In this article we explain two different operational interpretations of functional programs by two different logics. The programs are simply typed λ-terms with pairs, projections, if-then-else and least fixed point recursion. A logic for call-by-value evaluation and a logic for call-by-name evaluation are obtained as as extensions of a system which we call the basic logic of partial terms (BPT). This logic is suitable to prove properties of programs that are valid under both strict and non-strict evaluation. We use methods from denotational semantics to show that the two extensions of BPT are adequate for call-by-value and call-by-name evaluation. Neither the programs nor the logics contain the constant ‘undefined’.

We define a weak λ-calculus, λσw, as a subsystem of the full λ-calculus with explicit substitutions λσ[uArr]. We claim that λσw could be the archetypal output language of functional compilers, just as the λ-calculus is their universal input language. Furthermore, λσ[uArr] could be the adequate theory to establish the correctness of functional compilers. Here we illustrate these claims by proving the correctness of four simplified compilers and runtime systems modelled as abstract machines. The four machines we prove are the Krivine machine, the SECD, the FAM and the CAM. Thus, we give the first formal proofs of Cardelli's FAM and of its compiler.

Moggi's computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the Curry–Howard correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbert-style presentations of this logic and prove strong normalisation and confluence results.

A higher-order function is a function that takes another function as an argument or returns another function as a result. More specifically, a first-order function takes and returns base types, such as integers or lists. A kth-order function takes or returns a function of order k−1. Currying often artificially inflates the order of a function, so we will ignore all inessential currying. (Whether a particular instance of currying is essential or inessential is open to debate, but we expect that our choices will be uncontroversial.) In addition, when calculating the order of a polymorphic function, we instantiate all type variables with base types. Under these assumptions, most common higher-order functions, such as map and foldr, are second-order, so beginning functional programmers often wonder: What good are functions of order three or above? We illustrate functions of up to sixth-order with examples taken from a combinator parsing library.

Combinator parsing is a classic application of functional programming, dating back to at least Burge (1975). Most combinator parsers are based on Wadler's list-of-successes technique (Wadler, 1985). Hutton (1992) popularized the idea in his excellent tutorial Higher-Order Functions for Parsing. In spite of the title, however, he considered only functions of up to order three.