Modules and Logic Programming

We study conditions for a concurrent construction of proof-nets in the framework developed by Andreoli in recent papers. We define specific correctness criteria for that purpose. We first study closed modules (i.e. validity of the execution of a logic program), then extend the criterion to open modules (i.e. validity during the execution) distinguishing criteria for acyclicity and connectability in order to allow incremental verification

Encoding Hamiltonian circuits into multiplicative linear logic

10 pagesInternational audienceWe give a new proof of the NP-completeness of multiplicative linear logic without constants by a direct encoding of the Hamiltonian circuit decision problem

Rewritings in Polarized (Partial) Proof Structures

This paper is a first step towards a study for a concurrent construction of proof-nets in the framework of linear logic after Andreoli's works, by taking care of the properties of the structures. We limit here to multiplicative linear logic. We first give a criterion for closed modules (i.e. validity of polarized proof structures), then extend it to open modules (i.e. validity of partial proof structures) distinguishing criteria for acyclicity and connectability. The keypoint is an extensive use of the fundamental structural properties of the logics. We consider proof structures as built from n-ary bipolar objects and we show that strongly confluent (local) reductions on such objects are an elegant answer to the correctness problem. This has natural applications in (concurrent) logic programming

Rewritings for Polarized Multiplicative and Exponential Proof Structures

13 pagesInternational audienceWe study conditions for a concurrent construction of proof-nets in the framework of linear logic following Andreoli's works. We define specific correctness criteria for that purpose. We first study the multiplicative case and show how the correctness criterion given by Danos and decidable in linear time, may be extended to closed modules (i.e. validity of polarized proof structures). We then study the exponential case. This has natural applications in (concurrent) logic programming as validity of partial proof structures may be interpreted in terms of validity of a concurrent execution of clauses in an environment

Correctness of Multiplicative (and Exponential) Proof Structures is NL-Complete

15 pagesInternational audienceWe provide a new correctness criterion for unit-free MLL proof structures and MELL proof structures with units. We prove that deciding the correctness of a MLL and of a MELL proof structure is NL-complete. We also prove that deciding the correctness of an intuitionistic multiplicative essential net is NL-complete

Correctness of Linear Logic Proof Structures is NL-Complete

23 pagesInternational audienceWe provide new correctness criteria for all fragments (multiplicative, exponential, additive) of linear logic. We use these criteria for proving that deciding the correctness of a linear logic proof structure is NL-complete

Sublogarithmic uniform Boolean proof nets

Using a proofs-as-programs correspondence, Terui was able to compare two models of parallel computation: Boolean circuits and proof nets for multiplicative linear logic. Mogbil et. al. gave a logspace translation allowing us to compare their computational power as uniform complexity classes. This paper presents a novel translation in AC0 and focuses on a simpler restricted notion of uniform Boolean proof nets. We can then encode constant-depth circuits and compare complexity classes below logspace, which were out of reach with the previous translations.Comment: In Proceedings DICE 2011, arXiv:1201.034

An Interpretation of CCS into Ludics

Abstract Starting from works aimed at extending the Curry-Howard correspondence to process calculi through linear logic, we give another Curry-Howard counterpart for Milner's Calculus of Communicating Systems (CCS) by taking Ludics as the target system. Indeed interaction, Ludics' dynamic, allows to fully represent both the non-determinism and non-confluence of the calculus. We give an interpretation of CCS processes into carefully defined behaviours of Ludics using a new construction, called directed behaviour, that allows controlled interaction paths by using pruned designs. We characterize the execution of processes as interaction on behaviours, by implicitly representing the causal order and conflict relation of event structures. As a direct consequence, we are also able to interpret deadlocked processes, and identify deadlock-free ones

Concurrent processes as wireless proof nets

24 pagesWe present a proofs-as-programs correspondence between linear logic and process calculi that permits non-deterministic behaviours. Processes are translated into wireless proof nets, i.e. proof nets of multiplicative linear logic without cut wires. Typing a term using them consists in typing some of its possible determinisations in standard sequent calculus. Generalized cut-elimination steps in wireless proof nets is shown to correspond to executions that avoid deadlocks

A circuit uniformity sharper than DLogTime

18 pp.We consider a new notion of circuit uniformity based on the concept of rational relations, called Rational-uniformity and denoted Rat. Our goal is to prove it is sharper than DLogTime-uniformity, the notion introduced by Barrington et al. in 1992, denoted DLT, that is: 1) Rational-uniformity implies DLogTime-uniformity, 2) we have NC^0-Rat ⊊ NC^0-DLT ⊊ NC^1-Rat, 3) we have ∀k≥0, NC^k-DLT ⊆ NC^k+1-Rat, 4) Rational-uniformity preserves separation results known with DLogTime-uniformity. In other word, we obtain an interleaved hierarchy: NC^0-Rat ⊊ NC^0-DLT ⊊ NC^1-Rat ⊆ ... ⊆ NC^k-Rat ⊆ NC^k-DLT ⊆ NC^k+1-Rat ⊆ ... ⊆ NC, which implies NC-Rat = NC-DLT . We also prove that Reg ≠ NC^0-Rat. In other words, circuits build by rational relations compute relations not computable by rational relations. Finally, we consider circuits with unbounded fan-in, and we prove the standard result NC^k-Rat ⊆ AC^k-Rat ⊆ NC^k+1-Rat for all k≥0