Infinitary Rewriting: From Syntax to Semantics

Abstract

Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriting has also been a significant influence on the design and implementation of real programming languages, most notably the functional and logic programming families of languages. For a theoretical perspective on the place of rewriting in Computer Science, see for example [14]. For a programming language perspective, see for example [16]

Decomposing the Lattice of Meaningless Sets in the Infinitary Lambda Calculus

“…This addition has a number of useful consequences. The first is, as observed in [9], that with the extra axiom the calculus λ ∞ U is not only confluent and normalising, but also ω-compressible, just as the three standard instances that give rise to respectively the Böhm, Lévy-Longo and Berarducci trees. The second is that for any set of meaningless terms U the models M U and M U coincide, where Definition 3.…”

“…We will now define the notion of set of meaningless term. We follow the definition of [9]. This definition differs slightly from the earlier definition in [8,10,5] in that the axiom of closure under β-expansion has been added.…”

“…The first item of the following theorem is as Theorem 47 in [9] but the hypothesis of the second item has been restricted.…”

“…Although in the initial papers [6,8,10,5] on infinite lambda calculus only those three sample sets were presented as set of meaningless terms, these sets are not the only sets of meaningless terms. Only in the more recent papers [14,15,9] some aspects of the rich lattice of the sets of meaningless terms have been explored.…”

A Coinductive Confluence Proof for Infinitary Lambda-Calculus

“…The expansion axiom could probably be weakened slightly, but the present formulation is simple and it already appeared in the literature [42,41,25]. Sets of meaningless terms which do not satisfy the expansion axiom tend to be artificial.…”