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Models and theories of lambda calculus

Abstract : A quarter of century after Barendregt's main book, a wealth of interesting problems about models and theories of the untyped lambda-calculus are still open. In this thesis we will be mainly interested in the main semantics of lambda-calculus (i.e., the Scott-continuous, the stable, and the strongly stable semantics) but we will also define and study two new kinds of semantics: the relational and the indecomposable semantics. Models of the untyped lambda-calculus may be defined either as reflexive objects in Cartesian closed categories (categorical models) or as combinatory algebras satisfying the five axioms of Curry and the Meyer-Scott axiom (lambda-models). Concerning categorical models we will see that each of them can be presented as a lambda-model, even when the underlying category does not have enough points, and we will provide sufficient conditions for categorical models living in arbitrary cpo-enriched Cartesian closed categories to have H^* as equational theory. We will build a categorical model living in a non-concrete Cartesian closed category of sets and relations (relational semantics) which satisfies these conditions, and we will prove that the associated lambda-model enjoys some algebraic properties which make it suitable for modelling non-deterministic extensions of lambda-calculus. Concerning combinatory algebras, we will prove that they satisfy a generalization of Stone representation theorem stating that every combinatory algebra is isomorphic to a weak Boolean product of directly indecomposable combinatory algebras. We will investigate the semantics of lambda-calculus whose models are directly indecomposable as combinatory algebras (the indecomposable semantics) and we will show that this semantics is large enough to include all the main semantics and all the term models of semi-sensible lambda-theories, and that it is however largely incomplete. Finally, we will investigate the problem of whether there exists a non-syntactical model of lambda-calculus belonging to the main semantics which has an r.e. (recursively enumerable) order or equational theory. This is a natural generalization of Honsell-Ronchi Della Rocca's longstanding open problem about the existence of a Scott-continuous model of lambda-beta or lambda-beta-eta. Then, we introduce an appropriate notion of effective model of lambda-calculus, which covers in particular all the models individually introduced in the literature, and we prove that no order theory of an effective model can be r.e.; from this it follows that its equational theory cannot be lambda-beta or lambda-beta-eta. Then, we show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott-continuous semantics, we prove that no order theory of a graph model can be r.e. and that many effective graph models do not have an r.e. equational theory.
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Contributor : Giulio Manzonetto <>
Submitted on : Friday, July 6, 2012 - 2:52:18 PM
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Giulio Manzonetto. Models and theories of lambda calculus. Logic in Computer Science [cs.LO]. Université Paris-Diderot - Paris VII, 2008. English. ⟨tel-00715207⟩

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