A complete calculus for equational deduction in coalgebraic specification

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. Aditionally, 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 with respect to Birkhoff''s, and we suggest possible future extensions.

[1]  Desmond Fearnley-Sander,et al.  Universal Algebra , 1982 .

[2]  A. Selman.  Completeness of calculii for axiomatically defined classes of algebras , 1972 .

[3]  Horst Reichel,et al.  An approach to object semantics based on terminal co-algebras , 1995, Mathematical Structures in Computer Science.

[4]  Martin Wirsing,et al.  Behavioural and Abstractor Specifications , 1995, Sci. Comput. Program..

[5]  Fernando Orejas,et al.  Implementation and Behavioural Equivalence: A Survey , 1991, COMPASS/ADT.

[6]  Ulrich Hensel,et al.  A View on Implementing Processes: Categories of Circuits , 1995, COMPASS/ADT.

[7]  Michael Barr,et al.  Terminal Coalgebras in Well-Founded Set Theory , 1993, Theor. Comput. Sci..

[8]  Bart Jacobs,et al.  Inheritance and Cofree Constructions , 1996, ECOOP.

[9]  Horst Reichel,et al.  Defining Equations in Terminal Coalgebras , 1994, COMPASS/ADT.

[10]  Lawrence S. Moss,et al.  Coalgebraic Logic , 1999, Ann. Pure Appl. Log..

[11]  Bart Jacobs,et al.  Invariants, Bisimulations and the Correctness of Coalgebraic Refinements , 1997, AMAST.

[12]  Bart Jacobs,et al.  Objects and Classes, Co-Algebraically , 1995, Object Orientation with Parallelism and Persistence.