50 research outputs found
Comparing Böhm-Like Trees
Extending the infinitary rewriting definition of Böhm-like trees to infinitary Combinatory Reduction Systems (iCRSs), we show that each Böhm-like tree defined by means of infinitary rewriting can also be defined by means of a direct approximant function. In addition, we show that counterexamples exists to the reverse implication
Infinitary Combinatory Reduction Systems: Confluence
We study confluence in the setting of higher-order infinitary rewriting, in
particular for infinitary Combinatory Reduction Systems (iCRSs). We prove that
fully-extended, orthogonal iCRSs are confluent modulo identification of
hypercollapsing subterms. As a corollary, we obtain that fully-extended,
orthogonal iCRSs have the normal form property and the unique normal form
property (with respect to reduction). We also show that, unlike the case in
first-order infinitary rewriting, almost non-collapsing iCRSs are not
necessarily confluent
On Undefined and Meaningless in Lambda Definability
We distinguish between undefined terms as used in lambda definability
of partial recursive functions and meaningless terms as used in
infinite lambda calculus for the infinitary terms models that
generalise the Bohm model. While there are uncountable many known
sets of meaningless terms, there are four known sets of undefined
terms. Two of these four are sets of meaningless terms.
In this paper we first present set of sufficient conditions for a set
of lambda terms to serve as set of undefined terms in lambda
definability of partial functions. The four known sets of undefined
terms satisfy these conditions.
Next we locate the smallest set of meaningless terms satisfying these
conditions. This set sits very low in the lattice of all sets of
meaningless terms. Any larger set of meaningless terms than this
smallest set is a set of undefined terms. Thus we find uncountably
many new sets of undefined terms.
As an unexpected bonus of our careful analysis of lambda definability
we obtain a natural modification, strict lambda-definability, which
allows for a Barendregt style of proof in which the representation of
composition is truly the composition of representations
Counterexamples in Infinitary Rewriting with Non-Fully-Extended Rules
We show counterexamples exist to confluence modulo hypercollapsing subterms, fair normalisation, and the normal form property in orthogonal infinitary higher-order rewriting with non-fully-extended rules. This sets these systems apart from both fully-extended and finite systems, where no such counterexamples are possible
Nominal Coalgebraic Data Types with Applications to Lambda Calculus
We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus
Graph easy sets of mute lambda terms
Among the unsolvable terms of the lambda calculus, the mute ones are those having the highest degree of undefinedness. In this paper, we define for each natural number n, an infinite and recursive set M-n of mute terms, and show that it is graph-easy: for any closed term t of the lambda calculus there exists a graph model equating all the terms of M-n to t. Alongside, we provide a brief survey of the notion of undefinedness in the lambda calculus. (C) 2015 Elsevier B.V. All rights reserved