Summary: A Logic of Coequations
J. Ad'amek ?
Institute of Theoretical Computer Science
Abstract. By Rutten's dualization of the Birkhoff Variety Theorem,
a collection of coalgebras is a covariety (i.e., is closed under coprod
ucts, subcoalgebras, and quotients) iff it can be presented by a subset of
a cofree coalgebra. We introduce inference rules for these subsets, and
prove that they are sound and complete. For example, given a polynomial
endofunctor of a signature \Sigma , the cofree coalgebra consists of colored \Sigma
trees, and we prove that a set T of colored trees is a logical consequence
of a set S iff T contains every tree such that all recolorings of all its
subtrees lie in S. Finally, we characterize covarieties whose presentation
needs only n colors.
In the theory of systems as coalgebras (in the category of sets) presented for
example by Jan Rutten , cofree coalgebras C(k) consist of ``possible behavior
patterns'' of states of systems colored by k (observable) colors. Given a system A