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Abstract. We a give an intrinsic characterization of the class of functions which are computable in NC 1 that is by a uniform, logarithmic depth and polynomial size family circuit. Recall that the class of functions in ALogTime, that is in logarithmic time on an Alternating Turing Machine, is NC 1. Our characterization is in terms of first order functional programming languages. We define measure-tools called Supinterpretations, which allow to give space and time bounds and allow also to capture a lot of program schemas. This study is part of a research on static analysis in order to predict program resources. It is related to the notion of Quasi-interpretations and belongs to the implicit computational complexity line of research. 1

Abstract. In a previous paper [16], the sup-interpretation method was proposed as a new tool to control memory resources of first order functional programs with pattern matching by static analysis. Basically, a sup-interpretation provides an upper bound on the size of function outputs. In this former work, a criterion, which can be applied to terminating as well as non-terminating programs, was developed in order to bound polynomially the stack frame size. In this paper, we suggest a new criterion which captures more algorithms computing values polynomially bounded in the size of the inputs. Since this work is related to quasi-interpretations, we compare the two notions obtaining two main features. The first one is that, given a program, we have heuristics for finding a sup-interpretation when we consider polynomials of bounded degree. The other one consists in the characterizations of the set of function computable in polynomial time and in polynomial space. 1

Abstract. We show that in the context of orthogonal term rewriting systems, derivational complexity is an invariant cost model, both in innermost and in outermost reduction. This has some interesting consequences for (asymptotic) complexity analysis, since many existing methodologies only guarantee bounded derivational complexity. 1

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c t i v it y e p o r t 2009 Table of contents

c t i v it y e p o r t 2007 Table of contents

Abstract. We propose a new method to control memory resources by static analysis. For this, we introduce the notion of sup-interpretation which bounds from above the size of function outputs. We establish a criteria for which the stack frame size is polynomially bounded. The criteria analyses terminating as well as non-terminating programs. This method applies to first order functional programming with pattern matching. This work is related to quasi-interpretations but we are now able to determine resources of different algorithms and it is easier to perform an analysis with this new tools. 1

Abstract. In a previous work we introduced Dual light affine logic (DLAL) ([BT04]) as a variant of Light linear logic suitable for guaranteeing complexity properties on lambda calculus terms: all typable terms can be evaluated in polynomial time by beta reduction and all Ptime functions can be represented. In the present work we address the problem of typing lambda-terms in second-order DLAL. For that we give a procedure which, starting with a term typed in system F, determines whether it is typable in DLAL and outputs a concrete typing if there exists any. We show that our procedure can be run in time polynomial in the size of the original Church typed system F term. 1.