Christian Choffrut

Combinatorics of words

This is a survey on combinatorics of words to appear as a chapter in Handbook of Formal Languages. The topics covered in details are: defect effect, equations as properties of words, periodicity, finiteness conditions, avoidabilty and subword complexity.

On real-time cellular automata and trellis automata

SummaryIt is shown that f(n)-time one-way cellular automata are equivalent to f(n)-time trellis automata, the real-time one-way cellular automata languages are closed under reversal, the 2n-time one-way cellular automata are equivalent to real-time cellular automata and the latter are strictly more powerful than the real-time one-way cellular automata.

The Commutation of Finite Sets: a Challenging Problem

We prove that given a set X of two nonempty words, a set Y of nonempty words commutes with X if and only if either Y is a union of powers of X or

Long words: the theory of concatenation and o-power

X,Y\subseteq t^+

A Generalization of Ginsburg and Rose's Characterization of G-S-M Mappings

for some primitive word t. We also show that the same holds for certain special types of codes, but does not hold in general for sets of cardinality at least four.

On extendibility of unavoidable sets

It is shown that for any set A, the algebra of ordinal words on the alphabet A equipped with the operations of concatenation and ?-power is axiomatized by the equationsx·(y·z)=(x·y)·z,(x·y)?=x·(y·x)?,(xn)?=x?,n?1.Indeed, the algebra freely generated by A in the variety determined by these equations is the algebra of tail-finite ordinal words of length <?? on the alphabet A. It is further shown that this collection of identities cannot be replaced by any finite set. Last, a polynomial algorithm is given for recognizing when two terms denote the same tail-finite ordinal word.

Folding of the plane and the design of systolic arrays

We generalize Ginsburg and Roses characterization of g-s-m mappings to the broader family of so-called subsequential functions, introduced by M.P.Schutzenberger

Properties of Finite and Pushdown Transducers

Abstract A subset X of a free monoid A∗ is said to be unavoidable if all but finitely many words in A∗ contain some word of X as a subword. A. Ehrenfeucht has conjectured that every unavoidable set X is extendible in the sense that there exist x ϵ X and a ϵ A such that (X − {x}) ∪ {xa} is itself unavoidable. This problem remains open, we give some partial solutions and show how to efficiently test unavoidability, extendibility and other properties of X related to the problem.

Some decision problems on integer matrices

Systolic arrays were first introduced by Kung (see, e.g., [2] and [3]) as devices composed of processors of a few different types, which are regularly and locally connected. These processors are activated in a synchronous way by a unique clock which is the only global communication between them. This paper is a continuation of the work presented in [I] where ‘folding’ has been proposed as a technique for the design of systolic arrays. Here we first study the power of folding as a general geometric transformation. Then, as an application we show that two congruent sequences on a ‘regular’ grid can be identified by a limited number of foldings (cf. Theorem 3.5). This result can be used in the design of systolic arrays. Because of the importance of this motivation we start with an example of a systolic array which has been already briefly described in [ 11. Consider Kung and Leiserson’s hex-connected processor array for matrix multiplication [3, pp. 276-2801 modified in such a way that it applies to dense matrices. Fig. 1 illustrates the case where the dimension n of the matrix is equal to 3: the left-to-right and the right-to-left flows correspond to the two matrices A and B to be multiplied and the bottom-to-top flow to the product C = A x B. Each node represents an inner product step processor, i.e., a processor computing one step of a scalar product: s +s + ab (see Fig. 2). Assume we want to use this array to compute the different powers of a matrix A, which basically amounts to computing A, A X A = A2,. . . , Ak X Ak = A2k ).... One solution is to iteratively feed the different outputs of step k coming out in the upper part of the array, to both left and right inputs of step k + 1, i.e., to connect each yi to (Y~ and pi. However, in doing this we would create non-local connections and break the regularity of the layout. Instead, we can first fold the array (as one would fold a sheet of paper) along the axis 1, the righthand side coming on top of the left-hand side. The new array computes the same functions as the original one, very much in the same way as a Turing machine with a one way infinite working tape can simulate a Turing machine with a two-way infinite working tape, by folding the latter. Then, we can fold the new array along the axis 2 (the left-hand side on top of the right-hand side) and again along the axis 3 (the right-hand side on top of the left-hand side). The processors cxi, pi, yi will eventually occupy the same place and we must connect them to each other thus introducing only local connections. As a result the regularity of the initial lay-out is preserved. The price to pay for it is that the new array consists of up to 8 = 23 more

On Fatou properties of rational languages

We consider the subfamilies of rational and pushdown transducers and corresponding translations (relations) which are most frequently encountered in the literature. We survey some of the known results on the characterization, factorization, closure properties, decision problems and comparisons of classes and give new results on these properties using either direct proofs or results from other theories such as homomorphic equivalence.