Theoretical Informatics

We define several notions of language complexity for finite words, and use them to define and compute some new complexity functions for infinite sequences. In particular, they give a new characterization of Sturmian sequences and discriminate between Sturmian sequences with bounded or unbounded partial quotients.

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).

This paper is concerned with the computation of pseudovariety joins involving the pseudovariety LI of locally trivial semigroups. We compute, in particular, the join of LI with any subpseudovariety of CR h m N, the Mal'cev product of the pseudovariety of completely regular semigroups and the pseudovariety of nilpotent semigroups. Similar studies are conducted for the pseudovarieties K, D and N, where K (resp. D) is the pseudovariety of all semigroups S such that eS = e (resp. Se = e) for each idempotent e of S.

Throughout this paper, a semigroup S will denote a torsion free grading monoid, and it is a non-zero semigroup with 0. The operation is written additively. The aim of this paper is to study semigroup version of an integral domain ([1],[3],[4] and [티).

The definition of the Green-Kehayopulu relation H in le-semigroups mimics the definition of the usual Green relation H in plain semigroups. We show, however, that certain properties of H-classes essentially differ from those of H-classes: a non-singleton H-class cannot be a subgroup and an H-class H satisfying "Green's condition" (there exist a, b ∈ H such that ab ∈ H) need not constitute a subsemigroup. We provide various conditions that ensure that an H-class forms a subsemigroup.

The operator that constructs the pseudovariety generated by the idempotent-generated semi-groups of a given pseudovariety is investigated. Several relevant examples of pseudovarieties generated by their idempotent-generated elements are given, as well as some properties of this operator. Particular attention is paid to the pseudovarieties in {J, R, L, DA} concerning this operator and their generator ranks and idempotent-generator ranks.

This paper is concerned with the computation of pseudovariety joins involving the pseudovariety LI of locally trivial semigroups. We compute, in particular, the join of LI with any subpseudovariety of CR h m N, the Mal'cev product of the pseudovariety of completely regular semigroups and the pseudovariety of nilpotent semigroups. Similar studies are conducted for the pseudovarieties K, D and N, where K (resp. D) is the pseudovariety of all semigroups S such that eS = e (resp. Se = e) for each idempotent e of S.

In combinatorics of words, a concatenation of k consecutive equal blocks is called a power of order k. In this paper we take a different point of view and define an anti-power of order k as a concatenation of k consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. As a consequence, we show that every aperiodic uniformly recurrent word begins in an anti-power of order k, for any choice of k > 1. We further show that any infinite word avoiding anti-powers of order 3 is ultimately periodic, and that there exist aperiodic words avoiding anti-powers of order 4. We also show that there exist aperiodic recurrent words avoiding anti-powers of order 6.

In algebraic terms, the insertion of n-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality 1 x n. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity x n = 1. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under n-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from 1 x n in which both sides are regular elements with respect to the upper bound.

Rudimentary relations are those relations over natural integers that are deÿned by a ÿrst-order arithmetical formula, in which all quantiÿcations are bounded by some variable. Paris and Wilkie conjectured that the class R of rudimentary relations is not closed under modular counting or constant-bounded recursion. A consequence would be the proper containment of R in LINSPACE. It is now known that the closure of R under counting modulo an unsolvable non-abelian ÿnite group, or modulo the semigroup of all binary relations on a ÿnite set, contains ALINTIME. We characterize here the semigroups for which the counting is reducible to modular counting, or contains counting modulo an unsolvable non-abelian ÿnite group.

It is easy to see that if G is a non-abelian group of unitary matrices, then for no members A and B of G can the rank of AB − BA be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup S of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, S is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum G 1 ⊕ G 2 where G 1 is at most 3 × 3 and G 2 is abelian.

Weakly and strongly quasiperiodic morphisms are tools introduced to study quasiperiodic words. Formally they map respectively at least one or any non-quasiperiodic word to a quasiperiodic word. Considering them both on finite and infinite words, we get four families of morphisms between which we study relations. We provide algorithms to decide whether a morphism is strongly quasiperiodic on finite words or on infinite words.

Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.

Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. As a consequence, we characterize episturmian words having a unique directive word.

Weakly and strongly quasiperiodic morphisms are tools introduced to study quasiperiodic words. Formally they map respectively at least one or any non-quasiperiodic word to a quasiperiodic word. Considering them both on finite and infinite words, we get four families of morphisms between which we study relations. We provide algorithms to decide whether a morphism is strongly quasiperiodic on finite words or on infinite words.

In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor $w$ of an infinite word $x=x_1x_2x_3\cdots$ with each $x_i$ belonging to a fixed finite set $\mathbb{A},$ we say $w$ is closed if either $w\in \mathbb{A}$ or if $w$ is a complete first return to some factor $v$ of $x.$ Otherwise $w$ is said to be open. We show that for an aperiodic word $x\in \mathbb{A}^\mathbb{N},$ the complexity functions $Cl_x$ (resp. $Op_x)$ that count the number of closed (resp. open) factors of $x$ of each given length are both unbounded. More precisely, we show that if $x$ is aperiodic then $\liminf_{n\in \mathbb{N}} Op_x(n)=+\infty$ and $\limsup_{n\in S} Cl_x(n)=+\infty $ for any syndetic subset $S$ of $\mathbb{N}.$ However, there exist aperiodic infinite words $x$ verifying $\liminf_{n\in \mathbb{N}}Cl_x(n)<+\infty.$ Keywords: word complexity, periodicity, return words.

Colour refinement is a basic algorithmic routine for graph isomorphism testing, appearing as a subroutine in almost all practical isomorphism solvers. It partitions the vertices of a graph into "colour classes" in such a way that all vertices in the same colour class have the same number of neighbours in every colour class. Tinhofer [27], Ramana, Scheinerman, and Ullman [23] and Godsil [12] established a tight correspondence between colour refinement and fractional isomorphisms of graphs, which are solutions to the LP relaxation of a natural ILP formulation of graph isomorphism. We introduce a version of colour refinement for matrices and extend existing quasilinear algorithms for computing the colour classes. Then we generalise the correspondence between colour refinement and fractional automorphisms and develop a theory of fractional automorphisms and isomorphisms of matrices. We apply our results to reduce the dimensions of systems of linear equations and linear programs. Specifically, we show that any given LP L can efficiently be transformed into a (potentially) smaller LP L whose number of variables and constraints is the number of colour classes of the colour refinement algorithm, applied to a matrix associated with the LP. The transformation is such that we can easily (by a linear mapping) map both feasible and optimal solutions back and forth between the two LPs. We demonstrate empirically that colour refinement can indeed greatly reduce the cost of solving linear programs.

Recently, Berthé and the author developed a method to construct a multi-dimensional Sturmian word as a limit of a sequence of words on lattice domains. In the present paper, the reach of the approach is studied in the special case of Rauzy substitutions.

It is known that if trace is permutable on a semigroup S of complex matrices, i.e., tr(ABC) = tr(BAC) for all A, B, C in S, then S is triangularizable. We study an approximate version of this condition: | tr(ABC − BAC)| ≤ ερ(A)ρ(B)ρ(C) for all A, B, C in S, where ρ is the spectral radius. We show that this condition with ε < 3 yields commutativity for compact groups and triangularizabilty for certain groups including connected ones. For general semigroups additional assumptions are needed. Moreover, we show that any property on semigroups of matrices that satisfies certain pretriangularizing conditions, yields similar conclusions.

It is easy to see that if G is a non-abelian group of unitary matrices, then for no members A and B of G can the rank of AB − BA be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup S of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, S is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum G 1 ⊕ G 2 where G 1 is at most 3 × 3 and G 2 is abelian.

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).

In this paper we provide a new, abstract characterisation of classical Rees matrix semigroups over monoids with zero. The corresponding abstract class of semigroups is obtained by abstracting a number of algebraic properties from completely 0-simple semigroups: in particular, the relationship between arbitrary elements and idempotents.

