A new recursion-theoretic characterization of the polytime functions

Bellantoni, Stephen J.; Cook, Stephen A.

doi:10.1007/bf01201998

Cited by 329 publications

(422 citation statements)

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“…Some years ago, Bellantoni-Cook [BC92] and Leivant [Lei94] have introduced a notion of ramification. 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).…”

Section: Technical Approachesmentioning

confidence: 99%

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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%

“…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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“…Recent research [3,4,9] has provided many characterizations of Ptime (the class of all functions computable in polynomial time) by means of appropriate restrictions of the terms in Gödel's T [7]. Consider the following definition of an exponentially growing function.…”

Section: Introductionmentioning

confidence: 99%

“…Consider the following definition of an exponentially growing function. Approaches based on predicative recursion [19,3,16] argue that the exponential growth in this example is due to the way double is called: the previous value exp(ℓ) of the outer recursion is the recursive argument to the inner recursion.…”

Section: Introductionmentioning

confidence: 99%

A syntactical analysis of non-size-increasing polynomial time computation

Aehlig

Schwichtenberg²

Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332)

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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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A new recursion-theoretic characterization of the polytime functions

Cited by 329 publications

References 17 publications

Max-Plus Quasi-interpretations

Max-Plus Quasi-interpretations

A syntactical analysis of non-size-increasing polynomial time computation

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

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