Characterizations of Polynomial Complexity Classes with a Better Intensionality

ISBN : 978-1-60558-117-0International audienceIn this paper, we study characterizations of polynomial complexity classes using first order functional programs and we try to improve their intensionality, that is the number of natural algorithms captured. We use polynomial assignments over the reals. The polynomial assignments used are inspired by the notions of quasi-interpretation and sup-interpretation, and are decidable when considering polynomials of bounded degree ranging over real numbers. Contrarily to quasi-interpretations, the considered assignments are not required to have the subterm property. Consequently, they capture a strictly larger number of natural algorithms (including quotient, gcd, duplicate elimination from a list) than previous characterizations using quasi-interpretations

Synthesis of sup-interpretations: a survey

In this paper, we survey the complexity of distinct methods that allow the programmer to synthesize a sup-interpretation, a function providing an upper- bound on the size of the output values computed by a program. It consists in a static space analysis tool without consideration of the time consumption. Although clearly related, sup-interpretation is independent from termination since it only provides an upper bound on the terminating computations. First, we study some undecidable properties of sup-interpretations from a theoretical point of view. Next, we fix term rewriting systems as our computational model and we show that a sup-interpretation can be obtained through the use of a well-known termination technique, the polynomial interpretations. The drawback is that such a method only applies to total functions (strongly normalizing programs). To overcome this problem we also study sup-interpretations through the notion of quasi-interpretation. Quasi-interpretations also suffer from a drawback that lies in the subterm property. This property drastically restricts the shape of the considered functions. Again we overcome this problem by introducing a new notion of interpretations mainly based on the dependency pairs method. We study the decidability and complexity of the sup-interpretation synthesis problem for all these three tools over sets of polynomials. Finally, we take benefit of some previous works on termination and runtime complexity to infer sup-interpretations.Comment: (2012

Polynomials over the reals are safe for program interpretations

In the field of implicit computational complexity, we are con- sidering in this paper the fruitful branch of interpretation methods. Due to their good intensional properties, they have been widely developped. Among usual issues is the synthesis problem which has been solved by the use of Tarski's decision procedure, and consequently interpretations are usually chosen over the reals rather than over the integers. Doing so, one cannot use anymore the (good) properties of the natural (well-) ordering of N employed to bound the complexity of programs. We show that, actually, polynomials over the reals benefit from some properties that allows their safe use for complexity. We illustrate this by two char- acterizations, one of PTIME and one of PSPACE

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Global and local space properties of stream programs

The original publication is available at www.springerlink.comInternational audienceIn this paper, we push forward the approach proposed in [1] aiming at studying semantic interpretation criteria for the purpose of ensuring safety and complexity properties of programs working on streams. The paper improves the previous results by considering global and local upper bounds properties of both theoretical and practical interests guaranteeing that the size of each output stream element is bounded by a function in the maximal size of the input stream elements. Moreover, in contrast to previous studies, these properties also apply to a wide class of stream definitions, that is functions that do not have streams in the input but produce an output stream

The Derivational Complexity Induced by the Dependency Pair Method

We study the derivational complexity induced by the dependency pair method, enhanced with standard refinements. We obtain upper bounds on the derivational complexity induced by the dependency pair method in terms of the derivational complexity of the base techniques employed. In particular we show that the derivational complexity induced by the dependency pair method based on some direct technique, possibly refined by argument filtering, the usable rules criterion, or dependency graphs, is primitive recursive in the derivational complexity induced by the direct method. This implies that the derivational complexity induced by a standard application of the dependency pair method based on traditional termination orders like KBO, LPO, and MPO is exactly the same as if those orders were applied as the only termination technique

On Notions of Provability

In this thesis, we study notions of provability, i.e. formulas B(x,y) such that a formula ϕ is provable in T if, and only if, there is m ∈ N such that T ⊢ B(⌜ϕ⌝,m) (m plays the role of a parameter); the usual notion of provability, k-step provability (also known as k-provability), s-symbols provability are examples of notions of provability. We develop general results concerning notions of provability, but we also study in detail concrete notions. We present partial results concerning the decidability of kprovability for Peano Arithmetic (PA), and we study important problems concerning k-provability, such as Kreisel’s Conjecture and Montagna’s Problem: (∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Kreisel’s Conjecture] and Does PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ imply PA ⊢k steps ϕ? [Montagna’s Problem] Incompleteness, Undefinability of Truth, and Recursion are different entities that share important features; we study this in detail and we trace these entities to common results. We present numeral forms of completeness and consistency, numeral completeness and numeral consistency, respectively; numeral completeness guarantees that, whenever a Σb 1(S12 )-formula ϕ(⃗x ) is such that ⃗Q ⃗x .ϕ(⃗x ) is true (where ⃗Q is any array of quantifiers), then this very fact can be proved inside S12 , more precisely S12 ⊢ ⃗Q ⃗x .Prτ (⌜ϕ( •⃗ x )⌝). We examine these two results from a mathematical point of view by presenting the minimal conditions to state them and by finding consequences of them, and from a philosophical point of view by relating them to Hilbert’s Program. The derivability condition “provability implies provable provability” is one of the main derivability conditions used to derive the Second Incompleteness Theorem and is known to be very sensitive to the underlying theory one has at hand. We create a weak theory G2 to study this condition; this is a theory for the complexity class FLINSPACE. We also relate properties of G2 to equality between computational classes.O tema desta tese são noções de demonstração; estas últimas são fórmulas B(x,y) tais que uma fórmula ϕ é demonstrável em T se, e só se, existe m ∈ N tal que T ⊢ B(⌜ϕ⌝,m) (m desempenha o papel de um parâmetro). A noção usual de demonstração, demonstração em k-linhas (demonstração-k), demonstração em s-símbolos são exemplos de noções de demonstração. Desenvolvemos resultados gerais sobre noções de demonstração, mas também estudamos exemplos concretos. Damos a conhecer resultados parciais sobre a decidibilidade da demonstração-k para a Aritmética de Peano (PA), e estudamos dois problemas conhecidos desta área, a Conjectura de Kreisel e o Problema de Montagna: (∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Conjectura de Kreisel] e PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ implica PA ⊢k steps ϕ? [Problema de Montagna] A Incompletude, a Incapacidade de Definir Verdade, e Recursão são entidades que têm em comum características relevantes; nós estudamos estas entidades em detalhe e apresentamos resultados que são simultaneamente responsáveis pelas mesmas. Além disso, apresentamos formas numerais de completude e consistência, a completude numeral e a consistência numeral, respectivamente; a completude numeral assegura que, quando uma fórmula-Σb 1(S12) ϕ(⃗x ) é tal que ⃗Q ⃗x .ϕ(⃗x ) é verdadeira, então este facto pode ser verificado dentro de S12, mais precisamente S12 ⊢ ⃗Q ⃗x .Prτ (⌜ϕ( •⃗ x )⌝). Este dois resultados são analisados de um ponto de vista matemático onde apresentamos as condições mínimas para os demonstrar e apresentamos consequências dos mesmos, e de um ponto de vista filosófico, onde relacionamos os mesmos com o Programa de Hilbert. A condição de derivabilidade “demonstração implica demonstrabilidade da demonstração” é uma das condições usadas para derivar o Segundo Teorema da Incompletude e sabemos ser muito sensível à teoria de base escolhida. Nós criámos uma teoria fraca G2 para estudar esta condição; esta é uma teoria para a classe de complexidade FLINSPACE. Também relacionámos propriedades de G2 com igualdades entre classes de complexidade computacional