Blanchet-Sadri, F. (1992) Games, equations and dot-depth two monoids, Discrete Applied Mathematics 39, 99-111. Made available courtesy of Elsevier: http://www.elsevier.com ***Reprinted with permission. No further reproduction is authorized without written permission from Elsevier. This version of the document is not the version of record. Figures and/or pictures may be missing from this format of the document.***

to give, besides a typical result, an effectively verifiable sufficient condition for T and T-I to be nonisomorphic (not conjugate to their inverse) (Lemma 3). 9 By the same scheme he obtained a result on the nonisomorphism in the typical situation of measurable actions Tg and T~ where ~ is an arbitrary nontrivial automorphism sending a set of generators (gl ..... gk) of the group m r into (gl iI, .... gkik), respectively, where ij e {-i, i}.

We show that the interval of the lattice of semigroup pseudovarieties between the pseudovarieties generated by all semigroups of full and respectively, partial, order-preserving transformations of a finite chain, contains a chain isomorphic to the chain of real numbers (with the usual order). Similar results are proved for several related intervals.

We provide a classification of eventually periodic subshifts up to conjugacy and flow equivalence. We use our results to prove that each skew Sturmian subshift is conjugate to exactly one other skew Sturmian subshift and that all skew Sturmian subshifts are flow equivalent to one another.

In the present brief article we speculate about the mutual equivalence of Church-Turing and Zuse-Fredkin theses. Since the Church-Turing thesis is widely accepted while the Zuse-Fredkin thesis is not, we propose their “near-equivalence” as a strong argument in support of the Zuse-Fredkin thesis.

For each Schur function, or generalized matrix function, corresponding to a linear Corresponding to each permutation u in the symmetric group, S,, on N = (1,. . . , n) is the permutation matrix P, = (S,,,,j) in M, where 6 is the Kronecker delta. The group of all permutation matrices in M will be denoted by P.

We prove new results concerning the relation between bifix codes, epis-5 turmian words and subgroups of free groups. We study bifix codes in 6 factorial sets of words. We generalize most properties of ordinary max-7 imal bifix codes to bifix codes maximal in a recurrent set F of words 8 (F -maximal bifix codes). In the case of bifix codes contained in Sturmian 9 sets of words, we obtain several new results. Let F be a Sturmian set 10 of words, defined as the set of factors of a strict episturmian word. Our 11 results express the fact that an F -maximal bifix code of degree d behaves 12 just as the set of words of F of length d. An F -maximal bifix code of 13 degree d in a Sturmian set of words on an alphabet with k letters has 14 (k − 1)d + 1 elements. This generalizes the fact that a Sturmian set con-15 tains (k − 1)d + 1 words of length d. Moreover, given an infinite word x, 16 if there is a finite maximal bifix code X of degree d such that x has at 17 most d factors of length d in X, then x is ultimately periodic. Our main 18 result states that any F -maximal bifix code of degree d on the alphabet 19

It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly $(n+1)$ factors of length $n$. One possible generalization of Sturmian sequences is the set of infinite sequences over a $k$-letter alphabet, $k \geq 3$, which are closed under reversal and have at most one right special factor for each length. This is the set of episturmian sequences. These are not necessarily balanced over a $k$-letter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel's conjecture for the class of episturmian sequences. This conjecture was first introduced in number theory and has remained unsolved for more than 30 years. It states that for a fixed $k> 2$, there is only one way to cover $\Z$ by $k$ Beatty sequences. The problem can be translated to combinatorics on words: for a $k$-letter alphabet, there exists only one balanced sequence up to letter permutation that has different letter frequencies.

