Finite Words Research Articles

Foreword

This Special Issue of RAIRO, Theoretical Informatics and Applications is devoted to full versions of selected papers from the workshop Word Avoidability, Complexity and Morphisms which took place in Turku (Finland) on July 2004, as a satellite event of the conference ICALP'2004. The topics of this one day workshop concern particular aspects of Combinatorics on Words: string pattern avoidability, complexities of finite or infinite words, and free monoids morphims. The scientific program of the worshop consisted of two invited lectures given by A.E. Frid and J. Shallit as well as presentations of 8 contributed papers selected by the Program Committee from a total of 12 submissions. The Program Committee consisted of : A. Carpi (Perugia), J. Cassaigne (Marseille), J. Currie (Winnipeg), I. Petre (Turku), G. Richomme (Amiens), A. M. Shur (Ekaterinburg), and P. Séébold (Montpellier). This special issue contains the full text of five of the papers presented at the workshop. All of them were normally refereed and revised. I would like to thank everybody who contributed to the workshop and this issue including invited speakers, members of the program Committee, ICALP'2004's organization, authors and all referees of papers submitted to the worshop and this special issue. I am indebtful to J. Kärhumaki and J. Berstel for thinking of me to organize this workshop. Special thanks to P. Séébold for all his advices. Finally I am grateful to C. Choffrut, the editor-chief of the RAIRO-TIA for accepting to publish this special issue.

A generator of morphisms for infinite words

We present an algorithm which produces, in some cases, infinite words avoiding both large fractional repetitions and a given set of finite words. We use this method to show that all the ternary patterns whose avoidability index was left open in Cassaigne's thesis are 2-avoidable. We also prove that there exist exponentially many $\frac{7}{4}^+$ -free ternary words and $\frac{7}{5}^+$ -free 4-ary words. Finally we give small morphisms for binary words containing only the squares 2 , 1 2 and (01)² and for binary words avoiding large squares and fractional repetitions.

Iterated function systems, Ruelle operators, and invariant projective measures

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X X comes with a finite-to-one endomorphism r : X → X r\colon X\rightarrow X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in R d \mathbb {R}^d , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B , L B, L in R d \mathbb {R}^d of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) ( X , μ ) (X, \mu ) is based on a Markov process on certain paths. The probabilities are determined by a weight function W W on X X . From W W we define a transition operator R W R_W acting on functions on X X , and a corresponding class H H of continuous R W R_W -harmonic functions. The properties of the functions in H H are analyzed, and they determine the spectral theory of L 2 ( μ ) L^2(\mu ) . For affine IFSs we establish orthogonal bases in L 2 ( μ ) L^2(\mu ) . These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in R d \mathbb {R}^d .

Fuzzy regular languages over finite and infinite words

We consider finite automata over semirings and quemirings accepting finite and infinite words, respectively. We obtain Kleene theorems for fuzzy languages consisting of finite and infinite words. Furthermore, we introduce regular fuzzy grammars and linear fuzzy systems and we show that both of them specify the class of recognizable fuzzy languages consisting of finite and infinite words.

Some characterizations of finite Sturmian words

In this paper, we give some new characterizations of words which are finite factors of Sturmian words. An enumeration formula for primitive finite Sturmian words is given. Moreover, we provide two linear-time algorithms to recognize whether a finite word is Sturmian.

Handling Liveness Properties in (ω-)Regular Model Checking

Since the topic emerged several years ago, work on regular model checking has mostly been devoted to the verification of state reachability and safety properties. Though it was known that liveness properties could also be checked within this framework, little has been done about working out the corresponding details, and experimentally evaluating the approach. This paper addresses these issues in the context of regular model checking based on the encoding of states by finite or infinite words. It works out the exact constructions to be used in both cases, and solves the problem resulting from the fact that infinite computations of unbounded configurations might never contain the same configuration twice, thus making cycle detection problematic. Several experiments showing the applicability of the approach were successfully conducted.

Languages of finite words occurring infinitely many times in an infinite word

We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

A characterization of periodicity of bi-infinite words

A finite word is called bordered if it has a proper prefix which is also a suffix of that word. Costa proves in [Theoret. Comput. Sci. 290(3) (2003) 2053–2061] that a bi-infinite word w is of the form fgf ω ω , for some finite words f and g, if, and only if, there is a factorization w = suv , with u ∈ A * such that every factor s ′ uv ′ , with s ′ ≼ s and v ′ ⩽ v , is bordered. We present a shorter proof of that result in this paper.

Weak Bisimulation for Action-Type Coalgebras: (Extended Abstract)

We propose a coalgebraic definition of weak bisimulation for a class of coalgebras obtained from bifunctors over the category Set. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The transformation consists of two steps: First, the behaviour on actions is expanded to behaviour on finite words. Second, the behaviour on finite words is taken modulo the hiding of invisible actions, yielding behaviour on equivalence classes of words closed under silent steps. The coalgebraic definition is justified by two correspondence results, one for the classical notion of weak bisimulation of Milner and another for the notion of weak bisimulation for generative probabilistic transition systems as advocated by Baier and Hermanns.

