Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets

Roversi, Luca; Vercelli, Luca

doi:10.1007/978-3-642-02444-3_19

Cited by 5 publications

(9 citation statements)

References 11 publications

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“…This corresponds to the ?C node inside the boxes B i of Figure 13 We call nesting any restriction on linear logic which aims to tackle this kind of chains. The nesting in LLL [15], S LLL [20], mL 4 [3] and MS [27] is the absence of ?N node. Lemma 38 states that there are at most | E G | sequences of edges corresponding to copies of (B 2 , []), the sequence being entirely determined by the last edge 9 .…”

Section: Nestingmentioning

confidence: 99%

Paths-based criteria and application to linear logic subsystems characterizing polynomial time

Perrinel

2018

Information and Computation

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Several variants of linear logic have been proposed to characterize complexity classes in the proofs-as-programs correspondence. Light linear logic (LLL) ensures a polynomial bound on reduction time, and characterizes in this way polynomial time (Ptime). In this paper we study the complexity of linear logic proof-nets and propose three semantic criteria based on context semantics: stratification, dependence control and nesting. Stratification alone entails an elementary time bound, the three criteria entail together a polynomial time bound.These criteria can be used to prove the complexity soundness of several existing variants of linear logic. We define a decidable syntactic subsystem of linear logic: S DNLL. We prove that the proof-nets of SDNLL satisfy the three criteria, which implies that S DNLL is sound for Ptime. Several previous subsystems of linear logic characterizing polynomial time (LLL, mL 4 , maximal system of MS ) are embedded in S DNLL, proving its Ptime completeness.

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Section: Nestingmentioning

confidence: 99%

Paths-based criteria and application to linear logic subsystems characterizing polynomial time

Perrinel

2018

Information and Computation

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“…. In [27], this situation is called a chain of "spindle". We call "dependence control condition" any restriction on linear logic which aims to tackle this kind of spindle chains.…”

Section: Dependence Controlmentioning

confidence: 99%

“…This entails useless constraints on the strata corresponding to the auxiliary edges of boxes. Our dependence control condition is closer to M S. In [27], Roversi and Vercelli proposed to relax this discipline by considering a framework of logics, M S. M S is defined as a set of subsystems of ELL with indexes on ! and ?…”

Section: Dependence Controlmentioning

confidence: 99%

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On Paths-Based Criteria for Polynomial Time Complexity in Proof-Nets

Perrinel

2014

Foundational and Practical Aspects of Resource Analysis

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Girard's Light linear logic (LLL) characterized polynomial time in the proof-as-program paradigm with a bound on cut elimination. This logic relied on a stratification principle and a "one-door" principle which were generalized later respectively in the systems L 4 and L 3a . Each system was brought with its own complex proof of Ptime soundness. In this paper we propose a broad sufficient criterion for Ptime soundness for linear logic subsystems, based on the study of paths inside the proof-nets, which factorizes proofs of soundness of existing systems and may be used for future systems. As an additional gain, our bound stands for any reduction strategy whereas most bounds in the literature only stand for a particular strategy.

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“…We must say that the representation of SRN as a family of nets of LALL that we present in this work has been an alternative approach to the standard one, which would explore the relations between SRN and stratification by mapping a single term of SRN into a single net. That standard approach has been developed in [11,10,13]. Those works make some progress as compared to [8].…”

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confidence: 99%

Safe Recursion on Notation into a Light Logic by Levels

Roversi

Vercelli

2010

Electron. Proc. Theor. Comput. Sci.

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We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels (LALL), derived from the logic ML 4 . LALL is an intuitionistic deductive system, with a polynomial time cut elimination strategy. The embedding allows to represent every term t of SRN as a family of nets ⌈t⌉ l l∈N in LALL. Every net ⌈t⌉ l in the family simulates t on arguments whose bit length is bounded by the integer l. The embedding is based on two crucial features. One is the recursive type in LALL that encodes Scott binary numerals, i.e. Scott words, as nets. Scott words represent the arguments of t in place of the more standard Church binary numerals. Also, the embedding exploits the "fuzzy" borders of paragraph boxes that LALL inherits from ML 4 to "freely" duplicate the arguments, especially the safe ones, of t. Finally, the type of ⌈t⌉ l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of ⌈t⌉ l is a polynomial in l, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principles SRN relies on, and the proof theoretic one, called stratification, at the base of Light Linear Logic.

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Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets

Cited by 5 publications

References 11 publications

Paths-based criteria and application to linear logic subsystems characterizing polynomial time

Paths-based criteria and application to linear logic subsystems characterizing polynomial time

On Paths-Based Criteria for Polynomial Time Complexity in Proof-Nets

Safe Recursion on Notation into a Light Logic by Levels

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