Pierre Valarcher

• PhD-HDR
• Head of Department at Paris-Est Créteil University

We consider computable functionals mapping the Baire space into the set of integers. By continuity, the value of the functional on a given function depends only on a "critical" finite part of this function. Care: there is in general no way to compute this critical finite part without querying the function on an arbitrarily larger finite part! Never...

We formalize the set of algorithms computing elements of the set of primitive recursive functions (PR) as the set of abstract state machines (ASMs) which have their running lengths com- putable by PR functions. Then we show that there exists a programming language (only computing PR functions) which implements any previously defined algorithm prese...

We aim at extending the B language in order to build ASM programs which are correct with respect to B-like logical specifications. On the one hand, the main strengths of the B formal method are: i) the ability to express logical statements, and ii) the construction of a correct implementation by refinement. On the other hand, from our viewpoint, th...

It is well-known that Abstract State Machines (ASMs) can simulate "step-by-step" any type of machines (Turing machines, RAMs, etc.). We aim to overcome two facts: 1) simulation is not identification, 2) the ASMs simulating machines of some type do not constitute a natural class among all ASMs. We modify Gurevich's notion of ASM to that of EMA ("Evo...

We extend Meyer and Ritchie's Loop language with higher-order procedures and procedural variables and we show that the resulting programming language (called Loop ω ) is a natural imperative counterpart of Gödel System T. The argument is two-fold: (1) we define a translation of the Loop ω language into System T and we prove that this translation ac...

We give a complete characterization of the class of functions that are the intensional behaviours of primitive recursive (PR) algorithms. This class is the set of primitive recursive functions that have a null basic case of recursion. This result is obtained using the property of ultimate unarity and a geometrical approach of sequential functions...

The substitution of X by X^2 in binomial polynomials generates sequences of integers by Mahler's expansion. We give some properties of these integers and a combinatorial interpretation with covers by projection. We also give applications to the classification of boolean functions. This sequence arose from our previous research on classification and...

1. Introduction. This paper is devoted to the study of the expressive power of an elementary imperative programming language similar to the Loop language described by Meyer and Ritchie in [MR76]. While the λ-calculus is usually used to describe the denotational semantics of programming languages, we exploit it to encode the operational semantics of...

We investigate the structure of "worst-case" quasi reduced ordered decision diagrams and boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of "hard" boolean functions as functions whose QROBDD are "worst-case" ones. So...

HFE (Hidden Field Equations) is a public key cryptosystem using univariate polynomials over finite fields. It was proposed by J. Patarin in 1996. Well chosen parameters during the construction produce a system of quadratic multivariate polynomials over \mathbbF2{\mathbb{F}_2} as the public key. An enclosed trapdoor is used to decrypt messages. We...

We dealt with the cryptosystem HFE which we tried to cryptanalyse using BDD.

This paper is a contribution to the development of a theory of behaviour of programs. We study an intensional behaviour of system T of Gödel that is devoted to capturing not which function is computed but how it is computed. This intensional behaviour is captured by a denotational semantics in the domain of lazy natural numbers. It is shown that al...

The set of primitive recursive functions (pr) accepts many definitions. We study three languages (presented as rewriting rules) that compute pr functions and show that they do not have the same intensional behaviour (a new function associate to algorithm) from an input/output point of view (i.e. they do not compute in the same way): classical Primi...

We recall the definition of the class of primitive recursive algorithms (APRA) and we prove that there exists a functional programming lan-guage (primitive recursion with variable parameters or fragment of sys-tem T 1 of Gödel) that simulates all algorithms of APRA in lock-step (one step is simulated by a constant number of steps). The class AP RA...

We investigate the structure of "worst-case" quasi reduced ordered decision diagrams (or boolean graphs) and boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of "hard" boolean functions as functions whose boolean graph...

Sequential algorithms over countable data structures are mod-elized by Gurevich's Abstract State Machines as the iteration of a func-tional which goes from a finite product of function spaces into itself. ASM functionals have the following properties for some k: (1) k they modify their argument on at most k points, (2) k their modulus of continuity...