Bipolar Proof Nets for MALL

In this work we present a computation paradigm based on a concurrent and incremental construction of proof nets (de-sequentialized or graphical proofs) of the pure multiplicative and additive fragment of Linear Logic, a resources conscious refinement of Classical Logic. Moreover, we set a correspon- dence between this paradigm and those more pragmatic ones inspired to transactional or distributed systems. In particular we show that the construction of additive proof nets can be interpreted as a model for super-ACID (or co-operative) transactions over distributed transactional systems (typi- cally, multi-databases).Comment: Proceedings of the "Proof, Computation, Complexity" International Workshop, 17-18 August 2012, University of Copenhagen, Denmar

Generalized Connectives for Multiplicative Linear Logic

In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions. We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction ? and disjunction ?, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms

A new correctness criterion for multiplicative non commutative proof-nets

We introduce a new correctness criterion for multiplicative non commutative proof nets which can be considered as the non-commutative counterpart to the Danos-Regnier criterion for proof nets of linear logic. The main intuition relies on the fact that any switching for a proof net (obtained by mutilating one premise of each disjunction link) can be naturally viewed as a series-parallel order variety (a cyclic relation) on the conclusions of the proof net

Non decomposable connectives of linear logic

This paper studies the so-called generalized multiplicative connectives of linear logic, focusing on the question of finding the " non-decomposable " ones, i.e., those that may not be expressed as combinations of the default binary connectives of multiplicative linear logic, ⊗ (tensor) and (par). In particular, we concentrate on generalized connectives of a surprisingly simple form, called " entangled connectives " , and prove a characterization theorem giving a criterion for identifying the undecomposable ones

Construction of retractile proof structures

In this work we present a paradigm of focusing proof search based on an incremental construction of retractile (i.e, correct or sequen- tializable) proof structures of the pure (units free) multiplicative and ad- ditive fragment of linear logic. The correctness of proof construction steps (or expansion steps) is ensured by means of a system of graph retraction rules; this graph rewriting system is shown to be convergent, that is, terminating and confluent. Moreover, the proposed proof construction follows an optimal (parsimonious, indeed) retraction strategy that, at each expansion step, allows to take into account (abstract) graphs that are ”smaller” (w.r.t. the size) than the starting proof structures

A Proof of the Focusing Theorem via MALL Proof Nets

We present a demonstration of Andreoli’s focusing theorem for proofs of linear logic (MALL) that avoids directly reasoning on sequent calculus proofs. Following Andreoli-Maieli’s strategy, exploited in the MLL case, we prove the focusing theorem as a particular sequentialization strategy for MALL proof nets that are in canonical form. Canonical proof nets satisfy the property that asynchronous links are always ready to sequentialization while synchronous focusing links represent clusters of links that are hereditarily ready to sequentialization

Contractible Proof Structures

We present an application of a confluent rewriting system, based on local deformation steps (contrac- tions) of special graphs (proof structures), to the proof construction paradigm of the pure multiplicative and additive fragment of linear logic. This system allows to detect, among proof structures, those (correct) ones that correspond to very compact (bipolar and focussing) derivations of the sequent calculus of linear logic. In particular, a correct proof structure, called transitory net, is a proof structure that retracts to a special invariant graph, called normal form (i.e., a set of single nodes)

Non decomposable connectives of linear logic

This paper studies the so-called generalized multiplicative connectives of linear logic, focusing on the question of finding the non-decomposable" ones, i.e., those that may not be expressed as combinations of the default binary connectives of multiplicative linear logic, Tensor and Par. In particular, we concentrate on generalized connectives of a surprisingly simple form, called entangled connectives", and prove a characterization theorem giving a criterion for identifying the undecomposable ones