Expressive power of existential first-order sentences of Büchi's sequential calculus

The aim of this paper is to study the first-order theory of the successor, interpreted on finite words. More specifically, we are interested in the hierarchy based on quantifier alternations (or Σ n -hierarchy). It was known (J. Comput. Syst. Sci. 25 (1982) 360–375) that this hierarchy collapses at level 2, but the expressive power of the lower levels was not characterized effectively. We give a semigroup theoretic description of the expressive power of B Σ 1 , the boolean combinations of existential formulas. We also give an O ( n 7 ) -time algorithm to decide whether the language accepted by a deterministic n-state automaton is expressible by a first-order sentence (respectively, a B Σ 1 -sentence).

An Investigation of the Fields of Bounded Formal Power Series by Means of Theory of Cuts

We consider a class K of real closed fields F, |F|=|G|=ℵ1, where G is a group of Archimedean classes of F, and cofinality of each symmetric gap of F is ℵ1. We will show that this class is exactly a class of all bounded formal power series R〚G,ℵ1〛, where G is a divisible Abelian group, card(G)=ℵ1, under CH. A nonstandard real line *R, which is η1-set belongs to this class; we will also consider a construction R〚G(L,P),ℵ1〛 of fields from this class, where L is a totally ordered set, P is a totally ordered field, G(L,P) is a group of finite words. It will be describes symmetric gaps of such two fields in K, which are not η1-set.

Hereditary properties of words

Let P be a hereditary property of words, i.e. , an infinite class of finite words such that every subword (block) of a word belonging to P is also in P . Extending the classical Morse-Hedlund theorem, we show that either P contains at least n+1 words of length n for every n or, for some N , it contains at most N words of length n for every n . More importantly, we prove the following quantitative extension of this result: if P has m ≤ n words of length n then, for every k ≥ n + m , it contains at most ⌈( m + 1)/2⌉⌈( m + 1)/2⌈ words of length k .

Formula omitted]-Regular languages are testable with a constant number of queries

We continue the study of combinatorial property testing. For a property ψ, an ɛ-test for ψ, for 0<ɛ⩽1, is a randomized algorithm that given an input x, returns “yes’’ if x satisfies ψ, and returns “no’’ with high probability if x is ɛ-far from satisfying ψ, where ɛ-far essentially means that an ɛ-fraction of x needs to be changed in order for it to satisfy ψ. In (Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, 1999, pp. 645–655), Alon et al. show that regular languages are ɛ-testable with a constant (depends on ψ and ɛ and independent of x) number of queries. We extend the result in (Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, 1999, pp. 645–655) to ω-regular languages: given a nondeterministic Büchi automaton A on infinite words and a small ɛ>0, we describe an algorithm that gets as input an infinite lasso-shape word of the form x·yω, for finite words x and y, samples only a constant number of letters in x and y, returns “yes’’ if w∈L(A), and returns “no’’ with probability 23 if w is ɛ-far from L(A). We also discuss the applicability of property testing to formal verification, where ω-regular languages are used for the specification of the behavior of nonterminating reactive systems, and computations correspond to lasso-shape words.

Properties of the complexity function for finite words

The subword complexity function \(p_{w}\) of a finite word \(w\) over a finite alphabet \(A\) with \(\operatorname*{card}A=q\geq1\) is defined by \(p_{w}(n)=\operatorname*{card}(F(w)\cap A^{n})\) for \(n\in\mathbb{N},\) where \(F(w)\) represents the set of all the subwords or factors of \(w\). The shape of the complexity function, especially its piecewise monotonicity, is studied in detail.The function \(h\) defined as \(h(n)=\min\left\{ q^{n},N-n+1\right\} \) for \(n\in\{0,1,\) \(...,N\}\) has values greater than or equal to those of the complexity function \(p_{w}\) for any \(w\in A^{N}\), i.e., \(p_{w}(n)\leq h(n)\) for all \(n\in\{0,1,...,N\}\). As a first result regarding \(h\), it is proved that for each \(N\in\mathbb{N}\) there exist words of length \(N\) for which the maximum of their complexity function is equal to the maximum of the function \(h\); a way to construct such words is described. This result gives rise to a further question: for a given \(N\), is there a word of length \(N\) whose complexity function coincides with \(h\) for each \(n\in\{0,1,...,N\}?\) The problem is answered in affirmative, with different constructive proofs for binary alphabets (\(q=2\)) and for those with \(q&gt;2.\) This means that for each \(N\in\mathbb{N},\) there exist words \(w\) of length \(N\) whose complexity function is equal to the function \(h\). Such words are constructed using the de Bruijn graphs.

