Infinitary lambda calculus

Abstract

In a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. From the viewpoint of infinitary rewriting, the Bohm model of the lambda calculus can be seen as an infinitary term model. In contrast to term rewriting, there are several different possible notions of infinite tc1m, which give rise to different Bohm-like models, which embody different notions of lazy or cager computation.

Correctness of Copy in Calculi with Letrec

“…a lambda-calculus with letrec, seq, case, constructors, and por, which is equipped with a normal order reduction and a contextual semantics as definition of equality of expressions. First it defines the infinite trees corresponding to the unrolling of expressions as in the 111-calculus of [5] also adding constructors, case, seq and por. Then reduction on the infinite trees is defined, where the basic rules are the corresponding rules for (beta), seq, case and por, and the rule ∞ − → is a generalization of the (parallel) 1-reduction (see [2]); it can also be seen as an infinite development (see also [5]), however, the tree structure is a bit more general for LRCCPλ.…”

“…First it defines the infinite trees corresponding to the unrolling of expressions as in the 111-calculus of [5] also adding constructors, case, seq and por. Then reduction on the infinite trees is defined, where the basic rules are the corresponding rules for (beta), seq, case and por, and the rule ∞ − → is a generalization of the (parallel) 1-reduction (see [2]); it can also be seen as an infinite development (see also [5]), however, the tree structure is a bit more general for LRCCPλ. It is shown that convergence of expressions in the call-by-need lambda-calculus, as well as for the call-by-name calculus is equivalent to convergence of a normal-order-variant of (Tr)-reduction, i.e.…”

“…The infinite tree corresponding to an expression is intended to be the letrecunfolding of the expression with the extra condition that cyclic variable chains lead to local nontermination, represented by the symbol ⊥. This corresponds to the infinite trees in the 111-variant of the calculus in [5]. A rigorous definition is as follows, where we use the explicit binary application operator @, since it is easier to explain, but stick to the common notation in examples.…”

Standardization and Böhm Trees for Λμ-Calculus

“…In nitary λ-calculus has been introduced independently by Berarducci [3] and by Kennaway et al [16].…”

“…). In nitary λ-calculi have been considered in the literature [3,15,16,4,8] both to study in nite structures arising from lazy functional languages or to study consistency problems in the standard λ-calculus. Though, in nitary λ-calculi have been designed in a much di erent way from the in nitary calculus underlying Λµ-calculus: whereas in those frameworks, a reduction sequence may have trans nite length, terms have a (possibly in nite) depth which is bounded by ω.…”