scholarly journals Soft linear logic and polynomial time

2004 ◽  
Vol 318 (1-2) ◽  
pp. 163-180 ◽  
Author(s):  
Yves Lafont
Keyword(s):  
Polynomial Time ◽  
Linear Logic

Bounded Linear Logic: A Modular Approach to Polynomial Time Computability

1990 ◽  
pp. 195-209 ◽  
Author(s):  
Jean-Yves Girard ◽  
Andre Scedrov ◽  
Philip J. Scott
Keyword(s):  
Polynomial Time ◽  
Linear Logic ◽  
Modular Approach ◽  
Bounded Linear

FUNCTIONAL PEARL Linear lambda calculus and PTIME-completeness

2004 ◽  
Vol 14 (6) ◽  
pp. 623-633 ◽  
Author(s):  
HARRY G. MAIRSON
Keyword(s):  
Polynomial Time ◽  
Linear Logic ◽  
Lambda Calculus ◽  
Type Inference ◽  
Decision Problems ◽  
Logical Interpretation

We give transparent proofs of the PTIME-completeness of two decision problems for terms in the λ-calculus. The first is a reproof of the theorem that type inference for the simply-typed λ-calculus is PTIME-complete. Our proof is interesting because it uses no more than the standard combinators Church knew of some 70 years ago, in which the terms are linear affine – each bound variable occurs at most once. We then derive a modification of Church's coding of Booleans that is linear, where each bound variable occurs exactly once. A consequence of this construction is that any interpreter for linear λ-calculus requires polynomial time. The logical interpretation of this consequence is that the problem of normalizing proofnets for multiplicative linear logic (MLL) is also PTIME-complete.

scholarly journals Light logics and higher-order processes

2014 ◽  
Vol 26 (6) ◽  
pp. 969-992 ◽  
Author(s):  
UGO DAL LAGO ◽  
SIMONE MARTINI ◽  
DAVIDE SANGIORGI
Keyword(s):  
Polynomial Time ◽  
Linear Logic ◽  
Higher Order ◽  
Resource Control ◽  
Process Algebras ◽  
Π Calculus ◽  
Soft Processes

We show that the techniques for resource control that have been developed by the so-calledlight logicscan be fruitfully applied also to process algebras. In particular, we present a restriction of higher-order π-calculus inspired by soft linear logic. We prove that any soft process terminates in polynomial time. We argue that the class of soft processes may be naturally enlarged so that interesting processes are expressible, still maintaining the polynomial bound on executions.

scholarly journals Bounded linear logic: a modular approach to polynomial-time computability

1992 ◽  
Vol 97 (1) ◽  
pp. 1-66 ◽  
Author(s):  
Jean-Yves Girard ◽  
Andre Scedrov ◽  
Philip J. Scott
Keyword(s):  
Polynomial Time ◽  
Linear Logic ◽  
Modular Approach ◽  
Bounded Linear

scholarly journals Polynomial time in untyped elementary linear logic

2020 ◽  
Vol 813 ◽  
pp. 117-142
Author(s):  
Olivier Laurent
Keyword(s):  
Polynomial Time ◽  
Linear Logic

On session types and polynomial time

2015 ◽  
Vol 26 (8) ◽  
pp. 1433-1458 ◽  
Author(s):  
UGO DAL LAGO ◽  
PAOLO DI GIAMBERARDINO
Keyword(s):  
Polynomial Time ◽  
Linear Logic ◽  
Type System ◽  
Session Types ◽  
Intuitionistic Linear Logic

We show how systems of session types can enforce interactions to take bounded time for all typable processes. The type system we propose is based on Lafont's soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the existence, for every typable process, of a polynomial bound on the length of reduction sequences starting from it and on the size of its reducts.

Linear logic and polynomial time

2006 ◽  
Vol 16 (06) ◽  
pp. 947 ◽  
Author(s):  
DAMIANO MAZZA
Keyword(s):  
Polynomial Time ◽  
Linear Logic

Linear Logic Proof Games and Optimization

1996 ◽  
Vol 2 (3) ◽  
pp. 322-338 ◽  
Author(s):  
Patrick D. Lincoln ◽  
John C. Mitchell ◽  
Andre Scedrov
Keyword(s):  
Complexity Theory ◽  
Polynomial Time ◽  
Short Proof ◽  
Linear Logic ◽  
Constant Number ◽  
Proof Checker ◽  
Logarithmic Number ◽  
Surprising Discovery

§ 1. Introduction. Perhaps the most surprising recent development in complexity theory is the discovery that the class NP can be characterized using a form of randomized proof checker that only examines a constant number of bits of the “proof” that a string is in a language [6, 5, 31, 3, 4]. More specifically, writing ∣x∣ for the length of a string x, a language L in the class NP of languages recognizable in Nondeterministic polynomial time is traditionally given by a polynomial p and a polynomial-time predicate P such that a string x is in L iff there is some string y satisfying P(x, y), where ∣y∣ ≤ p (∣x∣). Intuitively, we can think of a string y as a possible proof that x ϵ L, with the predicate P some kind of proof checker that distinguishes good proofs from bad ones. A string x is therefore in a language L ϵ NP if there is a short proof that x ϵ L, and not in L otherwise. The surprising discovery is that the deterministic proof checker that reads the entire input and proof can be replaced by a probabilistic verifier that on input x and possible proof y, where y is presented in a certain way, flips at most O (log ∣x∣) coins and reads at most a constant number of bits of x and y. Obviously, a probabilistic verifier that does not read the whole proof will sometimes make mistakes. However, the surprising and essentially non-intuitive mathematical fact is that for each language L in NP, it is possible to find a polynomial q and verifier V flipping a logarithmic number of coins and reading a constant number of bits such that, for any x ϵ L, there exists y with ∣y∣ ≤ q(∣x∣) and with V (x, y) accepting with probability 1 and, for x ∉ L, there is no y with ∣y∣ ≤ q(∣x∣) and with V (x, y) accepting with probability ≥ 1/4. This idea canalsobeextended to PSPACE [10, 9], using a game that alternates between two players instead of a proof presented by a “single player.”

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