Maurice Nivat

Fiftieth volume of theoretical computer science

The collection of TCS issues is about 1 meter high, 17,000 pages long and it contains 1100 papers. When in 1974 Einar Fredriksson and myself started talking about the creation of a journal dedicated to Theoretical Computer Science we were very far from even dreaming that it could take such an extension within twelve years. We were also a bit shy: what could such a journal, very theoretical indeed and hard to read, be useful to, and who would read it? Fortunately, some people encouraged us and indeed helped us a lot, Mike Paterson who was at that time President of EATCS and who accepted to become Associate Editor, Albert Meyer who was a very active editor at the beginning, Arto Salomaa, who was to become President of EATCS shortly afterwards. Indeed, I should mention all the first members of the Editorial Board, for TCS would never have come to existence without them. Theoretical Computer Science is not a clearly defined discipline with neat borderlines: it is more a state of mind, the conviction that the observed computation phenomena can be formally described and analysed as any physical phenomenon; the conviction that such a formal description helps to understand these phenomena and to master them in order to design better algorithms, better computers, better systems. Our fundamental activity is not to prove theorems in strange mathematical theories, it is to model a complicated reality and in this respect it has to be compared with theoretical physics or what we call in French “Mecanique rationnelle”. This comparison can be pursued rather far, for we also use all possible mathematical concepts and methods and when we do not find appropriate ones in traditional mathematics we create them. The aim is quite clear: using the compact and unambiguous language of mathematics brings to life concepts and methods which will be useful to all designers, builders and users of computer systems, exactly in the same way as matrix calculus or Fourier series and transforms are useful to all engineers and technicians in the electric and electronic industry. And when one thinks about the amount of time it took to build the mathematical theory of matrices and to polish and simplify it up to the state in which it could be taught to all future engineers and become a tool in daily use, we can be extremely satisfied by the development of Theoretical Computer Science. It is true that concepts and methods which were still vague and unclear when TCS was created became essential tools for all industrial designers and manufacturers, in algorithmics, in semantics, in automata theory and control, etc. . . . Certainly, TCS can be proud to have contributed to this development. Coming back to what I was saying a few minutes ago, this contribution was made possible only by the miraculous fact that the first members of the Editorial Board were sharing the same conviction about the necessity of Theoretical Computer Science

Reconstructing convex polyominoes from horizontal and vertical projections

In [1], we studied the problem of reconstructing a discrete set 5 from its horizontal and vertical projections. We defined an algorithm that establishes the existence of a convex polyomino Λ whose horizontal and vertical projections are equal to a pair of assigned vectors (H,V), with H ∈ ℕ m and V ∈ ℕ n . Its computational cost is O(n4m4). In this paper, we introduce some operations for recontructing convex polyominoes by means of vectors Hs and Vs partial sums. These operations allows us to define a new algorithm whose complexity is less than O(n2m2).

Adherences of languages

Abstract This paper studies context-free sets of finite and infinite words. In particular, it gives a natural way of associating to a language a set of infinite words. It then becomes possible to begin a study of families of sets of infinite words rather similar to the classical studies of families of languages.

On translating one polyomino to tile the plane

Given a polyomino, we prove that we can decide whether translated copies of the polyomino can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider. If such a tiling exists the polyomino is called anexact polyomino. Further, every such tiling of the plane by translated copies of the polyomino is half-periodic. Moreover, all the possible surroundings of an exact polyomino are described in a simple way.

Metric interpretations of infinite trees and semantics of non deterministic recursive programs

Abstract In order to define semantics of non deterministic recursive programs we are led to consider infinite computations and to replace the structure of cpo on computation domain by the structure of complete metric space. In this setting we prove the two main theorems of semantics: 1. (i) equivalence between operational and denotational semantics, where this last one is defined 2. as a greatest fixed point for inclusion, 3. (ii) the one-many function computed by a program is the image of the set of trees computed by 4. the scheme associated with it.

