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.

The inhabitation problem for intersection types in the lambda-calculus is known to be undecidable. We study the problem in the case of non-idempotent intersection, considering several type assignment systems, which characterize the solvable or the strongly normalizing lambda-terms. We prove the decidability of the inhabitation problem for all the systems considered, by providing sound and complete inhabitation algorithms for them.

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.

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).

This paper presents a novel method for comparing computational properties of λ-terms that are typeable with intersection types, with respect to terms that are typeable with Curry types. We introduce a translation from intersection typing derivations to Curry typeable terms that is preserved by β-reduction: this allows the simulation of a computation starting from a term typeable in the intersection discipline by means of a computation starting from a simply typeable term. Our approach proves strong normalisation for the intersection system naturally by means of purely syntactical techniques. The paper extends the results presented in Bucciarelli et al. (1999) to the whole intersection type system of Barendregt, Coppo and Dezani, thus providing a complete proof of the conjecture, proposed in Leivant (1990), that all functions uniformly definable using intersection types are already definable using Curry types.

This paper presents a new method for comparing computation-properties of /spl lambda/-terms typeable with intersection types with respect to terms typeable with Curry types. In particular, strong normalization and /spl lambda/-definability are investigated. A translation is introduced from intersection typing derivations to Curry typeable terms; the main feature of the proposed technique is that the translation is preserved by /spl beta/-reduction. This allows to simulate a computation starting from a term typeable in the intersection discipline by means of a computation starting from a simply typeable term. Our approach naturally leads to prove strong normalization in the intersection system by means of purely syntactical techniques. In addition, the presented method enables us to give a proof of a conjecture proposed by Leivant in 1990, namely that all functions uniformly definable using intersection types are already definable using Curry types.

We introduce a technique based on logical relations, which, given two models M and N of a simply typed lambda-calculus L, allows us to construct a model M/N whose L-theory is a superset of both Th(M) and Th(N). By means of some examples, we show that classic bi-domain constructions lack this property, in general.

It is shown that Kahn-Plotkin sequentiality can be expressed by a preservation property similar to stability and that this kind of generalized stability can be extended to higher order. The main result is the construction of a model where all morphisms are functions and, at ground types, these functions are sequential.>