We study invertibility of λ-terms modulo λ-theories. Here a fundamental role is played by a class of λ-terms called finite hereditary permutations (FHP) and by their infinite generalisations (HP). More precisely, FHPs are the invertible elements in the least extensional λ-theory λ η and HPs are those in the greatest sensible λ-theory H^*. Our approach is based on inverse semigroups, algebraic structures that generalise groups and semilattices. We show that FHP modulo a λ-theory T is always an inverse semigroup and that HP modulo T is an inverse semigroup whenever T contains the theory of Böhm trees. An inverse semigroup comes equipped with a natural order. We prove that the natural order corresponds to η-expansion in FHP/T, and to infinite η-expansion in HP/T. Building on these correspondences we obtain the two main contributions of this work: firstly, we recast in a broader framework the results cited at the beginning; secondly, we prove that the FHPs are the invertible λ-terms in all the λ-theories lying between λ η and H^+. The latter is Morris' observational λ-theory, defined by using the β-normal forms as observables.

We propose an algebraization of classical and non-classical logics, based on factor varieties and decomposition operators. In particular, we provide a new method for determining whether a propositional formula is a tautology or a contradiction. This method can be automatized by defining a term rewriting system that enjoys confluence and strong normalization. This also suggests an original notion of logical gate and circuit, where propositional variables becomes logical gates and logical operations are implemented by substitution. Concerning formulas with quantifiers, we present a simple algorithm based on factor varieties for reducing first-order classical logic to equational logic. We achieve a completeness result for first-order classical logic without requiring any additional structure.

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A longstanding open problem is whether there exists a model of the untyped lambda calculus in the category CPO of complete partial orderings and Scott continuous functions, whose theory is exactly the least lambda-theory lambda-beta or the least extensional lambda-theory lambda-beta-eta. In this paper we analyze the class of reflexive Scott domains, the models of lambda-calculus living in the category of Scott domains (a full subcategory of CPO). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the beta-eta-calculus, i.e., there are equations not in lambda-beta-eta which hold in all extensional reflexive Scott domains.(ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott's information systems, called iweb-models. The class of iweb-models includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in CPO. Based on a fine-grained study of an ``effective'' version of Scott's information systems, we have shown that there are equations not in lambda-beta (resp. lambda-beta-eta) which hold in all (extensional) iweb-models.

A longstanding open problem is whether there exists a non-syntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for short). We introduce a notion of effective model of λ-calculus, which covers, in particular, all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ or λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. For Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim–Skolem theorem.

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the λβ or the least sensible λ-theory ℋ (which is generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there are a minimal λ-theory and a minimal sensible λ-theory represented by it. In this paper, we give a positive answer to this question for the class of graph models à la Plotkin, Scott and Engeler. In particular, we build two graph models whose theories are the set of equations satisfied in, respectively, any graph model and any sensible graph model. We conjecture that the least sensible graph theory, where ‘graph theory’ means ‘λ-theory of a graph model’, is equal to ℋ, while in one of the main results of the paper we show the non-existence of a graph model whose equational theory is exactly the λβ theory. Another related question is whether, given a class of lambda models, there is a maximal sensible λ-theory represented by it. In the main result of the paper, we characterise the greatest sensible graph theory as the λ-theory ℬ generated by equating λ-terms with the same Böhm tree. This result is a consequence of the main technical theorem of the paper, which says that all the equations between solvable λ-terms that have different Böhm trees fail in every sensible graph model. A further result of the paper is the existence of a continuum of different sensible graph theories strictly included in ℬ.

In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras. In every combinatory algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings), whose operations are defined by suitable combinators. Central elements are used to represent any combinatory algebra as a Boolean product of directly indecomposable combinatory algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial 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 lambda theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible: for every recursively enumerable lambda theory Tscr, there is a continuum of lambda theories including Tscr which are omitted by the indecomposable semantics

Sensible /spl lambda/-theories are equational extensions of the untyped lambda calculus that equate all the unsolvable /spl lambda/-terms and are closed under derivation. A longstanding open problem in lambda calculus is whether there exists a non-syntactic model whose equational theory is the least sensible /spl lambda/-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of models, there exist a minimal and maximal sensible /spl lambda/-theory represented by it. In This work we give a positive answer to this question for the semantics of lambda calculus given in terms of graph models. We conjecture that the least sensible graph theory, where "graph theory" means "/spl lambda/-theory of a graph model", is equal to H, while in the main result of the paper we characterize the greatest sensible graph theory as the lambda;-theory B generated by equating /spl lambda/-terms with the same Bohm tree. This result is a consequence of the fact that all the equations between solvable /spl lambda/-terms, which have different Bohm trees, fail in every sensible graph model. Further results of the paper are: (i) the existence of a continuum of different sensible graph theories strictly included in B (this result positively answers question 2 in [7, Section 6.3]); (ii) the non-existence of a graph model whose equational theory is exactly the minimal lambda theory /spl lambda//spl beta/ (this result negatively answers Question 1 in [7, Section 6.2] for the restricted class of graph models).

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.

Plotkin has conjectured that there exists an absolutely unorderable combinatory algebra, namely a combinatory algebra which cannot be embedded in another combinatory algebra admitting a nontrivial compatible partial order. In this paper we prove that a wide class of combinatory algebras admits extensions with a nontrivial compatible partial order.

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.

In this paper, we give a topological proof of the following result: there exist 2ℵ(ℵ/sub 0/) lambda theories of the untyped lambda calculus without a model in any semantics based on D.S. Scott's (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.

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.

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.

An analysis of relationships between Craig-style interpolation, compactness, and other related model-theoretic properties is carried out in the softer framework of categories of pre-institutions. While the equivalence between sentence interpolation and the Robinson property under compactness and Boolean closure is well known, a similar result under different assumptions (not involving compactness) is newly established for presentation interpolation. The standard concept of naturality of model transformation is enriched by a new property, termed restriction adequacy, which proves useful for the reduction of interpolation along pre-institution transformations. A distinct reduction theorem for the Robinson property is presented as well. A variant of the ultraproduct concept is further introduced, and the related closure property for pre-institutions is shown to be equivalent to compactness