It is well-known that Sturmian sequences are the non ultimately periodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly (n + 1) distinct factors of length n. A natural generalization of Sturmian sequences is the set of infinite episturmian sequences. These sequences are not necessarily balanced over a k-letter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel's conjecture for the special case of episturmian sequences. It appears that balanced episturmian sequences are all ultimately periodic and they can be classified in 3 families. * with the support of NSERC (Canada)

In this paper we study some classes of infinite words generalizing episturmian words, and analyse the relations occurring among such classes. In each case, the reversal operator R is replaced by an arbitrary involutory antimorphism ϑ of the free monoid A * . In particular, we define the class of ϑ-words with seed, whose "standard" elements (ϑ-standard words with seed) are constructed by an iterative ϑ-palindrome closure process, starting from a finite word u 0 called the seed. When the seed is empty, one obtains ϑ-words; episturmian words are exactly the R-words. One of the main theorems of the paper characterizes ϑ-words with seed as infinite words closed under ϑ and having at most one left special factor of each length n ≥ N (where N is some nonnegative integer depending on the word). When N = 0 we call such words ϑ-episturmian. Further results on the structure of ϑ-episturmian words are proved. In particular, some relationships between ϑ-words (with or without seed) and ϑ-episturmian words are shown.

Episturmian words are a suitable generalization to arbitrary alphabets of Sturmian words. In this paper we are interested in the problem of enumerating the palindromes in all episturmian words over a k-letter alphabet A k . We give a formula for the map g k giving for any n the number of all palindromes of length n in all episturmian words over A k . This formula extends to k > 2 a similar result obtained for k = 2 by the second and third author in 2006. The map g k is expressed in terms of the map P k counting for each n the palindromic prefixes of all standard episturmian words (epicentral words). For any n ≥ 0, P 2 (n) = ϕ(n + 2) where ϕ is the totient Euler function. The map P k plays an essential role also in the enumeration formula for the map λ k counting for each n the finite episturmian words over A k . Similarly to Euler's function, the behavior of P k is quite irregular. The first values of P k and of the related maps g k , and λ k for 3 ≤ k ≤ 6 have been calculated and reported in the Appendix. Some properties of P k are shown. In particular, broad upper and lower bounds for P k , as well as for n m=0 P k (m) and g k , are determined. Finally, some conjectures concerning the map P k are formulated.

The palindromization map ψ in a free monoid A * was introduced in 1997 by the first author in the case of a binary alphabet A, and later extended by other authors to arbitrary alphabets. Acting on infinite words, ψ generates the class of standard episturmian words, including standard Arnoux-Rauzy words. In this paper we generalize the palindromization map, starting with a given code X over A. The new map ψ X maps X * to the set PAL of palindromes of A * . In this way some properties of ψ are lost and some are saved in a weak form. When X has a finite deciphering delay one can extend ψ X to X ω , generating a class of infinite words much wider than standard episturmian words. For a finite and maximal code X over A, we give a suitable generalization of standard Arnoux-Rauzy words, called X-AR words. We prove that any X-AR word is a morphic image of a standard Arnoux-Rauzy word and we determine some suitable linear lower and upper bounds to its factor complexity.

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.

Episturmian sequences, or words, are a natural generalization of Sturmian sequences to arbitrary finite alphabets. In this paper we investigate various combinatorial properties of epicentral words, i.e., palindromic prefixes of standard episturmian words. We consider the action of certain involutions defined on epicentral words. We show that in some cases their fixed points may be characterized in terms of suitable faithful representations of epicentral words. We further show that various previously known results for central Sturmian words may be extended to epicentral words.

We study some classes of infinite words that generalize standard episturmian words, defined by replacing the reversal operator with an arbitrary involutory antimorphism ϑ of A * . An analysis of the relations occurring among such classes of words, and of the morphisms connecting them to standard episturmian words, is given. In particular, we analyse some structural properties of standard ϑ-episturmian words and their characteristic morphisms.

