Preface to the special issue: isomorphisms of types and invertibility of lambda terms

Abstract

Isomorphisms of types are computational witnesses of logical equivalence with additional properties. The types/formulas A and B are isomorphic if there are functions (in a certain formalism) f : A → B and g : B → A such that g ○ f and f ○ g are equal in a certain sense to the identity on A and B, respectively. Typical such formalisms are extensions of simply typed λ-calculus, with βη-convertibility as equality relation. Another view of a pair of functions f : A → B and g : B → A (besides establishing the logical equivalence of A and B) is that f is invertible with left-inverse g, and it is then natural to relax the above symmetric condition to just g ○ f being equal to the identity on A. In this situation, A is called a retract of B, which is thus a natural generalisation of the notion of an isomorphism, while both these notions are refinements of the concept of logical equivalence in operational terms, that is, in terms of computable functions.