsangue09akleene,

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

title = {A Kleene theorem for polynomial coalgebras},

booktitle = {In Foundations of Software Science and Computational Structures, 12th International Conference, FOSSACS 2009, volume 5504 of LNCS},

year = {2009},

pages = {122--136}

}

### OpenURL

### Abstract

Abstract. For polynomial functors G, we show how to generalize the classical notion of regular expression to G-coalgebras. We introduce a language of expressions for describing elements of the final G-coalgebra and, analogously to Kleene’s theorem, we show the correspondence between expressions and finite G-coalgebras. 1

### Citations

4242 |
Introduction to Automata Theory, Languages and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...etween regular expressions and (non-)deterministic automata has been widely studied and a translation between these two different formalisms is presented in most books on automata and language theory =-=[10,6]-=-. Formally, a deterministic automaton consists of a set of states S equipped with a transition function δ : S → 2 × S A determining for each state whether or not it is final and assigning to each inpu... |

402 |
Representation of events in nerve nets and finite automata, in Automata studies
- Kleene
- 1956
(Show Context)
Citation Context ... the final G-coalgebra and, analogously to Kleene’s theorem, we show the correspondence between expressions and finite G-coalgebras. 1 Introduction Regular expressions were first introduced by Kleene =-=[8]-=- to study the properties of neural networks. They are an algebraic description of languages, offering a declarative way of specifying the strings to be recognized and they define exactly the same clas... |

340 | Universal coalgebra: A theory of systems
- Rutten
- 2000
(Show Context)
Citation Context ...ting of a set of states S and a transition function g : S → GS , where the functor G determines the type of the dynamic system under consideration and is the base of the theory of universal coalgebra =-=[14]-=-. The central concepts in this theory are homomorphism of coalgebras, bisimulation equivalence and final coalgebra. These can be seen, respectively, as generalizations of automata homomorphism, langua... |

234 | Derivatives of regular expressions
- Brzozowski
- 1964
(Show Context)
Citation Context ...s both arise as the unique homomorphism into the final coalgebra of formal languages. The coalgebra structure on the set of regular expressions is determined by their so-called Brzozowski derivatives =-=[4]-=-. In the present paper, the set of expressions for the functor F (S) =2× S A differs from the classical definition in that we do not have Kleene star and full concatenation (sequential composition) bu... |

205 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ... as our ⊕ operator, and, more importantly, will not allow for an easy and modular axiomatization of bisimulation. Providing such a complete finite axiomatization generalizing the results presented in =-=[9,5]-=- is subject of our current research. This will provide a generalization of Kleene algebra to polynomial coalgebras. Further, we would like to deal with non-deterministic systems (which amounts to incl... |

167 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...ing w with the letter a. The notion of finality will play a key role later in providing a semantics to expressions. Let (S, f )and(T, g) betwoG-coalgebras. We call a relation R ⊆ S × T a bisimulation =-=[1]-=- if there exists a map e : R → GR such that the projections π1 and π2 are coalgebra homomorphisms, i.e. the following diagram commutes. S �� π1 R π2 �� T f ∃e GS GR Gπ1 Gπ2 �� We write s ∼G t whenever... |

140 |
Automata and Computability
- Kozen
- 1997
(Show Context)
Citation Context ...etween regular expressions and (non-)deterministic automata has been widely studied and a translation between these two different formalisms is presented in most books on automata and language theory =-=[10,6]-=-. Formally, a deterministic automaton consists of a set of states S equipped with a transition function δ : S → 2 × S A determining for each state whether or not it is final and assigning to each inpu... |

66 | Automata and coinduction (an exercise in coalgebra
- Rutten
- 1466
(Show Context)
Citation Context ...he functor G would be instantiated to 2 × Id A and the usual notions would be recovered. In particular, note that the final coalgebra for this functor is precisely the set 2A∗ of all languages over A =-=[15]-=-. Given the fact that coalgebras can be seen as generalizations of deterministic automata, it is natural to investigate whether there exists an appropriate ⋆ Partially supported by the Fundação para a... |

56 | Many-sorted coalgebraic modal logic: a model-theoretic study
- Jacobs
(Show Context)
Citation Context ...he case. Also the set of classical regular expressions has a join-semilattice structure, which provides also intuition for the differences in our definition of polynomial functors, when compared with =-=[13,7]-=-, in the use of a join-semilattice as constant and in the definition of +. If we want to generalize regular expressions to polynomial functors then we must guarantee that they also have such structure... |

36 | Coalgebras and modal logic
- Rößiger
- 2000
(Show Context)
Citation Context ...he case. Also the set of classical regular expressions has a join-semilattice structure, which provides also intuition for the differences in our definition of polynomial functors, when compared with =-=[13,7]-=-, in the use of a join-semilattice as constant and in the definition of +. If we want to generalize regular expressions to polynomial functors then we must guarantee that they also have such structure... |

18 | On the coalgebraic theory of Kleene algebra with tests
- Kozen
- 2008
(Show Context)
Citation Context ...expressions and their treatment in [15] can be viewed as a special instance of the present approach. This is also the case for the generalization of the results in [15] to automata on guarded strings =-=[11]-=-. Finally, the present paper extends the results in our FoSSaCS’08 paper [3], where a sound and complete specification language and a synthesis algorithm for Mealy machines is given. Mealy machines ar... |

15 |
Coalgebras and their logics
- Kurz
(Show Context)
Citation Context ...formalisms. Ordinary regular expressions are closed under intersection and complement. We would like to study whether a similar result can be obtained for our language. Coalgebraic modal logics (CML) =-=[12]-=- have been presented as a general theory for reasoning about transition systems. The connection between our language and CML is also subject of further study. Acknowledgements. The authors are gratefu... |

10 | A.: Coalgebraic logic and synthesis of Mealy machines
- Bonsangue, Rutten, et al.
- 2008
(Show Context)
Citation Context ...f the present approach. This is also the case for the generalization of the results in [15] to automata on guarded strings [11]. Finally, the present paper extends the results in our FoSSaCS’08 paper =-=[3]-=-, where a sound and complete specification language and a synthesis algorithm for Mealy machines is given. Mealy machines are coalgebras of the functor (B × Id) A ,whereA is a finite input alphabet an... |

5 |
Axiomatizing the equational theory of regular tree languages
- Ésik
- 2010
(Show Context)
Citation Context ... as our ⊕ operator, and, more importantly, will not allow for an easy and modular axiomatization of bisimulation. Providing such a complete finite axiomatization generalizing the results presented in =-=[9,5]-=- is subject of our current research. This will provide a generalization of Kleene algebra to polynomial coalgebras. Further, we would like to deal with non-deterministic systems (which amounts to incl... |

2 | Regular expressions for polynomial coalgebras
- Bonsangue, Rutten, et al.
- 2007
(Show Context)
Citation Context ...then take εs = ε. Moreover, s ∼ εs, because the relation RG = {〈εs, s〉 |s ∈ S} is a bisimulation (for every functor G). Due to space restrictions we omit the proof of this fact, which can be found in =-=[2]-=-. Let us illustrate the construction above by some examples. Consider the following deterministic automaton over a two letter alphabet A = {a, b}, whose transition function is depicted in the followin... |