Dominique Perrin

Two-way string-matching

A new string-matching algorithm is presented, which can be viewed as an intermediate between the classical algorithms of Knuth, Morris, and Pratt on the one hand and Boyer and Moore, on the other hand. The algorithm is linear in time and uses constant space as the algorithm of Galil and Seiferas. It presents the advantage of being remarkably simple which consequently makes its analysis possible. The algorithm relies on a previously known result in combinatorics on words, called the Critical Factorization Theorem,which relates the global period of a word to Its local repetitions of blocks

First-order logic and star-free sets

formulas are built up in the usual way by means of the connectives -I, v , A and the quantifiers 3 and V bounding up both types of variables. Now, we say that a word w on the alphabet A satisfies such a sentence 4 if 6 is true when variables are interpreted as integers, set-variables are interpreted as set of integers and the formula X,X is interpreted as “the letter in position x in w is an a.” McNaughton [3] was the first to consider the case where the set of formulas is restricted to first-order, that is, when set-variables are ignored. He proved that the languages defined in this way are precisely the star-free languages, that is, all languages obtained from finite languages by boolean operations and concatenation product. Later on, star-free languages have been considerably studied. First, a fundamental result of Schiitzenberger shows that star-free languages are exactly the languages recognized by an aperiodic finite monoid (i.e., a monoid all of whose groups are trivial). Further on, a great number of subclasses of star-free languages have been studied [6]. Among the most famous, let us quote the locally testable languages studied by McNaughton and Brzozowski and Simon and the piecewise testable languages, introduced by Simon. Star-free languages are defined by two types of operations: boolean operations on one hand and concatenation product on the other hand. This naturally defines a hierarchy based on the alternative use of these operations. The hierarchy was originally introduced by Brzozowski who showed with Knast [ 11 that the inclusion was proper on each level. Furthermore the class of locally testable languages 393 0022~0000/86

The origins of combinatorics on words

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On the expressive power of temporal logic

We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a by-product of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this two-sided interaction.

Symbolic dynamics and finite automata

Abstract We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the “restricted” temporal logic (RTL), obtained by discarding the “until” operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads to a polynomial time algorithm to check whether the formal language accepted by an n -state deterministic automaton is RTL-expressible.

A note on the Burrows-Wheeler transformation

Symbolic dynamics is a field which was born with the work in topology of Marston Morse at the beginning of the 1920s [44]. It is, according to Morse, an “algebra and geometry of recurrence”. The idea is the following. Divide a surface into regions named by certain symbols. We then study the sequences of symbols obtained by scanning the successive regions while following a trajectory starting from a given point. A further paper by Morse and Hedlund [45] gave the basic results of this theory. Later, the theory was developed by many authors as a branch of ergodic theory (see for example the collected works in [59] or [12]). One of the main directions of research has been the problem of the isomorphism of shifts of finite type (see below the definition of these terms). This problem is not yet completely solved although the latest results of Kim and Roush [35] indicate a counterexample to a long-standing conjecture formulated by F. Williams [61].

Chains and Superchains for ω-Rational Sets, Automata and Semigroups

We relate the Burrows-Wheeler transformation with a result in combinatorics on words known as the Gessel-Reutenauer transformation.

On the generating sequences of regular languages on k symbols

We introduce several equivalent notions that generalize ones introduced by Klaus Wagner for finite Muller automata under the name of chains and superchains. We define such objects in relation to ω-rational sets, Muller automata or also ω-semigroups. We prove their equivalence and derive some basic properties of these objects. In a subsequent paper, we show how these concepts allow us to derive a new presentation of the hierarchy due to K. Wagner and W. Wadge.

Codes, unambiguous automata and sofic systems

The main result is a characterization of the generating sequences of the length of words in a regular language on <i>k</i> symbols. We say that a sequence <i>s</i> of integers is regular if there is a finite graph <i>G</i> with two vertices <i>i, t</i> such that <i>s</i>_<i>n</i> is the number of paths of length <i>n</i> from <i>i</i> to <i>t</i> in <i>G</i>. Thus the generating sequence of a regular language is regular. We prove that a sequence <i>s</i> is the generating sequence of a regular language on <i>k</i> symbols if and only if both sequences <i>s</i> = (<i>s</i>_<i>n</i>)_<i>n</i>≥0 and <i>t</i> = (<i>k</i>^<i>n</i> − <i>s</i>_<i>n</i>)_<i>n</i>≥0 are regular.

A hierarchy of shift equivalent sofic shifts

We study the relationship between codes and unambiguous automata inside a sofic system. We show that a recognizable set is a code in a sofic system if and only if a particular automaton associated to the set and the shift is unambiguous. We discuss an example of a finite complete code in a sofic system in connection with the factorization conjecture.