## A Complete Calculus for Equational Deduction in Coalgebraic Specification (1997)

Venue: | Recent Trends in Algebraic Development Techniques, WADT 97, volume 1376 of LNCS |

Citations: | 17 - 0 self |

### BibTeX

@TECHREPORT{Corradini97acomplete,

author = {Andrea Corradini and Andrea Corradini},

title = {A Complete Calculus for Equational Deduction in Coalgebraic Specification},

institution = {Recent Trends in Algebraic Development Techniques, WADT 97, volume 1376 of LNCS},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

The use of coalgebras for the specification of dynamical systems with a hidden state space is receiving more and more attention in the years, as a valid alternative to algebraic methods based on observational equivalences. However, to our knowledge, the coalgebraic framework is still lacking a complete equational deduction calculus which enjoys properties similar to those stated in Birkhoff's completeness theorem for the algebraic case. In this paper we present a sound and complete equational calculus for coalgebras of a restricted class of polynomial functors. This restriction allows us to borrow some "algebraic" notions for the formalization of the calculus. Additionally, we discuss the notion of colours as a suitable dualization of variables in the algebraic case. Then the completeness result is extended to the "non-ground" or "coloured" case, which is shown to be expressive enough to deal with equations of hidden sort. Finally we discuss some weaknesses of the proposed results wit...

### Citations

941 | Categories for the working mathematician - Lane |

307 | Universal coalgebra: a theory of systems
- Rutten
(Show Context)
Citation Context ...sonable to expect that colouring functions from the carrier of a coalgebra to a given set of colours could play for coalgebras a role dual to that of variables and assignments for algebras (following =-=[Rut96]). Then we-=- argue that the deduction rules of Section 4 are actually for ground equations, and we present new rules for the "non-ground" or coloured case, for which we prove again soundness and complet... |

131 | Terminal coalgebras in well-founded set theory - Barr - 1993 |

103 |
Universal Algebra
- |Meinke, Tucker
- 1992
(Show Context)
Citation Context ...ncomplete way, some basic notions of algebraic specification that we shall use as guidelines for setting up the coalgebraic counterpart. In this paragraph we use standard algebraic terminology (e.g., =-=[MT92]-=-); for example, by equation we mean the traditional ones (that we assume familiar to the reader) and not those of Definition 11. Every algebraic specification h\Sigma ; Ei where E is a set of equation... |

93 |
An approach to object semantics based on terminal coalgebras
- Reichel
- 1995
(Show Context)
Citation Context ...amples from other papers we refer to in this section are often presented with a different terminology. We take the freedom of recasting them according to our syntax. 14 of a simple database system in =-=[Rei95]-=-, or that of a memory in [Jac97]. Our hope is that the results presented in this paper generalize smoothly to the conditional case, as it happens in the algebraic case (see [Sel72]): this is a topic f... |

68 | Objects and Classes, Coalgebraically
- Jacobs
- 1996
(Show Context)
Citation Context ...-defined homomorphism. Its uniqueness follows from the finality of Z \Pi . Alternative characterizations of the carrier of the final coalgebra satisfying a set of equations are proposed in [HR95] and =-=[Jac96b]-=-. In [HR95] the set Z \Pi#E is determined as the collection of all elements ! A (y) 2 Z \Pi for A 2 Coalg(\Pi; E) and y 2 XA ; in [Jac96b], instead, the same set is determined as the carrier of the la... |

34 | Behavioural and abstractor specifications - Bidoit, Hennicker, et al. - 1995 |

28 | A.: Implementation and behavioural equivalence : a survey - Orejas, Navarro, et al. - 1993 |

26 | Inheritance and cofree constructions
- Jacobs
- 1996
(Show Context)
Citation Context ...diately other equations which are not legal according to our definition. This can happen for various reasons, that we shall discuss in turn. Equations with variables of input sorts. As an example, in =-=[Jac96a]-=- the co-signature 13 BA of Example 2 is extended with attribute name : X ! String and with method ch-name: X \Theta String ! X with the obvious meaning. Then, among others, equation bal(ch-name(s; x))... |

24 |
Coalgebraic logic, Annals of Pure and Applied Logic 96
- Moss
- 1999
(Show Context)
Citation Context ...; t 0 n ). 10 Note that this definition of the entailment relation subsumes both weakening and contraction. 11 The soundness of a rule equivalent to contradiction is remarked also by Lawrence Moss in =-=[Mos97]-=- in a much richer logical framework. Theorem 15 below shows that, as far as an equational calculus is concerned, such a rule contributes in an essential way to completeness. A similar rule was suggest... |

12 | Invariants, Bisimulations and the Correctness of of Coalgebraic Refinements
- Jacobs
- 1997
(Show Context)
Citation Context ...r to in this section are often presented with a different terminology. We take the freedom of recasting them according to our syntax. 14 of a simple database system in [Rei95], or that of a memory in =-=[Jac97]-=-. Our hope is that the results presented in this paper generalize smoothly to the conditional case, as it happens in the algebraic case (see [Sel72]): this is a topic for future work. Equations of hid... |

10 |
Reichel Defining Equations in Terminal Coalgebras
- Hensel, Horst
- 1995
(Show Context)
Citation Context ...ing equations only. Using the standard final semantics, defining equations determine a sub-coalgebra of the final coalgebra, containing only the behaviours of interest. In particular, as explained in =-=[HR95]-=-, properties expressible by equations in this framework are safety or invariant properties, i.e., properties that must hold in any possible state of a coalgebra. A natural question is whether for equa... |

7 | A view on implementing processes: Categories of circuits - Hensel, Spooner - 1995 |

2 |
Completeness of Calculii for Axiomatically Defined Classes of Algebras, Algebra Universalis 2
- Selman
- 1972
(Show Context)
Citation Context ... we see various directions for possible generalizations of the results presented here. Firstly, we would like to consider conditional equations as well, looking for a complete calculus in the line of =-=[Sel72]-=-: conditional equations seem sufficiently expressive to capture most kinds of coalgebraic equations we have found in literature (always sticking to a purely coalgebraic setting). Secondly, one could c... |