Serge Burckel

The wellordering on positive braids

Abstract This paper studies Artins braid monoids using combinatorial methods. More precisely, we investigate the linear ordering defined by Dehornoy. Laver has proved that the restriction of this ordering to positive braids is a wellordering. In order to study this order, we develop a natural wellordering ⪡ on the free monoid on infinitely many generators by representing words as trees. Our construction leads to a (new) normal form for (positive) braids. Our main result is that the restriction of our order ⪡ to the normal braid words coincides with the restriction of Dehornoys ordering to positive braids. Our method gives an alternative proof of Lavers result using purely combinatorial arguments and gives the order type, namely ω ω ω .

Closed iterative calculus

Abstract We prove that any mapping on {0,1} n can be computed by an iterative calculus only involving the arguments.

Functional equations associated with congruential functions

Abstract It has been proved by Conway that the general problem of whether the Collatz type functions converge is undecidable. We present a modified proof of this result, which enables us to state new undecidability properties concerning functional equations.

Three generators for minimal writing-space computations

We construct, for each integer n , three functions from {0,1} n to {0,1} such that any boolean mapping from {0,1} n to {0,1} n can be computed with a finite sequence of assignations only using the n input variables and those three functions.

Computation of the Ordinal of Braids

There exists a linear ordering on braids coming from left distributivity assumptions. The restriction of this order to positive braids is a wellordering. We present here some tools and an effective algorithm for the computation of the rank of any positive braid in this wellordering.

Sequential computation of linear Boolean mappings

We prove that any linear Boolean mapping of dimension n can be computed with a double sequence of linear assignments of the n variables. The proof is effective and gives a decomposition of Boolean matrices and directed graphs.

Mapping Computation with No Memory

We investigate the computation of mappings from a set S n to itself with in situ programs , that is using no extra variables than the input, and performing modifications of one component at a time. We consider several types of mappings and obtain effective computation and decomposition methods, together with upper bounds on the program length (number of assignments). Our technique is combinatorial and algebraic (graph coloration, partition ordering, modular arithmetics). For general mappings, we build a program with maximal length 5n *** 4, or 2n *** 1 for bijective mappings. The length is reducible to 4n *** 3 when |S | is a power of 2. This is the main combinatorial result of the paper, which can be stated equivalently in terms of multistage interconnection networks as: any mapping of {0,1} n can be performed by a routing in a double n -dimensional Benes network. Moreover, the maximal length is 2n *** 1 for linear mappings when S is any field, or a quotient of an Euclidean domain (e.g. ***/s ***). In this case the assignments are also linear, thereby particularly efficient from the algorithmic viewpoint. The in situ trait of the programs constructed here applies to optimization of program and chip design with respect to the number of variables, since no extra writing memory is used. In a non formal way, our approach is to perform an arbitrary transformation of objects by successive elementary local transformations inside these objects only with respect to their successive states.

Quadratic Sequential Computations of Boolean Mappings

Abstract This paper proposes a constructive proof that any mapping on n boolean variables can be computed by a straight-line program made up of n2 assignments of the n input variables.

Syntactical Methods for Braids of Three Strands

We investigate the three-strand positive braid monoid. We propose a syntactical linear time algorithm for the word problem, a nearly canonical rewriting system and the computation of the growth function.

In Situ Design of Register Operations

We present methods to design programs or electronic circuits, for performing any operation on k registers of any sizes in a processor, in such a way that one uses no other working memory (such as other registers or external memories). In this way, any operation is performed with at most 4k - 3 assignments of these registers, or 2k - 1 when the operation is linear or bijective.