Lambda abstraction algebras: representation theorems

Mathematical Foundations of Computer Science 2010

2010

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Abstract. We initiate a purely algebraic study of Ehrhard and Regnier's resource λ-calculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambda-algebras and resource lambda-abstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λ-calculus. We also show that the ideal completion of a resource combinatory (resp. lambda-, lambda-abstraction) algebra induces a "classical" combinatory (resp. lambda-, lambda-abstraction) algebra, and that any model of the classical λ-calculus raising from a resource lambda-algebra determines a λ-theory which equates all terms having the same Böhm tree.

Section: The Rλ-calculus From the Algebraic Point Of Viewmentioning

confidence: 91%

Section: Proof Suppose Bt (M ) = Bt (N ) Then Obviously T (Bt (M ))mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Resource Combinatory Algebras

Carraro

Ehrhard

Mathematical Foundations of Computer Science 2010

2010

Self Cite

“…Lambda abstraction algebras are meant to axiomatize those identities between contexts that are valid for the lambda calculus and were studied by Goldblatt, Pigozzi and Salibra in a series of papers [23,24,25,27,28,29]. We now give the formal definition of a lambda abstraction algebra.…”

Section: Lambda Abstraction Algebrasmentioning

confidence: 99%

“…In [28] Salibra has shown that the variety (i.e., equational class) generated by the term algebra of λβ is axiomatized by the finite schema of identities characterizing lambda abstraction algebras (LAA's). The equational theory of lambda abstraction algebras, introduced by Pigozzi and Salibra in [23] and [24], is intended as an alternative to combinatory logic in this regard since it is a first-order algebraic description of lambda calculus, which keeps the lambda notation and hence all the functional intuitions. Lambda abstraction algebras are axiomatized by the equations that hold between contexts of the lambda calculus (i.e., λ-terms with 'holes' [4, Def.…”

Section: Introductionmentioning

confidence: 99%

The Lattice of Lambda Theories

Lusin

Journal of Logic and Computation

2004

Lambda theories are equational extensions of the untyped lambda calculus that are closed under derivation. The set of lambda theories is naturally equipped with a structure of complete lattice, where the meet of a family of lambda theories is their intersection, and the join is the least lambda theory containing their union. In this paper we study the structure of the lattice of lambda theories by universal algebraic methods. We show that nontrivial quasi-identities in the language of lattices hold in the lattice of lambda theories, while every nontrivial lattice identity fails in the lattice of lambda theories if the language of lambda calculus is enriched by a suitable finite number of constants. We also show that there exists a sublattice of the lattice of lambda theories which satisfies: (i) a restricted form of distributivity, called meet semidistributivity; and (ii) a nontrivial identity in the language of lattices enriched by the relative product of binary relations.

A Finite Equational Axiomatization of the Functional Algebras for the Lambda Calculus

Information and Computation

Goldblatt

1999

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A lambda theory satisfies an equation between contexts, where a context is a *-term with some``holes'' in it, if all the instances of the equation fall within the lambda theory. In the main result of this paper it is shown that the equations (between contexts) valid in every lambda theory have an explicit finite equational axiomatization. The variety of algebras determined by the above equational theory is characterized as the class of isomorphic images of functional lambda abstraction algebras. These are algebras of functions and naturally arise as the``coordinatizations'' of environment models or lambda models, the natural combinatory models of the lambda calculus. The main result of this paper is also applied to obtain a completeness theorem for the infinitary lambda calculus recently introduced by Berarducci.