An algebraic interpretation of the λβK-calculus and a labelled λ-calculus

Abstract

The interpretation <I, N>, although strongly inspired by Scott's theory of computation, is purely algebraic. Here, we do not have a definition of application as in Scott [10,11] or Welch [15]. But with the help of the labelled calculus, any expression can be considered as the limit of expressions having a normal form. If we think of λ-expressions as programs, the interpretation <I,N> seems to be the minimal one to consider.Thus we expect that <I,N> is some kind of free interpretation.This is proved by Welch for “continuous semantics”,i.e. roughly speaking for interpretations where the Wadsworth theorem is true.Welch did it for his model but his interpretation seems to be equivalent to the one we used here. Hyland [4 ],who independently considered also the same interpretation, proved that there is an extensional equivalence relation corresponding to equality in I. Furthermore, he showed for any λ-expressions M,M′ that M ⊂ M′ iff M ⊂/Pω M′ where Pω is Scott's model [11]. Another question is to take into account extensionality and build an algebraic interpretation where the η-rule is valid.This is done by Hyland [4 ] Finally,the labelled λ-calculus seems interesting in itself [6 ],since we can capture the history of any reduction in the labels.