Claudia Faggian

Basic Logic: Reflection, Symmetry, Visibility

We introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic. quantum and non-modal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterize B positively: reflection, symmetry and visibility. A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection. To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cut-elimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube.

L-Nets, strategies and proof-nets

We consider the setting of L-nets, recently introduced by Faggian and Maurel as a game model of concurrent interaction and based on Girards Ludics. We show how L-nets satisfying an additional condition, which we call logical L-nets, can be sequentialized into traditional tree-like strategies, and vice-versa.

From Basic Logic to Quantum Logics with Cut-Elimination

The results presented in this paper wereobtained in the framework of basic logic, a new logicaiming at the unification of several logical systems.The first result is a sequent formulation for orthologic which allows the use of methods of proof theoryin quantum logic. Such a formulation admits a verysimple procedure of cut-elimination and hence, becauseof the subformula property, also a method of proof search and an effective decision procedure. Byusing the framework of basic logic, we also obtain acut-free formulation for orthologic with implication,for linear orthologic, and, more in generally for a wide range of new quantum-like logics. Theselogics meet some requirements expressed by physicistsand computer scientists. In particular, we propose agood candidate for a linear quantum logic withimplication.

Partial Orders, Event Structures and Linear Strategies

We introduce a Game Semantics where strategies are partial orders, and composition is a generalization of the merging of orders. Building on this, to bridge between Game Semantics and Concurrency, we explore the relation between Event Structures and Linear Strategies. The former are a true concurrency model introduced by Nielsen, Plotkin, Winskel, the latter a family of linear innocent strategies developed starting from Girards work in the setting of Ludics. We extend our construction on partial orders to classes of event structures, showing how to reduce composition of event structures to the simple definition of merging of orders. Finally, we introduce a compact closed category of event structures which embeds Linear Strategies.

Interactive observability in Ludics: the geometry of tests

Ludics [J.-Y. Girard, Locus solum, Math. Structures in Comput. Sci. 11 (2001) 301-506] is a recent proposal of analysis of interaction, developed by abstracting away from proof-theory. It provides an elegant, abstract setting in which interaction between agents (proofs/programs/processes) can be studied at a foundational level, together with a notion of equivalence from the point of view of the observer.An agent should be seen as some kind of black box. An interactive observation on an agent is obtained by testing it against other agents.In this paper we explore what can be observed interactively in this setting. In particular, we characterize the objects that can be observed in a single test: the primitive observables of the theory.Our approach builds on an analysis of the geometrical properties of the agents, and highlights a deep interleaving between two partial orders underlying the combinatorial structures: the spatial one and the temporal one.

Jump from parallel to sequential proofs: multiplicatives

We introduce a new class of multiplicative proof nets, J-proof nets, which are a typed version of Faggian and Maurel’s multiplicative L-nets. In J-proof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints. As a byproduct, we obtain a simple proof of the sequentialisation theorem.

Proof nets sequentialisation in multiplicative linear logic

a b s t r a c t We provide an alternative proof of the sequentialisation theorem for proof nets of multiplicative linear logic. Namely, we show how a proof net can be transformed into a sequent calculus proof simply by properly adding to it some special edges, called sequential edges, which express the sequentiality constraints given by sequent calculus.

Classical Proofs via Basic Logic

Cut-elimination, besides being an important tool in proof-theory, plays a central role in the proofs-as-programs paradigm. In recent years this approach has been extended to classical logic (cf. Girard 1991, Parigot 1991, and recently Danos Joinet Schellinx 1997). This paper introduces a new sequent calculus for (propositional) classical logic, indicated by ⊥C. Both, the calculus and the cut-elimination procedure for ⊥C extend those for basic logic (Sambin Battilotti Faggian 1997).

Interactive Observability in Ludics

In Ludics [7] a proof/program (called design) can be thought of as a black box, to be studied by making it interact with other designs.

A term calculus for unitary approach to normalization

The correspondence between logical constructive systems and several A-calculi, known as the Curry-Howard isomorphllm, has played and holds a leading role in functional programming. The relevance for computer science of intuitionistic and linear logic is vicll esstablished; in recent years a significant amount of work has also been directed to cl-sical logic as a computational system. Whilst the interpretation and the computational content is different in the different logical systems, the central role is always played by the cut-elimination (normalization) procedure. This paper is concerned with the study of a unitary approach, having on normalization its main focus. Our approach stems from the program proposed in [Sambin, Battilotti, Faggian 971 which introduces basic logic as common denominator of several logical systems, with the final aim of their study in a uniform enviroment, with a uniform method. Our paper is closely connected with a work in preparation with Giovanni Sambin, and leading to a unified system named UB. Having this a8 “logical foundation” of our work, we introduce a term calculus, called Aue, that is a (fully symmetric) extension of the simple typed X-calculus. Subsystems of XUB are in a Formulas-as-Types correspondence with a large variety of logical systems, among which classical logic, basic logic (and thus non-distributive logics), linear logic (MALL), intuitionistic logic, and its dual (cf. paraconsistent logic, e.g. [5]). The reduction rulks are unique. Therefore, there is a single procedure, which is, at the same time, a normalization procedure for each of the subsystems.