The complexity of first-order and monadic second-order logic revisited

The model-checking problem for a logic L on a class C of structures asks whether a given L-sentence holds in a given structure in C. In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic. We show that unless PTIME=NP, the model-checking problem for monadic second-order logic on finite words is not solvable in time f( k)· p( n), for any elementary function f and any polynomial p. Here k denotes the size of the input sentence and n the size of the input word. We establish a number of similar lower bounds for the model-checking problem for first-order logic, for example, on the class of all trees.

Well-quasi-orders and regular ω-languages

In “ On regularity of context-free languages” [Theoret. Comput. Sci. 27 (1983) 311], Ehrenfeucht et al. showed that a set L of finite words is regular if and only if L is ⩽-closed under some monotone well-quasi-order (WQO) ⩽ over finite words. We extend this result to regular ω-languages. That is, (1) an ω-language L is regular if and only if L is ≼-closed under a periodic extension ≼ of some monotone WQO over finite words, and (2) an ω-language L is regular if and only if L is ≼-closed under a WQO ≼ over ω-words that is a continuous extension of some monotone WQO over finite words.

Recursion and topology on 2 ⩽ ω for possibly infinite computations

In the context of possibly infinite computations yielding finite or infinite (binary) outputs, the space 2 ⩽ω = 2 ∗ ∪ 2 ω appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2 ω , this space can be viewed (Section 3.5.2) as the simplest compact space native to computer science. In this paper we study some of its properties involving topology and computability. Though 2 ⩽ ω can be considered as a computable metric space in the sense of computable analysis, a direct and self-contained study, based on its peculiar properties related to words, is much illuminating. It is well known that computability for maps 2 ω → 2 ω reduces to continuity with recursive modulus of continuity. With 2 ⩽ ω , things get less simple. Maps 2 ω → 2 ⩽ω or 2 ⩽ω → 2 ⩽ω induced by input/output behaviours of Turing machines on finite or infinite words—which we call semicomputable maps—are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 ⩽ ω . Continuity asks for a stronger notion of computability. We prove for (semi)continuous and (semi)computable maps F : I→ O with I, O∈{2 ω,2 ⩽ω} a detailed representation theorem (Theorem 81) via functions f : 2 ∗ → 2 ∗ following two approaches: bottom-up from f to F and top-down from F to f.

Harmonic and gold Sturmian words

In the combinatorics of Sturmian words an essential role is played by the set PER of all finite words w on the alphabet A={a,b} having two periods p and q which are coprime and such that | w|= p+ q−2. As is well known, the set St of all finite factors of all Sturmian words equals the set of factors of PER. Moreover, the elements of PER have many remarkable structural properties. In particular, the relation Stand = A∪ PER {ab,ba} holds, where Stand is the set of all finite standard Sturmian words. In this paper we introduce two proper subclasses of PER that we denote by Harm and Gold. We call an element of Harm a harmonic word and an element of Gold a gold word. A harmonic word w beginning with the letter x is such that the ratio of two periods p/ q, with p< q, is equal to its slope, i.e., (| w| y +1)/(| w| x +1), where { x, y}={ a, b}. A gold word is an element of PER such that p and q are primes. Some characterizations of harmonic words are given and the number of harmonic words of each length is computed. Moreover, we prove that St is equal to the set of factors of Harm and to the set of factors of Gold. We introduce also the classes Harm and Gold of all infinite standard Sturmian words having infinitely many prefixes in Harm and Gold, respectively. We prove that Gold ∩ Harm contain continuously many elements. Finally, some conjectures are formulated.

A Hierarchical Completeness Proof for Propositional Interval Temporal Logic with Finite Time

We present a completeness proof for Propositional Interval Temporal Logic (PITL) with finite time which avoids certain difficulties of conventional methods. It is more gradated than previous efforts since we progressively reduce reasoning within the original logic to simpler reasoning in sublogics. Furthermore, our approach benefits from being less constructive since it is able to invoke certain theorems about regular languages over finite words without the need to explicitly describe the associated intricate proofs. A modified version of regular expressions called Fusion Expressions is used as part of an intermediate logic called Fusion Logic. Both have the same expressiveness as PITL but are lower-level notations which play an important role in the hierarchical structure of the overall completeness proof. In particular, showing completeness for PITL is reduced to showing completeness for Fusion Logic. This in turn is shown to hold relative to completeness for conventional linear-time temporal logic with finite time. Logics based on regular languages over finite words and !-words offer a promising but elusive framework for formal specification and verification. A number of such logics and decision procedures have been proposed. In addition, various researchers have obtained complete axiom systems by embedding and expressing the decision procedures directly within the logics. The work described here contributes to this topic by showing how to exploit some interesting links between regular languages and interval-based temporal logics.

Symbolic analysis of finite words: the complexity function

We present several properties of the complexity function of finite words, the function counting the number of different factors in a word, for each length. To establish a first set of properties, we use the de Bruijn graphs and the suffix tree representations of a word. This allows us to show some inequalities that control the variation as well as the maximal value of the complexity function. Motivated by the applications, we discuss the change of the complexity function when sliding or increasing the size of a window laid down on a sequence to be analysed.