9,627 research outputs found
A Simple Semantics for ML Polymorphism
We give a framework for denotational semantics for the polymorphic core of the programming language ML. This framework requires no more semantic material than what is needed for modeling the simple type discipline. In our view, the terms of ML are pairs consisting of a raw (untyped) lambda term and a type-scheme that ML\u27s type inference system can derive for the raw term. We interpret type-schemes as sets of simple types. Then, given any model M of the simply typed lambda calculus, the meaning of an ML term will be a set of pairs, each consisting of a simple type Ï„ and an element of M of type Ï„.
Hence, there is no need to interpret all raw terms, as was done in Milner\u27s original semantic framework. In comparison to Mitchell and Harper\u27s analysis, we avoid having to provide a very large type universe in which generic type-schemes are interpreted. Also, we show how to give meaning to ML terms rather than to derivations in the ML type inference system (which can be several for the same term).
We give an axiomatization for the equational theory that corresponds to our semantic framework and prove the analogs of the compeleteness theorems that Friedman proved for the simply typed lambda calculus. The framework can be extended to languages with constants, type constructors and recursive types (via regular trees). For the extended language, we prove a theorem that allows the transfer of certain full abstraction results from languages based on the typed lambda calculus to ML-like languages
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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