A call-by-name lambda-calculus machine

We present, in this paper, a particularly simple lazy machine which runs programs written in λ-calculus. It was introduced by the present writer more than twenty years ago. It has been, since, used and implemented by several authors, but remained unpublished. In the first section, we give a rather informal, but complete, description of the machine. In the second part, definitions are formalized, which allows us to give a proof of correctness for the execution of λterms. Finally, in the third part, we build an extension for the machine, with a control instruction (a kind of call-by-name call/cc) and with continuations. This machine uses weak head reduction to execute λ-calculus, which means that the active redex must be at the very beginning of the λterm. Thus, computation stops if there is no redex at the head of the λ-term. In fact, we reduce at once a whole chain λx1 . . . λxn. Therefore, execution also stops if there are not enough arguments. The first example of a λ-calculus machine is P. Landin’s celebrated SECD-machine [8]. The one presented here is quite different, in particular because it uses call-by-name. This needs some explanation, since functional programming languages are, most of the time, implemented through call-by-value. Here is the reason for this choice : Starting in the sixties, a fascinating domain has been growing between logic and theoretical computer science, that we can designate as the Curry-Howard correspondence. Succinctly, this correspondence permits the transformation of a mathematical proof into a program, which is written : in λ-calculus if the proof is intuitionistic and only uses logical axioms ; in λ-calculus extended with a control instruction, if one uses the law of excluded middle [4] and the axioms of Zermelo-Frænkel set theory [6], which is most often the case. Other instructions are necessary if one uses additional axioms, such as