Pratt [22] defines action algebras as Kleene algebras with residuals and action logic as the equational theory of action algebras. In opposition to Kleene algebras, action algebras form a (finitely based) variety. Jipsen [9] proposes a Gentzen-style sequent system for action logic but leaves it as an open question if this system admits cut-elimination and if action logic is decidable. We show t...

Abs t rac t . The notion of institution is dissected into somewhat weaker notions. We introduce a novel notion of institution morphism, and characterize preservation of institution properties by corresponding properties of such morphisms. Target of this work is the stepwise construction of a general framework for translating logics, and algebraic specifications using logical systems. Earlier tr...

This paper studies the notion of “freshness” that often occurs in the meta-theory of computer science languages involving various kinds of names. Nominal Equational Logic is an extension of ordinary equational logic with assertions about the freshness of names. It is shown to be both sound and complete for the support interpretation of freshness and equality provided by the Gabbay-Pitts nominal...

Coalgebra develops a general theory of transition systems, parametric in a functor T ; the functor T specifies the possible one-step behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T , a logic for T -coalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing one-step trans...

We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established.

We introduce the concept of semantic paramodulation as a \semantic" de nition of paramodulation (in the sense that it applies to any model, not only to the term algebra) within the framework of category-based equational logic (introduced by [8, 9]). This not only generalises the traditional syntactic approaches to paramodulation, but also provides an abstract framework for a uni ed treatment of...

Birkhoff’s completeness theorem of equational logic asserts the coincidence of the model-theoretic and proof-theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for finitary many-sorted equational deduction, generalized the completeness theorem of Birkhoff to the completeness of finitary many-sorted equational logic and provided simultaneo...

There are currently no fewer than four dedicated logics for equality reasoning over nominal sets: nominal algebra, nominal equational logic, nominal equational logic with equality only, and permissive-nominal algebra. In this survey and research paper we present these logics side-by-side in a common notation, survey their similarities and differences, discuss their proofand model-theories, and ...