## An algebra for Kripke polynomial coalgebras (2009)

Venue: | 24TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE |

Citations: | 7 - 7 self |

### BibTeX

@MISC{Bonsangue09analgebra,

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

title = { An algebra for Kripke polynomial coalgebras},

year = {2009}

}

### OpenURL

### Abstract

Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of so-called Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be characterized up to bisimulation by a specification language that generalizes Kleene’s regular expressions for finite automata. In this paper, we equip this specification language with an axiomatization and prove it sound and complete with respect to bisimulation, using a purely coalgebraic argument. We demonstrate the usefulness of our framework by providing a finite equational system for (non-)deterministic finite automata, labelled transition systems with explicit termination, and automata on guarded strings.

### Citations

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- Rutten
- 1996
(Show Context)
Citation Context ...a pair (S , f ) consisting of a set of states S together with a function f : S → GS . The functor G , together with the function f , determines the transition structure or dynamics of the G-coalgebra =-=[14]-=-. Classical examples of coalgebras are deterministic automata, infinite streams, non-deterministic automata and partial automata, which are, respectively, coalgebras for the functors D , St , N and P ... |

235 | Domain theory in logical form
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- 1991
(Show Context)
Citation Context ...ure of our proof of soundness and completeness. The third group of axioms, relating the joinsemilattice structure with the functor specific operators, are similar to the ones coming from domain logic =-=[1]-=- or coalgebraic modal logic [4]. The main novelty of our work compared to [1, 4, 11, 12] is the inclusion of fixed-points. Acknowledgements. The authors are indebted to F. Bonchi and E. de Vink for us... |

200 | A completeness theorem for Kleene algebras and the algebra of regular events
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- 1994
(Show Context)
Citation Context ...e would also like to study the precise connection between our language and coalgebraic modal logics [12, 11]. Further, we want to investigate the relation with bialgebras [15, 7]. 7.1 Related work In =-=[9]-=- a sound a complete axiomatization for regular expressions was presented. There, regular expressions form a Kleene algebra. ı.e an idempotent semiring. Because we do not have sequential composition we... |

165 |
A final coalgebra theorem
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ...ion of coalgebras. We will also use bisimulation as a semantic equivalence for our language of expressions. Let (S , f ) and (T , g) be two G-coalgebras. A relation R ⊆ S × T is called a bisimulation =-=[3]-=- if there exists a map e : R → G(R) such that the projections π1 and π2 are coalgebra homomorphisms, i.e. the following diagram commutes. S f R π2 //π1oo ∃e T g FS FR Fπ1 // Fπ2 oo FT We writ... |

142 | Towards a Mathematical Operational Semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...uations as rewriting rules. We would also like to study the precise connection between our language and coalgebraic modal logics [12, 11]. Further, we want to investigate the relation with bialgebras =-=[15, 7]-=-. 7.1 Related work In [9] a sound a complete axiomatization for regular expressions was presented. There, regular expressions form a Kleene algebra. ı.e an idempotent semiring. Because we do not have ... |

56 | Many-sorted coalgebraic modal logic: a model-theoretic study - Jacobs |

55 | Coalgebraic modal logic: soundness, completeness and decidability, Theoret - Pattinson - 2003 |

35 | Coalgebras and modal logic
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- 2000
(Show Context)
Citation Context ...ely, the type of deterministic automata, infinite streams, non-deterministic automata and partial deterministic automata. Our definition of Kripke polynomial functors slightly differs from the one of =-=[13, 6]-=- in the use of a join-semilattice as constant functor and in the definition of +. This small variation plays an important technical role in giving a full coalgebraic treatment of the language of expre... |

29 | Presenting functors by operations and equations - Bonsangue, Kurz - 2006 |

26 | A finite model construction for coalgebraic modal logic - Schröder - 2006 |

18 | On the coalgebraic theory of Kleene algebra with tests
- Kozen
- 2008
(Show Context)
Citation Context ...s axiomatization is very similar to the one presented in [2]. The differences are only in the fact that we consider action prefixing instead of sequential composition. AGS. It has recently been shown =-=[10]-=- that automata on guarded strings (acceptors of the join irreducible elements of the free Kleene algebra with tests on generators Σ,T ) are coalgebras for the functor B × IdAt×Σ, where At is the set o... |

13 |
A bialgebraic review of deterministic automata, regular expressions and languages
- Jacobs
- 2006
(Show Context)
Citation Context ...uations as rewriting rules. We would also like to study the precise connection between our language and coalgebraic modal logics [12, 11]. Further, we want to investigate the relation with bialgebras =-=[15, 7]-=-. 7.1 Related work In [9] a sound a complete axiomatization for regular expressions was presented. There, regular expressions form a Kleene algebra. ı.e an idempotent semiring. Because we do not have ... |

13 | CIRC : A Circular Coinductive Prover
- Lucanu, Rosu
- 2007
(Show Context)
Citation Context ...nvestigate automated reasoning about equality of expressions. This can be done either in a purely coalgebraic manner, by implementing Kleene’s theorem in, an existing coinductive prover, such as Circ =-=[8]-=-, or in an algebraic manner, by using the equations as rewriting rules. We would also like to study the precise connection between our language and coalgebraic modal logics [12, 11]. Further, we want ... |

13 | Coalgebraic Automata Theory: Basic Results
- Kupke, Venema
(Show Context)
Citation Context ...ve prover, such as Circ [8], or in an algebraic manner, by using the equations as rewriting rules. We would also like to study the precise connection between our language and coalgebraic modal logics =-=[12, 11]-=-. Further, we want to investigate the relation with bialgebras [15, 7]. 7.1 Related work In [9] a sound a complete axiomatization for regular expressions was presented. There, regular expressions form... |

10 | A Kleene theorem for polynomial coalgebras
- Bonsangue, Rutten, et al.
- 2009
(Show Context)
Citation Context ...artially supported by the Fundação para a Ciência e a Tecnologia, Portugal, under grant number SFRH/BD/27482/2006. by finite automata and those represented by regular expressions. It was showed in =-=[5]-=- that Kleene’s theorem can be generalized to other types of transition systems. Finite automata are replaced by G-coalgebras (for a polynomial Set endofunctor G), regular expressions by a language Exp... |

8 | Beyond Rank 1: Algebraic Semantics and Finite Models for Coalgebraic Logics
- Pattinson, Schröder
- 2008
(Show Context)
Citation Context ...ve prover, such as Circ [8], or in an algebraic manner, by using the equations as rewriting rules. We would also like to study the precise connection between our language and coalgebraic modal logics =-=[12, 11]-=-. Further, we want to investigate the relation with bialgebras [15, 7]. 7.1 Related work In [9] a sound a complete axiomatization for regular expressions was presented. There, regular expressions form... |