Multidimensional generalized automatic sequences and shape-symmetric morphic words

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

A word w is called repetitive if it contains two consecutive equal factors ; otherwise w is nonrepetitive. Thus the word abacacb is repetitive, and abcacbabcbac is nonrepetitive. There is no nonrepetitive word of length 4 over a two letter alphabet ; on the contrary, there exist infinite nonrepetitive words over a three letter alphabet. Most of the explicitly known infinite nonrepetitive words are constructed by iteration of a morphism. In this paper, we show that it is decidable whether an infinite word over a three letter alphabet obtained by iterating a morphism is nonrepetitive. We also investigate nonrepetitive morphisms, i.e. morphisms preserving nonrepetitive words, and we show that it is decidable whether a morphism (over an arbitrary finite alphabet) is nonrepetitive.

Every infinite regular language contains a regular subset of the formuv+w for some words u, v, w, where v is not the empty word. The regularsubsets of the above form are called regular components. Some well-knowncontext-free languages, such as the Dyck language and the balanced language(over an alphabet X with |X| = 2), are decomposed as disjointunions of disjunctive languages. In this paper, we investigate thedecompositions of some of the regular languages and the context-freelanguages as disjoint unions of regular components. An infinite language iscalled regular component splittable if it can be expressed as a disjointunion of regular components and a finite set. We show that the Dycklanguage, the balanced language and some disjunctive context-free splittablelanguages are regular component splittable. We also present an example toshow that there is a disjunctive context-free language which is not regularcomponent splittable.

Union-free expressions are used in union normal form to decompose any regular language to a finite union of union-free languages. Based on the automata characterisation of the union-free languages, by restricting the 1CFPAs not to have transitions by the empty word, or to be deterministic, the n-union-free and the deterministic union-free languages are defined. Union-complexity as a measure of descriptional complexity of regular languages was introduced recently. By the minimum number of union-free/n-union-free/deterministic union-free languages needed to get a regular language as their union, its union-complexity/n-union-complexity/d-union-complexity is defined. It is already known that union-complexity and n-union-complexity are finite for every regular language, however there are regular languages with infinite d-union-complexity. Operational union-complexity, that is, to predict the union-complexity of a language obtained by a language operation from languages with known union-complexity is an important and interesting question belonging to the field of descriptional complexity of formal systems. In the present paper, the Kleene plus, the positive Kleene closure operator is studied. As the Kleene star and plus operations have very different effects on the union-free languages, it is an interesting problem to investigate how the union-complexities may change under this operation. In particular, we show that the union-complexity of a regular language is not growing when this operation is being applied on it. On the other hand, the n-union-complexity of the Kleene plus of an n-union-free language remains 1, but the n-union-complexity of the Kleene plus of other regular languages may grow. Further, the deterministic union-complexity may jump to an infinite value even if the original language had a relatively small deterministic union-complexity, e.g., 4.Keywordsunion-complexityunion-free languagesregular expressionsKleene closure

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragmenti?ź$$\varSigma _2$$ over infinite words. Here, $$\varSigma _2$$ consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is $$\mathbb {B}\varSigma _2$$. Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an $$\omega $$-regular language L, it is decidable whether L is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun's recent decidability result for $$\mathbb {B}\varSigma _2$$ from finite to infinite words.

We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.

Given an infinite word with values in a finite alphabet that admit frequencies for all its factors, the smallest frequency function associates with a given positive integer the smallest frequency for factors of this length, and the recurrence function measures the gaps between successive occurrences of factors. We develop in this paper the relations between the growth orders of the smallest frequency and of the recurrence functions. As proved by M. Boshernitzan, linearly recurrent words are characterized in terms of the smallest frequency function. In this paper, we see how to extend this result beyond the case of linearly recurrent words. We first establish a general relation between the smallest frequency and the recurrence functions. Given a lower bound for the smallest frequency function, this relation provides an upper bound for the recurrence function which involves the primitive of this lower bound. We then see how to improve this relation in the case of Sturmian words where the product of the smallest frequency and the recurrence function is bounded for most of the lengths of factors. We also indicate how to construct Sturmian words having a prescribed behaviour for the smallest frequency function, and for the product of the smallest frequency and the recurrence function.

Automated reasoning about systems with infinite domains requires an extension of regular automata to infinite alphabets. Existing formalisms of such automata cope with the infiniteness of the alphabet by adding to the automaton a set of registers or pebbles, or by attributing the alphabet by labels from an auxiliary finite alphabet that is read by an intermediate transducer. These formalisms involve a complicated mechanism on top of the transition function of automata over finite alphabets and are therefore difficult to understand and to work with. We introduce and study variable finite automata over infinite alphabets (VFA). VFA form a natural and simple extension of regular (and ω-regular) automata, in which the alphabet consists of letters as well as variables that range over the infinite alphabet domain. Thus, VFAs have the same structure as regular automata, only that some of the transitions are labeled by variables. We compare VFA with existing formalisms, and study their closure properties and classical decision problems. We consider the settings of both finite and infinite words. In addition, we identify and study the deterministic fragment of VFA. We show that while this fragment is sufficiently strong to express many interesting properties, it is closed under union, intersection, and complementation, and its nonemptiness and containment problems are decidable. Finally, we describe a determinization process for a determinizable subset of VFA.

The aim of this chapter is to present some recent research work in the combinatorial aspects of the theory of semigroups which are of great interest both for Algebra and Theoretical Computer Science. This research mainly concerns that part of combinatorics of finite and infinite words over a finite alphabet, which is usually called the theory of “unavoidable” regularities. The book by Lothaire [77] can be considered an excellent introduction to this theory.

We consider several problems of infinite words over a finite alphabet. In particular, we describe a few automata-theoretic methods to define infinite words. Properties of infinite words studied in more details are repetition-freeness, periodicity and different kinds of complexity issues. Examples are used to illustrate the power of infinite words in many applications, as well as illustrations of problems from different areas of mathematics.

Automata on linear orderings

An algebraic approach to the theory of regular languages of finite and infinite words (∞-languages) is presented. It extends the algebraic theory of regular languages of finite words, which is based on finite semigroups. Their role is taken over by a structure called right binoid. A variety theorem is proved: there is a one-to-one correspondence between varieties of ∞-languages and pseudovarieties of right binoids. The class of locally threshold testable languages and several natural subclasses (such as the class of locally testable languages) as well as classes of the Borel hierarchy over the Cantor space (restricted to regular languages) are investigated as examples for varieties of ∞-languages. The corresponding pseudovarieties of right binoids are characterized and in some cases defining equations are derived. The connections with the algebraic description and classification of regular languages of infinite words in terms of finite semigroups are pointed out.

State complexity of pattern matching in regular languages

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game.

Cost functions are defined as mappings from a domain like words or trees to \(\mathbb{N} \cup \left\{{\infty}\right\}\), modulo an equivalence relation ≈ which ignores exact values but preserves boundedness properties. Cost logics, in particular cost monadic second-order logic, and cost automata, are different ways to define such functions. These logics and automata have been studied by Colcombet et al. as part of a “theory of regular cost functions”, an extension of the theory of regular languages which retains robust equivalences, closure properties, and decidability. We develop this theory over infinite words, and show that the classical results FO = LTL and MSO = WMSO also hold in this cost setting (where the equivalence is now up to ≈). We also describe connections with forms of weak alternating automata with counters.