Sur diverses familles de langages fermées par transduction rationnelle

SummaryThe main theorem gives a sufficient condition for an AFL to be the closure under union of the set of images under rational transductions of any of its sets of generators. All the AFLs known to have this property satisfy the given condition. As an application we give a short proof of the fact that every generator of the AFL of algebraic (context-free) languages is a faithful generator, i.e. can be mapped onto every algebraic language by a faithful (& free) rational transduction.

Ensembles reconnaissables de mots biinfinis

The purpose of automata theory is to study and classify those properties of words that may be defined by a finite structure, say a finite automaton or a finite monoid. It seems natural to consider the same problem for infinite words. This amounts to studying the asymptotic behaviour of finite automata. As is well-known, this breaks the equivalence between determinism and non-determinism of finite automata. The study of the infinite behaviour of finite automata is based on a deep theorem due to B-&-uuml;chi and Mc Naughton: the recognizable sets of infinite words are the finite boolean combinations of deterministic ones (i.e. recognized by deterministic automata). The aim of this paper is to build an analogous theorem for two-sided infinite sequences. We define a biinfinite word as the equivalence class under the shift of a two-sided infinite sequence. The recognizable sets of biinfinite words are defined in a natural way and one is led to a two-sided notion of determinism. This notion seems to be new and justifies the consideration of biinfinite words. The main result of this paper is the extension to biinfinite words of the theorem of B-&-uuml;chi and Mc Naughton: the recognizable sets of biinfinite words are the finite boolean combinations of deterministic ones (Theorem 3.1). There exist three available proofs of B-&-uuml;chi-Mc Naughtons theorem. The original one by Mc Naughton [4] is hard to read. The proof given by Eilenberg in his book [2] has been constructed by Sch-&-uuml;tzenberger and Eilenberg from Mc Naughtons proof; it is similar to that of Rabin [5]. Finally, Sch-&-uuml;tzenberger gave a further proof in [6], which makes the argument more direct by using the methods of the theory of finite monoids. The proof of our main result follows closely Sch-&-uuml;tzenbergers method. This method allows to reduce the two-sided case to the one-sided case, although this seems very difficult to obtain directly. In the first section, we briefly recall the theory of the one-sided infinite behaviour of finite automata. In particular, we give Sch-&-uuml;tzenbergers proof of B-&-uuml;chi-Mc Naughtons theorem. The elements of this proof are used in the proof of our main result. In the second section we define the notions of biinfinite word, biautomaton and deterministic biautomaton. The last section contains the proof of our main result.

The algebraic semantics of recursive program schemes

This is a survey of general properties of recursive program schemes and classes of interpretations.

Reversal-Bounded Acceptors and Intersections of Linear Languages

A Turing machine whose behavior is restricted so that each read-write head can change its direction only a bounded number of times is reversal-bounded. Here we consider nondeterministic multitape acceptors which are both reversal-bounded and also operate in linear time. Our main result shows that such an acceptor need have only three pushdown stores as auxiliary storage, each pushdown store need make only one reversal, and the acceptor can operate in real time.

The number of convex polyominoes reconstructible from their orthogonal projections

Abstract Many problems of computer-aided tomography, pattern recognition, image processing and data compression involve a reconstruction of bidimensional discrete sets from their projections. [3–5,10,12,16,17]. The main difficulty involved in reconstructing a set Λ starting out from its orthogonal projections ( V,H ) is the ‘ambiguity’ arising from the fact that, in some cases, many different sets have the same projections ( V,H ). In this paper, we study this problem of ambiguity with respect to convex polyominoes, a class of bidimensional discrete sets that satisfy some connection properties similar to those used by some reconstruction algorithms. We determine an upper and lower bound to the maximum number of convex polyominoes having the same orthogonal projections ( V,H ), with V ∈ N n and H ∈ N m . We prove that under these connection conditions, the ambiguity is sometimes exponential. We also define a construction in order to obtain some convex polyominoes having the same orthogonal projections.