Antonio Bucciarelli

Not enough points is enough

Models of the untyped λ-calculus may be defined either as applicative structures satisfying a bunch of first-order axioms (λ-models), or as reflexive objects in cartesian closed categories (categorical models). In this paper we show that any categorical model of λ-calculus can be presented as a λ-model, even when the underlying category does not have enough points. We provide an example of an extensional model of λ-calculus in a category of sets and relations which has not enough points. Finally, we present some of its algebraic properties which make it suitable for dealing with non-deterministic extensions of λ-calculus.

On phase semantics and denotational semantics: the exponentials

Abstract We extend to the exponential connectives of linear logic the study initiated in Bucciarelli and Ehrhard (Ann. Pure. Appl. Logic 102 (3) (2000) 247). We define an indexed version of propositional linear logic and provide a sequent calculus for this system. To a formula A of indexed linear logic, we associate an underlying formula A of linear logic, and a family 〈A〉 of elements of | A | , the interpretation of A in the category of sets and relations. Then A is provable in indexed linear logic iff the family 〈A〉 is contained in the interpretation of some proof of A . We extend to this setting the product phase semantics of indexed multiplicative additive linear logic introduced in Bucciarelli and Ehrhard (2000) , defining the symmetric product phase spaces. We prove a soundness result for this truth-value semantics and show how a denotational model of linear logic can be associated to any symmetric product phase space. Considering a particular symmetric product phase space, we obtain a new coherence space model of linear logic, which is non-uniform in the sense that the interpretation of a proof of !A−∘B contains informations about the behavior of this proof when applied to “chimeric” arguments of type A (for instance: booleans whose value can change during the computation). In this coherence semantics, an element of a web can be strictly coherent with itself, or two distinct elements can be “neutral” (that is, neither strictly coherent, nor strictly incoherent).

Categorical Models for Simply Typed Resource Calculi

We introduce the notion of differential @l-category as an extension of Blute-Cockett-Seelys differential Cartesian categories. We prove that differential @l-categories can be used to model the simply typed versions of: (i) the differential @l-calculus, a @l-calculus extended with a syntactic derivative operator; (ii) the resource calculus, a non-lazy axiomatisation of Boudols @l-calculus with multiplicities. Finally, we provide two concrete examples of differential @l-categories, namely, the category MRel of sets and relations, and the category MFin of finiteness spaces and finitary relations.

The Minimal Graph Model of Lambda Calculus

A longstanding open problem in lambda-calculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambda-calculus whose theory is exactly the beta-theory or the beta-eta-theory. A related question, raised recently by C.Berline, is whether, given a class of lambda-models, there is a minimal equational theory represented by it.

The Inhabitation Problem for Non-Idempotent Intersection Types

The inhabitation problem for intersection types is known to be undecidable. We study the problem in the case of non-idempotent intersection, and we prove decidability through a sound and complete algorithm. We then consider the inhabitation problem for an extended system typing the λ-calculus with pairs, and we prove the decidability in this case too. The extended system is interesting in its own, since it allows to characterize solvable terms in the λ-calculus with pairs.

A Relational Model of a Parallel and Non-deterministic λ -Calculus

We recently introduced an extensional model of the pureλ -calculus living in a canonical cartesian closedcategory of sets and relations [6]. In the present paper, we studythe non-deterministic features of this model. Unlike mosttraditional approaches, our way of interpreting non-determinismdoes not require any additional powerdomain construction. We showthat our model provides a straightforward semantics ofnon-determinism (may convergence) by means ofunions of interpretations, as well as ofparallelism (must convergence) by means of abinary, non-idempotent operation available on the model, which isrelated to the mix rule of Linear Logic. More precisely,we introduce a λ -calculus extended withnon-deterministic choice and parallel composition, and we defineits operational semantics (based on the may andmust intuitions underlying our two additional operations).We describe the interpretation of this calculus in our model andshow that this interpretation is sensible with respect to ouroperational semantics: a term converges if, and only if, it has anon-empty interpretation.

Intersection types and λ-definability

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.

On linear information systems

Scott’s information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a newSeely category), with a “set-theoretic” interpretation of exponentials that recovers Scott continuous functions via the co-Kleisli construction. From a domain theoretic point of view, linear information systems are equivalent to prime algebraic Scott domains, which in turn generalize prime algebraic lattices, already known to provide a model of classical linear logic.

Graph easy sets of mute lambda terms

Among the unsolvable terms of the lambda calculus, the mute ones are those having the highest degree of undefinedness. In this paper, we define for each natural number n, an infinite and recursive set M n of mute terms, and show that it is graph-easy: for any closed term t of the lambda calculus there exists a graph model equating all the terms of M n to t. Alongside, we provide a brief survey of the notion of undefinedness in the lambda calculus.

Minimal lambda-theories by ultraproducts

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the least lambda-theory lb or the least sensible lambda-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there is a minimal lambda-theory represented by it. In this paper, we give a general tool to answer positively to this question and we apply it to a wide class of webbed models: the i-models. The method then applies also to graph models, Krivine models, coherent models and filter models. In particular, we build an i-model whose theory is the set of equations satisfied in all i-models.