Antonino Salibra

Equational type logic

Equational type logic is an extension of (conditional) equational logic, that enables one to deal in a single, unified framework with diverse phenomena such as partiality, type polymorphism and dependent types. In this logic, terms may denote types as well as elements, and atomic formulae are either equations or type assignments. Models of this logic are type algebras, viz. universal algebras equipped with a binary relation—to support type assignment. Equational type logic has a sound and complete calculus, and initial models exist. The use of equational type logic is illustrated by means of simple examples, where all of the aforementioned phenomena occur. Formal notions of reduction and extension are introduced, and their relationship to free constructions is investigated. Computational aspects of equational type logic are investigated in the framework of conditional term rewriting systems, generalizing known results on confluence of these systems. Finally, some closely related work is reviewed and future research directions are outlined in the conclusions.

On the algebraic models of Lambda calculus

The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a first-order algebraic description of lambda calculus, which allows to keep the lambda notation and hence all the functional intuitions. In this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus; for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety. For example, the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras. This result is applied to obtain a generalization of the genericity lemma of finitary lambda calculus to the infinitary lambda calculus. Another result of the paper is an algebraic proof of consistency of the infinitary lambda calculus. Finally, some algebraic constructions by Krivine are generalized to lambda abstraction algebras.

Topological incompleteness and order incompleteness of the lambda calculus

A model of the untyped lambda calculus univocally induces a lambda theory (i.e., a congruence relation on λ-terms closed under α- and β-conversion) through the kernel congruence relation of the interpretation function. A semantics of lambda calculus is (equationally) incomplete if there exists a lambda theory that is not induced by any model in the semantics. In this article, we introduce a new technique to prove in a uniform way the incompleteness of all denotational semantics of lambda calculus that have been proposed so far, including the strongly stable one, whose incompleteness had been conjectured by Bastonero, Gouy and Berline. We apply this technique to prove the incompleteness of any semantics of lambda calculus given in terms of partially ordered models with a bottom element. This incompleteness removes the belief that partial orderings with a bottom element are intrinsic to models of the lambda calculus, and that the incompleteness of a semantics is only due to the richness of the structure of representable functions. Instead, the incompleteness is also due to the richness of the structure of lambda theories. Further results of the article are: (i) an incompleteness theorem for partially ordered models with finitely many connected components (= minimal upward and downward closed sets); (ii) an incompleteness theorem for topological models whose topology satisfies a suitable property of connectedness; (iii) a completeness theorem for topological models whose topology is non-trivial and metrizable.

A Soft Stairway to Institutions

The notion of institution is dissected into somewhat weaker notions. We introduce a novel notion of institution morphism, and characterize preservation of institution properties by corresponding properties of such morphisms. Target of this work is the stepwise construction of a general framework for translating logics, and algebraic specifications using logical systems. Earlier translations of order-sorted conditional equational logic and of conditional equational logics for partial algebras into equational type logic are revisited in this light. Model-theoretic results relating to compactness are presented as well.

The Lattice of Lambda Theories

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.

Applying Universal Algebra to Lambda Calculus

The aim of this article is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra, there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e. algebras that cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e. the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the article we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

Lambda abstraction algebras: representation theorems

Abstract Lambda abstraction algebras (LAAs) are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. Like combinatory algebras they can be defined by true identities and thus form a variety in the sense of universal algebra, but they differ from combinatory algebras in several important respects. The most natural LAAs are obtained by coordinatizing environment models of the lambda calculus. This gives rise to two classes of LAAs of functions of finite arity: functional LAAs (FLA) and point-relativized functional LAAs (RFA). It is shown that RFA is a variety and is the smallest variety including FLA. Dimension-complemented LAAs constitute the widest class of LAAs that can be represented as an algebra of functions and are known to have a natural intrinsic characterization. We prove that every dimension-complemented LAA is isomorphic to RFA. This is the crucial step in showing that RFA is a variety.

Lambda Abstraction Algebras: Coordinatizing Models of Lambda Calculus

Lambda abstraction algebras are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order logic; they are intended as an alternative to combinatory algebras in this regard. Like combinatory algebras they can be defined by true identities and thus form a variety in the sense of universal algebra. One feature of lambda abstraction algebras that sets them apart from combinatory algebras is the way variables in the lambda calculus are abtracted; this provides each lambda abstraction algebra with an implicit coordinate system. Another peculiar feature is the algebraic reformulation of (β)-conversion as the definition of abstract substitution. Functional lambda abstraction algebras arise as the “coordinatizations” of environment models or lambda models, the natural combinatory models of the lambda calculus. As in the case of cylindric and polyadic algebras, questions of the functional representation of various subclasses of lambda abstraction algebras are an important part of the theory. The main result of the paper is a stronger version of the functional representation theorem for locally finite lambda abstraction algebras, the algebraic analogue of the completeness theorem of lambda calculus. This result is used to study the connection between the combinatory models of the lambda calculus and lambda abstraction algebras. Two significant results of this kind are the existence of a strong categorical equivalence between lambda algebras and locally finite lambda abstraction algebras, and between lambda models and rich, locally finite lambda abstraction algebras.

A continuum of theories of lambda calculus without semantics

In this paper, we give a topological proof of the following result: there exist 2¿(/spl aleph//sub 0/) lambda theories of the untyped lambda calculus without a model in any semantics based on D.S. Scotts (1972, 1981) view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we positively solve the conjecture, stated by O. Bastonero and X. Gouy (1999) and by C. Berline (2000), that the strongly stable semantics is incomplete.

Easiness in graph models

We generalize Baeten and Boerbooms method of forcing to show that there is a fixed sequence (uk)k∈ω of closed (untyped) λ-terms satisfying the following properties: (a) For any countable sequence (gk)k∈ω of Scott continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that (λx.xx)(λx.xx)uk represents gk in the model. (b) For any countable sequence (tk)k∈ω of closed λ-terms there is a graph model that satisfies (λx.xx)(λx.xx)uk = tk for all k. We apply these two results, which are corollaries of a unique theorem, to prove the existence of (1) a finitely axiomatized λ-theory L such that the interval lattice constituted by the λ-theories extending L is distributive; (2) a continuum of pairwise inconsistent graph theories (= λ-theories that can be realized as theories of graph models); (3) a congruence distributive variety of combinatory algebras (lambda abstraction algebras, respectively).