Order-Sorted Congruence Closure

In this paper, an algorithm for testing the unsatisfiability of a set of ground order-sorted equational Horn clauses (for coherent signatures) is presented. This result follows from the fact that the concept of congruence closure extends to finite sets of ground order-sorted equational Horn clauses. We show how to compute the order-sorted congruence closure and obtain an algorithm running in O(η2)

Modal Rules are Co-Implications

In [13], it was shown that modal logic for coalgebras dualises—concerning definability— equational logic for algebras. This paper establishes that, similarly, modal rules dualise implications:It is shown that a class of coalgebras is definable by modal rules iff it is closed under H (images) and Σ (disjoint unions). As a corollary the expressive power of rules of infinitary modal logic on Kripke frames is characterised

Modal Rules are Co-Implications

In [13], it was shown that modal logic for coalgebras dualises—concerning definability— equational logic for algebras. This paper establishes that, similarly, modal rules dualise implications:It is shown that a class of coalgebras is definable by modal rules iff it is closed under H (images) and Σ (disjoint unions). As a corollary the expressive power of rules of infinitary modal logic on Kripke frames is characterised

The algebra of functions with antidomain and range

We give complete, finite quasiequational axiomatisations for algebras of unary partial functions under the operations of composition, domain, antidomain, range and intersection. This completes the extensive programme of classifying algebras of unary partial functions under combinations of these operations. We look at the complexity of the equational theories and provide a nondeterministic polynomial upper bound. Finally we look at the problem of finite representability and show that finite algebras can be represented as a collection of unary functions over a finite base set provided that intersection is not in the signature