33 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Deciding Conjugacy of a Rational Relation

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    A rational relation is conjugate if every pair of related words are conjugates. It is shown that checking whether a rational relation is conjugate is decidable. For this, we generalise the Lyndon-Sch\"utzenberger's theorem from word combinatorics. A consequence of the generalisation is that a set of pairs generated by a sumfree rational expression is conjugate if and only if there is a word witnessing the conjugacy of all the pairs

    A structure theorem for streamed information

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    We identify the free half shuffle algebra of Sch\"utzenberger (1958) with an algebra of real-valued functionals on paths, where the half shuffle emulates integration of a functional against another. We then provide two, to our knowledge, new identities in arity 3 involving its commutator (area), and show that these are sufficient to recover the Zinbiel and Tortkara identities of Dzhumadil'daev (2007). We use these identities to prove that any element of the free half shuffle algebra can be expressed as a polynomial over iterated areas. Moreover, we consider minimal sets of iterated integrals defined through the recursive application of the half shuffle on Hall trees. Leveraging the duality between this set of Hall integrals and classical Hall bases of the free Lie algebra, we prove using combinatorial arguments that any element of the free half shuffle algebra can be written uniquely as a polynomial over Hall integrals. We interpret this result as a structure theorem for streamed information, loosely analogous to the unique prime factorisation of integers, allowing to split any real valued function on streamed data into two parts: a first that extracts and packages the streamed information into recursively defined atomic objects (Hall integrals), and a second that evaluates a polynomial function in these objects without further reference to the original stream. The question of whether a similar result holds if Hall integrals are replaced by Hall areas is left as an open conjecture. Finally, we construct a canonical, but to our knowledge, new decomposition of the free half shuffle algebra as shuffle power series in the greatest letter of the original alphabet with coefficients in a sub-algebra freely generated by a new alphabet with an infinite number of letters. We use this construction to provide a second proof of our structure theorem

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

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    We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism α such that α(xy) ≠ α(yx) for given variables x and y. </p

    Branching densities of cube-free and square-free words

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    Binary cube-free language and ternary square-free language are two “canonical” represen-tatives of a wide class of languages defined by avoidance properties. Each of these two languages can be viewed as an infinite binary tree reflecting the prefix order of its elements. We study how “homogenious” these trees are, analysing the following parameter: the density of branching nodes along infinite paths. We present combinatorial results and an efficient search algorithm, which together allowed us to get the following numerical results for the cube-free language: the minimal density of branching points is between 3509/9120 ≈ 0.38476 and 13/29 ≈ 0.44828, and the maximal density is between 0.72 and 67/93 ≈ 0.72043. We also prove the lower bound 223/868 ≈ 0.25691 on the density of branching points in the tree of the ternary square-free language. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.This research was funded by Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2020-1537/1)

    The word problem and combinatorial methods for groups and semigroups

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    The subject matter of this thesis is combinatorial semigroup theory. It includes material, in no particular order, from combinatorial and geometric group theory, formal language theory, theoretical computer science, the history of mathematics, formal logic, model theory, graph theory, and decidability theory. In Chapter 1, we will give an overview of the mathematical background required to state the results of the remaining chapters. The only originality therein lies in the exposition of special monoids presented in §1.3, which uni.es the approaches by several authors. In Chapter 2, we introduce some general algebraic and language-theoretic constructions which will be useful in subsequent chapters. As a corollary of these general methods, we recover and generalise a recent result by Brough, Cain & Pfei.er that the class of monoids with context-free word problem is closed under taking free products. In Chapter 3, we study language-theoretic and algebraic properties of special monoids, and completely classify this theory in terms of the group of units. As a result, we generalise the Muller-Schupp theorem to special monoids, and answer a question posed by Zhang in 1992. In Chapter 4, we give a similar treatment to weakly compressible monoids, and characterise their language-theoretic properties. As a corollary, we deduce many new results for one-relation monoids, including solving the rational subset membership problem for many such monoids. We also prove, among many other results, that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. In Chapter 5, we study context-free graphs, and connect the algebraic theory of special monoids with the geometric behaviour of their Cayley graphs. This generalises the geometric aspects of the Muller-Schupp theorem for groups to special monoids. We study the growth rate of special monoids, and prove that a special monoid of intermediate growth is a group

