Occurrences Of Subwords Research Articles

Counting subwords in circular words and their Parikh matrices

The word inference problem is to determine languages such that the information on the number of occurrences of those subwords in the language can uniquely identify a word. A considerable amount of work has been done on this problem, but the same cannot be said for circular words despite growing interests on the latter due to their applications—for example, in splicing systems. Meanwhile, Parikh matrices are useful tools and well established in the study of subword occurrences. In this work, we propose two ways of counting subword occurrences in circular words. We then extend the idea of Parikh matrices to the context of circular words and investigate this extension. Motivated by the word inference problem, we study ambiguity in the identification of a circular word by its Parikh matrix. Accordingly, two rewriting rules are developed to generate ternary circular words which share the same Parikh matrix.

Generalized Parikh matrices of picture array

The theory of Parikh matrices and subword occurrences has led to extensive research in combinatorics on words. The extension of Parikh matrix of a word into picture array is an another interesting problem. Since the Parikh matrix does not say about the number of subword occurrences in the array. Therefore, in this paper we introduce the generalized Parikh matrix of a picture array which gives the number of subwords occurrences in the array. We also discuss some properties of generalized Parikh matrices.

Freeness Problem for Matrix Semigroups of Parikh Matrices

Since the undecidability of the mortality problem for 3 × 3 matrices over integers was proved using the Post Correspondence Problem, various studies on decision problems of matrix semigroups have emerged. The freeness problem in particular has received much attention but decidability remains open even for 2 × 2 upper triangular matrices over nonnegative integers. Parikh matrices are upper triangular matrices introduced as a generalization of Parikh vectors and have become useful tools in studying of subword occurrences. In this work, we focus on semigroups of Parikh matrices and study the freeness problem in this context.

2D Words and Generalized Parikh Matrices

The extension of Parikh vector is Parikh matrix which is a useful tool in arithmetizing words by numbers. Another interesting problem is the extension of Parikh matrix of a words to 2D words. However, Parikh matrix fails to tell about the number of subword occurrences in 2D word. Therefore, in this paper we introduce the notion of generalized Parikh matrix of 2D words which fulfill the number of subwords occurrences in an 2D words and also we obtain the properties of the generalized Parikh matrices.

Generalized Parikh matrices of picture array

The theory of Parikh matrices and subword occurrences has led to extensive research in combinatorics on words. The extension of Parikh matrix of a word into picture array is an another interesting problem. Since the Parikh matrix does not say about the number of subword occurrences in the array. Therefore, in this paper we introduce the generalized Parikh matrix of a picture array which gives the number of subwords occurrences in the array. We also discuss some properties of generalized Parikh matrices.

Computing the k-binomial complexity of the Thue–Morse word

Two words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both the abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word x maps the integer n to the number of classes in the quotient, by this k-binomial equivalence relation, of the set of factors of length n occurring in x. This complexity measure has not been investigated very much. In this paper, we characterize the k-binomial complexity of the Thue–Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue–Morse word is aperiodic, its k-binomial complexity eventually takes only two values. In this paper, we first express the number of occurrences of subwords appearing in iterates of the form Ψℓ(w) for an arbitrary morphism Ψ. We also thoroughly describe the factors of the Thue–Morse word by introducing a relevant new equivalence relation.

Strong (2 ⋅ t) and Strong (3 ⋅ t) Transformations for Strong M-Equivalence

Parikh matrices have been extensively investigated due to their usefulness in studying subword occurrences in words. Due to the dependency of Parikh matrices on the ordering of the alphabet, the notion of strong [Formula: see text]-equivalence was proposed as an order-independent alternative to [Formula: see text]-equivalence in studying words possessing the same Parikh matrix. This paper introduces and studies the notions of strong [Formula: see text] and strong [Formula: see text] transformations in determining when two ternary words are strongly [Formula: see text]-equivalent. The irreducibility of strong [Formula: see text] transformations is then scrutinized, further exemplified by a structural characterization of irreducible strong [Formula: see text] transformations. The common limitation of these transformations in characterizing strong [Formula: see text]-equivalence is then addressed.

RANDOMNESS AND NON‐RANDOMNESS PROPERTIES OF PIATETSKI‐SHAPIRO SEQUENCES MODULO m

We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.

Separating words by occurrences of subwords

We obtain some lower bounds for the complexity of word separation by occurrences of subwords. In the case of length 1 subwords, we show that the bound is exact up to a constant factor. Connection with the problem of separating words by automata is considered.