Let w be a finite word and n the least non-negative integer such that w has no right special factor of length n and its right factor of length n is unrepeated. We prove that if all the factors of another word v up to the length n + 1 are also factors of w, then v itself is a factor of w. A similar result for ultimately periodic infinite words is established. As a consequence, some 'uniqueness conditions' for ultimately periodic words are obtained as well as an upper bound for the rational exponents of the factors of uniformly recurrent non-periodic infinite words. A general formula is derived for the 'critical exponent' of a power-free Sturmian word. In particular, we effectively compute the 'critical exponent' of any Sturmian sequence whose slope has a periodic development in a continued fraction.

We say that two finite words $u$ and $v$ are abelian equivalent if and only if they have the same number of occurrences of each letter, or equivalently if they define the same Parikh vector. In this paper we investigate various abelian properties of words including abelian complexity, and abelian powers. We study the abelian complexity of the Thue-Morse word

We define a new induction algorithm for k-interval exchange transformations associated to the "symmetric" permutation i → k − i + 1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of generalized partial quotients given by an infinite path in a graph whose vertices, or states, are certain trees we call trees of relations. This induction is self-dual for the duality between the usual Rauzy induction and the da Rocha induction. We use it to describe those words obtained by coding orbits of points under a symmetric interval exchange, in terms of the generalized partial quotients associated with the vector of lengths of the k intervals. As a consequence, we improve a bound of Boshernitzan in a generalization of the three-distances theorem for rotations. However, a variant of our algorithm, applied to a class of interval exchange transformations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations.

We define a new induction algorithm for k-interval exchange transformations associated to the "symmetric" permutation i → k − i + 1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of generalized partial quotients given by an infinite path in a graph whose vertices, or states, are certain trees we call trees of relations. This induction is self-dual for the duality between the usual Rauzy induction and the da Rocha induction. We use it to describe those words obtained by coding orbits of points under a symmetric interval exchange, in terms of the generalized partial quotients associated with the vector of lengths of the k intervals. As a consequence, we improve a bound of Boshernitzan in a generalization of the three-distances theorem for rotations. However, a variant of our algorithm, applied to a class of interval exchange transformations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations.

A syntactic characterization of strictly locally testable languages is given by means of the concept of constant. If S is a semigroup and X a subset of S, an element c of S is called constant for X if fi)r all p, q, r, s ~ S ~ [pcq, rcs eX =. pcs c X]. The main result of the paper states that a recognizable subset X of a free semigroup A ~ is strictly locally testable if and only if all the idempotents of the syntactic semigroup S(X) of X are constants for X' ~ X~, where ,6 : A +-~ S(X) is the syntactic morphism. By this result some remarkable consequences are derived for recognizable subsemigroups of A". In particular we prove that if X is a recognizable free subsemigroup of A ~ and Y ~ X/X ~ its base then the following conditions are equivalent : (1) X is strictly locally testable.

If M is a maximal (proper) subsemigroup of a finite semigroup S, then M contains all but one J-class J(M) of S. When J(M)is non-regular J(M)n M = ~ so M = S --J(M). When J(M) is regular either J(M) c3 M = ~ or M n J(M) has a natural form with respect to the Green-Rees coordinates in J(M). Specifically, there exist an isomorphism j : J(M) ~ ~ dl~ B, G, C) of J(M) ~ with a Rees regular matrix semigroup so that j(M c~ J(M)) = G" x A x B, where G' is a maximal subgroup of G or j(M n J(M)) is the complement of a "rectangle" of ,,~-classes of.Z/~ B, G, C). In the first case, (M c3 J(M)) o is a maximal subsemigroup of J(M) ~ In the second, (M n J(M)) o is maximal in J(M) ~ when j(M n J(M)) = r162176 B, G, C) --(G • A' x B ~) for proper subsets A' and B' of A and B, respectively, but need not be when j(Mn J(M)) = G X A X B' or j(M n J(M)) = G X A' X B. The notation of this paper, with slight variations, follows [1]. ~'~ B, G, C) denotes a Rees matrix semigroup with finite index sets A, B, finite group G and C : B • A ~ G o the structure matrix. If J is a J-class of a semigroup S, we denote by j0 the semigroup J w {0}, 0 r J, with multiplication tjlj~ ,