    Solving One Variable Word Equations in the Free Group in Cubic Time

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    A word equation with one variable in a free group is given as U=VU = V, where both UU and VV are words over the alphabet of generators of the free group and X,X1X, X^{-1}, for a fixed variable XX. An element of the free group is a solution when substituting it for XX yields a true equality (interpreted in the free group) of left- and right-hand sides. It is known that the set of all solutions of a given word equation with one variable is a finite union of sets of the form {αwiβ:iZ}\{\alpha w^i \beta \: : \: i \in \mathbb Z \}, where α,w,β\alpha, w, \beta are reduced words over the alphabet of generators, and a polynomial-time algorithm (of a high degree) computing this set is known. We provide a cubic time algorithm for this problem, which also shows that the set of solutions consists of at most a quadratic number of the above-mentioned sets. The algorithm uses only simple tools of word combinatorics and group theory and is simple to state. Its analysis is involved and focuses on the combinatorics of occurrences of powers of a word within a larger word.Comment: 52 pages, accepted to STACS 202

    On Special k-Spectra, k-Locality, and Collapsing Prefix Normal Words

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    The domain of Combinatorics on Words, first introduced by Axel Thue in 1906, covers by now many subdomains. In this work we are investigating scattered factors as a representation of non-complete information and two measurements for words, namely the locality of a word and prefix normality, which have applications in pattern matching. In the first part of the thesis we investigate scattered factors: A word u is a scattered factor of w if u can be obtained from w by deleting some of its letters. That is, there exist the (potentially empty) words u1, u2, . . . , un, and v0,v1,...,vn such that u = u1u2 ̈ ̈ ̈un and w = v0u1v1u2v2 ̈ ̈ ̈unvn. First, we consider the set of length-k scattered factors of a given word w, called the k-spectrum of w and denoted by ScatFactk(w). We prove a series of properties of the sets ScatFactk(w) for binary weakly-0-balanced and, respectively, weakly-c-balanced words w, i.e., words over a two- letter alphabet where the number of occurrences of each letter is the same, or, respectively, one letter has c occurrences more than the other. In particular, we consider the question which cardinalities n = | ScatFactk (w)| are obtainable, for a positive integer k, when w is either a weakly-0- balanced binary word of length 2k, or a weakly-c-balanced binary word of length 2k ́ c. Second, we investigate k-spectra that contain all possible words of length k, i.e., k-spectra of so called k-universal words. We present an algorithm deciding whether the k-spectra for given k of two words are equal or not, running in optimal time. Moreover, we present several results regarding k-universal words and extend this notion to circular universality that helps in investigating how the universality of repetitions of a given word can be determined. We conclude the part about scattered factors with results on the reconstruction problem of words from scattered factors that asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word w P {a, b} ̊ can be reconstructed from the number of occurrences of at most min(|w|a, |w|b) + 1 scattered factors of the form aib, where |w|a is the number of occurrences of the letter a in w. Moreover, we generalise the result to alphabets of the form {1, . . . , q} by showing that at most ∑q ́1 |w|i (q ́ i + 1) scattered factors suffices to reconstruct w. Both results i=1 improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here. In the second part we consider patterns, i.e., words consisting of not only letters but also variables, and in particular their locality. A pattern is called k-local if on marking the pattern in a given order never more than k marked blocks occur. We start with the proof that determining the minimal k for a given pattern such that the pattern is k-local is NP- complete. Afterwards we present results on the behaviour of the locality of repetitions and palindromes. We end this part with the proof that the matching problem becomes also NP-hard if we do not consider a regular pattern - for which the matching problem is efficiently solvable - but repetitions of regular patterns. In the last part we investigate prefix normal words which are binary words in which each prefix has at least the same number of 1s as any factor of the same length. First introduced in 2011 by Fici and Lipták, the problem of determining the index (amount of equivalence classes for a given word length) of the prefix normal equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix normal words can be split into certain subproblems
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