Ugo de'Liguoro

Intersection and Union Types

Type assignment systems with intersection and union types are introduced. Although the subject reduction property with respect to s-reduction does not hold for a natural deduction-like system, we manage to overcome this problem in two, different ways. The first is to adopt a notion of parallel reduction, which is a refinement of Gross-Knuth reduction. The second is to introduce type theories to refine the system, among which is the theory called ? that induces an assignment system preserving s-reduction. This type assignment system further clarifies the relation with the intersection discipline through the decomposition, first, of a disjunctive type into a set of conjunctive types and, second, of a derivation in the new type assignment system into a set of derivations in the intersection type assignment system. For this system we propose three semantics and prove soundness and completeness theorems.

On progress for structured communications

We propose a new typing system for the π-calculus with sessions, which ensures the progress property, i.e. once a session has been initiated, typable processes will never starve at session channels. In the current literature progress for session types has been guaranteed only in the case of nested sessions, disallowing more than two session channels interfered in a single thread. This was a severe restriction since many structured communications need combinations of sessions. We overcome this restriction by inferring the order of channel usage, but avoiding any tagging of channels and names, neither explicit nor inferred. The simplicity of the typing system essentially relies on the session typing discipline, where sequencing and branching of communications are already structured by types. The resulting typing enjoys a stronger progress property than that one in the literature: it assures that for each well-typed process P which contains an open session there is an irreducible process Q such that the parallel composition P|Q is well-typed too and it always reduces, also in presence of interfered sessions.

Sessions and session types: an overview

We illustrate the concepts of sessions and session types as they have been developed in the setting of the π-calculus. Motivated by the goal of obtaining a formalisation closer to existing standards and aiming at their enhancement and strengthening, several extensions of the original core system have been proposed, which we survey together with the embodying of sessions into functional and object-oriented languages, as well as some implementations.

A Filter Model for Concurrent

Type-free lazy

\lambda

\lambda

-Calculus

-calculus is enriched with angelic parallelism and demonic nondeterminism. Call-by-name and call-by-value abstractions are considered and the operational semantics is stated in terms of a must convergence predicate. We introduce a type assignment system with intersection and union types, and we prove that the induced logical semantics is fully abstract.

Nondeterministic Extensions of Untyped λ-Calculus

The main concern of this paper is the interplay between functionality and nondeterminism. We ask whether the analysis of parallelism in terms of sequentiality and nondeterminism, which is usual in the algebraic treatment of concurrency, remains correct in the presence of functional application and abstraction, We argue in favour of a distinction between nondeterminism and parallelism, due to the conjunctive nature of the former in contrast to the disjunctive character of the latter. This is the basis of our analysis of the operational and denotational semantics of the nondeterministic ?-calculus, which is the classical calculus plus a choice operator, and of our election of bounded indeterminacy as the semantic counterpart of conjunctive nondeterminism. This leads to operational semantics based on the idea of must preorder, coming from the classical theory of solvability and from the theory of process algebras. To characterize this relation, we build a model using the inverse limit construction over nondeterministic algebras, and we prove it fully abstract using a generalization of Bohm trees. We further prove conservativity theorems for the equational theory of the model and for other theories related to nondeterministic ?-calculus with respect to classical ?-theories.

Filter models for conjunctive-disjunctive l-calculi

Abstract The distinction between the conjunctive nature of non-determinism as opposed to the disjunctive character of parallelism constitutes the motivation and the starting point of the present work. λ-calculus is extended with both a non-deterministic choice and a parallel operator; a notion of reduction is introduced, extending β-reduction of the classical calculus. We study type assignment systems for this calculus, together with a denotational semantics which is initially defined constructing a set semimodel via simple types. We enrich the type system with intersection and union types, dually reflecting the disjunctive and conjunctive behaviour of the operators, and we build a filter model. The theory of this model is compared both with a Morris-style operational semantics and with a semantics based on a notion of capabilities.

Sub-behaviour relations for session-based client/server systems

We propose a refinement and a simplification of the behavioural semantics of session types, based on the concepts of compliance and sub-behaviour from the theory of web contracts. We introduce three relations on a suitable class of behaviours with higher-order input/output, called ‘session behaviours’. Such relations, depending on each other, represent the idea of sub-behaviour from the point of view of a client, a server or a peer, respectively. A restriction of the intersection of the first two relations characterizes the ‘usual’ sub-behaviour relation from the literature. We then device an algorithmic formal system for three subtyping relations (dubbed CSP-subtyping) for session types that takes into account the role played by a user of a channel during an interaction, so extending Gay and Hole subtyping theory. We show that our session behaviours and sub-behaviour relations provide sound and complete semantics for CSP-subtyping, and for Gay and Hole subtyping as a by-product.

Toward the interpretation of non-constructive reasoning as non-monotonic learning

We study an abstract representation of the learning process, which we call learning sequence, aiming at a constructive interpretation of classical logical proofs, that we see as learning strategies, coming from Coquands game theoretic interpretation of classical logic. Inspired by Golds notion of limiting recursion and by the Limit-Computable Mathematics by Hayashi, we investigate the idea of learning in the limit in the general case, where both guess retraction and resumption are allowed. The main contribution is the characterization of the limits of non-monotonic learning sequences in terms of the extension relation between guesses.