Virgile Mogbil

Soft lambda-Calculus: A Language for Polynomial Time Computation

Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambda-calculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time. We then extend the type system of Soft logic with recursive types. This allows us to consider non-standard types for representing lists. Using these datatypes we examine the concrete expressiveness of Soft lambda-calculus with the example of the insertion sort algorithm.

A polytime functional language from light linear logic

We introduce a typed functional programming language LPL (acronym for Light linear Programming Language) in which all valid programs run in polynomial time, and which is complete for polynomial time functions. LPL is based on lambda-calculus, with constructors for algebraic data-types, pattern matching and recursive definitions, and thus allows for a natural programming style. The validity of LPL programs is checked through typing and a syntactic criterion on recursive definitions. The higher order type system is designed from the ideas of Light linear logic: stratification, to control recursive calls, and weak exponential connectives §, !, to control duplication of arguments.

Non-deterministic Boolean proof nets

Abstract. We introduce Non-deterministic Boolean proof nets to study the correspondence with Boolean circuits, a parallel model of computation. We extend the cut elimination of Non-deterministic Multiplicative Linear logic to a parallel procedure in proof nets. With the restriction of proof nets to Boolean types, we prove that the cut-elimination procedure corresponds to Non-deterministic Boolean circuit evaluation and reciprocally. We obtain implicit characterization of the complexity classes NP and NC (the efficiently parallelizable functions).

Uniform Circuits, & Boolean Proof Nets

The relationship between Boolean proof nets of multiplicative linear logic ( APN) and Boolean circuits has been studied [Ter04] in a non-uniform setting. We refine this results by taking care of uniformity: the relationship can be expressed in term of the (Turing) polynomial hierarchy. We give a proofs-as-programs correspondence between proof nets and deterministic as well as non-deterministic Boolean circuits with a uniform depth-preserving simulation of each other. The Boolean proof nets class m&BN(poly) is built on multiplicative and additive linear logic with a polynomial amount of additive connectives as the non-deterministic circuit class NNC(poly) is with non-deterministic variables. We obtain uniform-APN= NCand m& BN(poly) = NNC(poly)=NP.

Correctness of linear logic proof structures is NL-complete

We 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.

Proofs as executions

This paper proposes a new interpretation of the logical contents of programs in the context of concurrent interaction, wherein proofs correspond to valid executions of a processes. A type system based on linear logic is used, in which a given process has many different types, each typing corresponding to a particular way of interacting with its environment and cut elimination corresponds to executing the process in a given interaction scenario. A completeness result is established, stating that every lock-avoiding execution of a process in some environment corresponds to a particular typing. Besides traces, types contain precise information about the flow of control between a process and its environment, and proofs are interpreted as composable schedulings of processes. In this interpretation, logic appears as a way of making explicit the flow of causality between interacting processes.

Rewritings for Polarized Multiplicative and Exponential Proof Structures

We study conditions for a concurrent construction of proof-nets in the framework of linear logic following Andreolis 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 and give a correctness criterion by means of a contraction relation that helps to discover frontiers of exponential boxes.

Correctness of multiplicative (and exponential) proof structures is NL-complete

We 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 Multiplicative Additive Proof Structures is NL-Complete

The authors revisit the correctness criterion for the multiplicative additive fragment of linear logic. We prove that deciding the correctness of corresponding proof structures is NL-complete.

Encoding Hamiltonian circuits into multiplicative linear logic

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