1,894 research outputs found
Relational Graph Models at Work
We study the relational graph models that constitute a natural subclass of
relational models of lambda-calculus. We prove that among the lambda-theories
induced by such models there exists a minimal one, and that the corresponding
relational graph model is very natural and easy to construct. We then study
relational graph models that are fully abstract, in the sense that they capture
some observational equivalence between lambda-terms. We focus on the two main
observational equivalences in the lambda-calculus, the theory H+ generated by
taking as observables the beta-normal forms, and H* generated by considering as
observables the head normal forms. On the one hand we introduce a notion of
lambda-K\"onig model and prove that a relational graph model is fully abstract
for H+ if and only if it is extensional and lambda-K\"onig. On the other hand
we show that the dual notion of hyperimmune model, together with
extensionality, captures the full abstraction for H*
Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion
We study the semantics of a resource-sensitive extension of the lambda
calculus in a canonical reflexive object of a category of sets and relations, a
relational version of Scott's original model of the pure lambda calculus. This
calculus is related to Boudol's resource calculus and is derived from Ehrhard
and Regnier's differential extension of Linear Logic and of the lambda
calculus. We extend it with new constructions, to be understood as implementing
a very simple exception mechanism, and with a "must" parallel composition.
These new operations allow to associate a context of this calculus with any
point of the model and to prove full abstraction for the finite sub-calculus
where ordinary lambda calculus application is not allowed. The result is then
extended to the full calculus by means of a Taylor Expansion formula. As an
intermediate result we prove that the exception mechanism is not essential in
the finite sub-calculus
Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators
The symmetric interaction combinators are an equally expressive variant of
Lafont's interaction combinators. They are a graph-rewriting model of
deterministic computation. We define two notions of observational equivalence
for them, analogous to normal form and head normal form equivalence in the
lambda-calculus. Then, we prove a full abstraction result for each of the two
equivalences. This is obtained by interpreting nets as certain subsets of the
Cantor space, called edifices, which play the same role as Boehm trees in the
theory of the lambda-calculus
Using models to model-check recursive schemes
We propose a model-based approach to the model checking problem for recursive
schemes. Since simply typed lambda calculus with the fixpoint operator,
lambda-Y-calculus, is equivalent to schemes, we propose the use of a model of
lambda-Y-calculus to discriminate the terms that satisfy a given property. If a
model is finite in every type, this gives a decision procedure. We provide a
construction of such a model for every property expressed by automata with
trivial acceptance conditions and divergence testing. Such properties pose
already interesting challenges for model construction. Moreover, we argue that
having models capturing some class of properties has several other virtues in
addition to providing decidability of the model-checking problem. As an
illustration, we show a very simple construction transforming a scheme to a
scheme reflecting a property captured by a given model.Comment: Long version of a paper presented at TLCA 201
Syntactic definitions of undefined: On defining the undefined
In the lambda-calculus, there is a standard notion of what terms should be considered to be “undefined”: the unsolvable terms. There are various equivalent characterisations of this property of terms. We attempt to find a similar notion for orthogonal term rewrite systems. We find that in general the properties of terms analogous to the various characterisations of solvability differ. We give two axioms that a notion of undefinedness should satisfy, and explore some of their consequences. The axioms lead to a concept analogous to the Böhm trees of the λ-calculus. Each term has a unique B5hm tree, and the set of such trees forms a domain which provides a denotational semantics for the rewrite system. We consider several particular notions of undefinedness satisfying the axioms, and compare them
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