The height of piecewise-testable languages and the complexity of the logic of subwords

Abstract

The height of a piecewise-testable language L is the maximum length of the words needed to define L by excluding and requiring given subwords. The height of L is an important descriptive complexity measure that has not yet been investigated in a systematic way. This article develops a series of new techniques for bounding the height of finite languages and of languages obtained by taking closures by subwords, superwords and related operations.As an application of these results, we show that FO 2 pA ˚, Ďq, the two-variable fragment of the first-order logic of sequences with the subword ordering, can only express piecewisetestable properties and has elementary complexity.

Combinatorial Algorithms for Subsequence Matching: A Survey

“…The notion of k-universality coincides to that of k-richness introduced in [36,37]. We use the name k-universality rather than k-richness, as richness of words is also used with other meanings, see, e. g., [19,45].…”

“…The concept of subsequences is employed in many different areas of computer science. Subsequences appear in areas of theoretical computer science such as, for instance, in formal languages and logics (e. g., where they are used in relation to piecewise testable languages [54,55,35,36,37], or to define the subword order and downward closures [31,41,40,60]) or in combinatorics on words, where they are used to define the notions of binomial equivalence and binomial complexity, or to introduce the notion of subword histories, [49,23,43,42,51,47,50]; however, subsequences are also used in more applied settings, e. g., for modelling concurrency [48,52,13], or in database theory (especially event stream processing [5,30,61]). Moreover, many classical algorithmic problems are based on subsequences, e. g., longest common subsequence [6] or shortest common supersequence [46], and, in particular, such problems have recently regained interest in the context of fine-grained complexity (see [11,12,1,2]).…”

Matching Patterns with Variables in Approximate Settings

“…Subsequences are also a heavily investigated topic in the area of word combinatorics, string algorithms, and combinatorial pattern matching, and are connected to other areas of computer science (see, e.g., in the Chapter Subwords by J. Sakarovitch and I. Simon of the standard textbook [123] or the survey [107] and the references therein). In theoretical computer science, one can often encounter subsequences and their generalizations; for instance, in logic of automata theory, subsequences are used in the context of piecewise testability [156,157], in particular, to the height of piecewise testable languages [94,95,96], subword order [84,113,112], or downward closures [167]. In combinatorics on words, many concepts were developed around the idea of counting the occurrences of particular subsequences of a word, such as the k-binomial equivalence [144,68,119,117], subword histories [153], and Parikh matrices [130,146].…”

“…Two strings w 1 and w 2 are k-Simon congruent (w 1 ∼ k w 2 ), iff S k (w 1 ) = S k (w 2 ). Simon's congruence, which was originally introduced to study piecewise testable events, is now established in the study of piecewise testable languages, a special class of regular languages with applications in learning theory, databases theory, or linguistics (see, e.g., [96] and the references therein). More information and a broad overview of the work of Imre Simon can be found in the two survey papers [140,141] Next to many interesting combinatorial properties, Simon's congruence is a string similarity measure.…”

“…A specific congruence class defined by ∼ k frequently arises in the literature: the class of ksubsequence universal words consists of words that contain all possible k-length words over a fixed alphabet as subsequences. The maximal k for which a word w is k-subsequence universal is denoted by the universality index ι(w) = k. This particular class was first studied in [95,97], and is related to the more general problem of fixed-length universality: for a given k and alphabet Σ, is Σ k described/generated/accepted by a mechanism which describes/generates/accepts languages? Such problems were investigated in contexts related to and motivated by formal languages, automata theory, or combinatorics, where the notion of universality is central [17,50,63,106,2,3,152,64].…”