9,904 research outputs found
Expressibility in the Lambda Calculus with Letrec
We investigate the relationship between finite terms in lambda-letrec, the
lambda calculus with letrec, and the infinite lambda terms they express. As
there are easy examples of lambda-terms that, intuitively, are not unfoldings
of terms in lambda-letrec, we consider the question: How can those infinite
lambda terms be characterised that are lamda-letrec-expressible in the sense
that they can be obtained as infinite unfoldings of terms in lambda-letrec?
For 'observing' infinite lambda-terms through repeated 'experiments' carried
out at the head of the term we introduce two rewrite systems (with rewrite
relations) -reg-> and -reg+-> that decompose the term, and produce 'generated
subterms' in two notions. Thereby the sort of the step can be observed as well
as its target, a generated subterm. In both systems there are four sorts of
decomposition steps: -lambda-> steps (decomposing a lambda-abstraction), -@0>
and -@1> steps (decomposing an application into its function and argument), and
respectively, -del-> steps (delimiting the scope of an abstraction, for
-reg->), and -S-> (delimiting of scopes, for -reg+->). These steps take place
on infinite lambda-terms furnished with a leading prefix of abstractions for
gathering previously encountered lambda-abstractions and keeping the generated
subterms closed. We call an infinite lambda-term 'regular'/'strongly regular'
if its set of -reg-> -reachable / -reg-> -reachable generated subterms is
finite. Furthermore, we analyse the binding structure of lambda-terms with the
concept of 'binding-capturing chain'.
Using these concepts, we answer the question above by providing two
characterisations of lambda-letrec-expressibility. For all infinite
lambda-terms M, the following statements are equivalent: (i) M is
lambda-letrec-expressible; (ii) M is strongly regular; (iii) M is regular, and
it only has finite binding-capturing chains.Comment: 79 pages, 25 figure
A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of<br> Automata
Model checking properties are often described by means of finite automata.
Any particular such automaton divides the set of infinite trees into finitely
many classes, according to which state has an infinite run. Building the full
type hierarchy upon this interpretation of the base type gives a finite
semantics for simply-typed lambda-trees.
A calculus based on this semantics is proven sound and complete. In
particular, for regular infinite lambda-trees it is decidable whether a given
automaton has a run or not. As regular lambda-trees are precisely recursion
schemes, this decidability result holds for arbitrary recursion schemes of
arbitrary level, without any syntactical restriction.Comment: 23 page
Semantics of a Typed Algebraic Lambda-Calculus
Algebraic lambda-calculi have been studied in various ways, but their
semantics remain mostly untouched. In this paper we propose a semantic analysis
of a general simply-typed lambda-calculus endowed with a structure of vector
space. We sketch the relation with two established vectorial lambda-calculi.
Then we study the problems arising from the addition of a fixed point
combinator and how to modify the equational theory to solve them. We sketch an
algebraic vectorial PCF and its possible denotational interpretations
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*
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