Initial Algebras of Terms with Binding and Algebraic Structure

Abstract

One of the many results which makes Joachim Lambek famous is: an initial algebra of an endofunctor is an isomorphism. This fixed point result is often referred to as "Lambek's Lemma". In this paper, we illustrate the power of initiality by exploiting it in categories of algebra-valued presheaves EM(T ) N , for a monad T on Sets. The use of presheaves to obtain certain calculi of expressions (with variable binding) was introduced by Fiore, Plotkin, and Turi. They used setvalued presheaves, whereas here the presheaves take values in a category EM(T ) of Eilenberg-Moore algebras. This generalisation allows us to develop a theory where more structured calculi can be obtained. The use of algebras means also that we work in a linear context and need a separate operation ! for replication, for instance to describe strength for an endofunctor on EM(T ). We apply the resulting theory to give systematic descriptions of non-trivial calculi: we introduce nondeterministic and weighted lambda terms and expressions for automata as initial algebras, and we formalise relevant equations diagrammatically.

From urelements to Computation

“…Nowadays, these developments are described as "nominal computation", and there is active research on its applications. Classical research questions, such as computability via Turing machines, are finding their way in this new area of computation theory [1,22], as well as algebraic description of terms with binding over richer algebraic structures than just pure names [20], coalgebraic interpretations of nominal regular expressions [23], alternative, more powerful interpretations of freshness [42,35], theories of languages of infinite words aimed at automata-theoretic model checking [9], and several other developments scattered along many research groups. This is just the surface of a much richer theory.…”