Simply typed fixpoint calculus and collapsible pushdown automata

Simply typed λ-calculus with fixpoint combinators, λY-calculus, offers an interesting method for approximating program semantics. The Böhm tree of a λY-term represents the meaning of the program up to the meaning of built-in constants. It is much easier to reason about properties of such trees than properties of interpreted programs. Moreover, some interesting properties of programs are already expressible on the level of these trees.Collapsible pushdown automata (CPDA) give another way of generating the same class of trees as λY-terms. We clarify the relationship between the two models. In particular, we present two relatively simple translations from λY-terms to CPDA using Krivine machines as an intermediate step. The latter are general machines for describing computation of the weak head normal form in the λ-calculus. They provide the notions of closure and environment that facilitate reasoning about computation.

An environment machine for the λμ-calculus

We introduce a natural deduction-like formalisation of Parigot's λμ-calculus. From this, we derive an environment machine that allows any well-typed λμ-term to be reduced to its weak head normal form. The soundness and completeness of the machine is proved.