Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time

Leivant, Daniel

doi:10.1007/978-1-4612-2566-9_11

Cited by 99 publications

(145 citation statements)

References 10 publications

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“…In [BC92] this is expressed as a distinction between normal and safe arguments and a restriction on the way primitive recursion can be applied and functions composed ([Lei94] introduces a related notion of tier). By complying to this programming discipline the programmer is relieved from the problem of explicitly providing a bound.…”

Section: Technical Approachesmentioning

confidence: 99%

Max-Plus Quasi-interpretations

Amadio

2003

Lecture Notes in Computer Science

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Abstract/RésuméQuasi-interpretations are a tool to bound the size of the values computed by a first-order functional program (or a term rewriting system) and thus a mean to extract bounds on its computational complexity. We study the synthesis of quasi-interpretations selected in the space of polynomials over the max-plus algebra determined by the non-negative rationals extended with −∞ and equipped with binary operations for the maximum and the addition. We prove that in this case the synthesis problem is NP-hard, and in NP for the particular case of multi-linear quasi-interpretations when programs are specified by rules of bounded size.The relevance of multi-linear quasi-interpretations is discussed by comparison to certain syntactic and type theoretic conditions proposed in the literature to control time and space complexity.Keywords: Functional languages and term rewriting. Function algebras and implicit computational complexity. Static analysis. Polynomial interpretations and max-plus algebras.Les quasi-interprétations sont un outil pour borner la taille des valeurs calculées par un programme fonctionnel du premier ordre (ou un système de réécriture de termes) et ainsi un moyen pour extraire des bornes sur sa complexité. Nousétudions la synthése des polynômes sur l'algèbre max-plus determinée par le rationnels non-negatifsétendus avec −∞. Nous demontrons que dans ce cas le problème de la synthèse est NP-difficile et dans NP pour le cas particulier des quasi-interpretations multi-linéaires quand les programmes sont specifiés par des règles de taille bornée. L'intérêt des quasi-interprétation multi-linéaires est discuté par comparaisoǹ a certaines restrictions syntaxiques et de typage proposées dans la litterature pour contrôler la complexité en temps et en espace.

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Section: Technical Approachesmentioning

confidence: 99%

Max-Plus Quasi-interpretations

Amadio

2003

Lecture Notes in Computer Science

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“…but instead by restricting the primitives or the features of the languages. Various approaches have been used for that, primarily in logic and in functional languages: restrictions of the comprehension scheme in second-order logic [Lei91,Lei02]; ramification in logic or in recursion [Lei94,BC92]; read-only functional programs [Jon01]; variants of linear logic [Gir98,Laf04]. .…”

Section: Introductionmentioning

confidence: 99%

Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy

Baillot

2011

Programming Languages and Systems

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Abstract. Elementary linear logic is a simple variant of linear logic, introduced by Girard and which characterizes in the proofs-as-programs approach the class of elementary functions (computable in time bounded by a tower of exponentials of fixed height). Other systems, like light linear logic have then been defined to capture in a similar way polynomial time functions, but at the price of either a more complicated syntax or of more involved encodings. Such logical systems can then be the basis of type systems to guarantee statically complexity properties on lambdacalculus. Our goal here is to show that despite its simplicity, elementary linear logic can nevertheless be used as a common framework to characterize the different levels of a hierarchy of deterministic time complexity classes, within elementary time. We consider a variant of this logic with type fixpoints and weakening (elementary affine logic with fixpoints). The key ingredients are then the choice of specific types and a finer study of the normalization procedure on proof-nets. We characterize in this way the class P of polynomial time predicates and more generally the hierarchy of classes k-EXP, for k ≥ 0, where k-EXP is the union of DTIME(2 n i k ), for i ≥ 1.

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“…So P M LTS belongs to the same complexity class as Multiplicative Linear Logic (MLL), but it extends MLL, since some Church numerals and some common combinators on them exist as proof nets of P M LTS , for every M. Related works. Our multimodal framework for polynomial time computations may recall ramified systems [5,6]. The formal relation is still officially unclear, even though the results claimed in [7] support the idea that modalities will result in a formal, logically founded, refinement of ramification.…”

Section: Introductionmentioning

confidence: 99%

Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets

Roversi¹,

Vercelli

2009

Lecture Notes in Computer Science

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Abstract. We introduce a multimodal stratified framework MS that generalizes an idea hidden in the definitions of Light Linear/Affine logical systems: "More modalities means more expressiveness". MS is a set of building-rule schemes that depend on parameters. We interpret the values of the parameters as modalities. Fixing the parameters yields deductive systems as instances of MS, that we call subsystems. Every subsystem generates stratified proof nets whose normalization preserves stratification, a structural property of nodes and edges, like in Light Linear/Affine logical systems. A first result is a sufficient condition for determining when a subsystem is strongly polynomial time sound. A second one shows that the ability to choose which modalities are used and how can be rewarding. We give a family of subsystems as complex as Multiplicative Linear Logic -they are linear time and space sound -that can represent Church numerals and some common combinators on them.

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Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time

Cited by 99 publications

References 10 publications

Max-Plus Quasi-interpretations

Max-Plus Quasi-interpretations

Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy

Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets

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