4 research outputs found
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus calculus, which covers in particular all the models
individually introduced in the literature. We prove that the order theory of an
effective model is never r.e.; from this it follows that its equational theory
cannot be l-beta or l-beta-eta. We then show that no effective model living in
the stable or strongly stable semantics has an r.e. equational theory.
Concerning Scott's semantics, we investigate the class of graph models and
prove that no order theory of a graph model can be r.e., and that there exists
an effective graph model whose equational/order theory is minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34
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