Several extensions of the Parikh matrix L-morphism

The Parikh matrix mapping is a morphism assigning to each word w over a k-letter alphabet a (k+1)×(k+1) upper triangular matrix with entries expressing the number of occurrences in w of some specific subwords. To tackle the problem of ambiguity of this mapping two new mappings have been proposed in literature, assigning to words matrices with polynomial entries (q-matrices). One is a more subtle, but still ambiguous morphism, the other is unambiguous but not a morphism. We show that the former mapping can be extended to match even a fairly general extension of the original Parikh matrix morphism. Then we introduce an unambiguous q-matrix morphism based on the same general Parikh matrix mapping. Finally, we consider the problem of incomplete information on word symbols and show that the general Parikh matrix mapping can be further extended to deal with counts of fuzzy subword occurrences.

Subword occurrences, weighted automata and iterated morphisms, especially the Fibonacci morphism

The paper investigates the number of subword occurrences in a word, in particular, in a word obtained by iterating a morphism. Associated graphs or automata are used as a tool. Special attention is paid to the Fibonacci morphism. Explicit formulas in terms of n are obtained for the number of occurrences of subwords of length at most 2 in the nth iteration of the Fibonacci morphism.

Criteria for the matrix equivalence of words

This paper investigates the criteria for deciding whether two words are matrix equivalent. Certain upper diagonal matrices, generally referred to as the Parikh matrices, have been widely investigated because of their usefulness in computing the numbers of subword occurrences and thereby characterizing words by numbers. However, apart from the binary alphabet, not much is known about the properties of the matrix equivalent words, that is, words possessing the same matrix. The paper investigates both the general criteria, as well as the criteria valid in natural special cases. An exhaustive solution is obtained for ternary alphabets.

USEFULNESS OF DIRECTED ACYCLIC SUBWORD GRAPHS IN PROBLEMS RELATED TO STANDARD STURMIAN WORDS

The class of finite Sturmian words consists of words having particularly simple compressed representation, which is a generalization of the Fibonacci recurrence for Fibonacci words. The subword graphs of these words (especially their compacted versions) have a very special regular structure. In this paper we investigate this structure in more detail than in previous papers and show how several syntactical properties of Sturmian words follow from their graph properties. Consequently simple alternative graph-based proofs of several known facts are presented. The very special structure of subword graphs leads also to special easy algorithms computing some parameters of Sturmian words: the number of subwords, the critical factorization point, lexicographically maximal suffixes, occurrences of subwords of a fixed length, and right special factors. These algorithms work in linear time with respect to n, the size of the compressed representation of the standard word, though the words themselves can be of exponential size with respect to n. Some of the computed parameters can be also of exponential size, however we provide their linear size compressed representations. We introduce also a new concept related to standard words: Ostrowski automata.

Independence of certain quantities indicating subword occurrences

When words are characterized in terms of numerical quantities, awkward considerations due to the noncommutativity of words are avoided. The numerical quantity investigated in this paper is | w | u , the number of occurrences of a word u as a (scattered) subword of a word w . Parikh matrices recently introduced have these quantities as their entries. According to the main result in this paper, no entry in a Parikh matrix, no matter how high the dimension, can be computed in terms of the other entries. Consequences concerning various inference problems between numbers | w | u themselves, as well as of the word w from these numbers, are obtained.

Connections between subwords and certain matrix mappings

Parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of the theory of words. In particular, many inequalities between (scattered) subword occurrences have been obtained as consequences of the properties of the matrices. This paper continues the investigation of Parikh matrices and subword occurrences. In particular, we study certain inequalities, as well as information about subword occurrences sufficient to determine the whole word uniquely. Some algebraic considerations, facts about forbidden subwords, as well as some open problems are also included.

MATRIX INDICATORS FOR SUBWORD OCCURRENCES AND AMBIGUITY

The paper investigates inequalities between the numbers of different (scattered) subword occurrences. The Parikh matrix recently introduced is an efficient tool. We give various characterizations for Parikh matrices. Of special interest is the case where the matrix determines the word uniquely. We investigate such matrix unambiguous words. The considerations are extended to concern languages. Several open problems and problem areas are indicated.

Subword histories and Parikh matrices

Parikh matrices recently introduced give much more information about a word than just the number of occurrences of each letter. In this paper we introduce the closely related notion of a subword history and obtain a sequence of general results: elimination of products, decidability of equivalence, and normal form. We also investigate overall methods for proving the validity of such results. A general inequality of “Cauchy type” for subword occurrences is established.