arXiv subjects

Štěpán Holub

Publications and source records attributed to Štěpán Holub.

At least 19 recordsLinked to original sources

arXiv2026

Formalization of Fragments of the Theory of Hereditarily Finite Sets

The axiomatization of the theory of hereditarily finite sets in first-order classical logic is systematically explored and formalized in Isabelle/HOL. The formalization uses a hierarchy of locales, each corresponding to a fragment of the theory given by a particular collection of axioms. An inductive definition of first-order definable predicates is introduced and used to formalize axiom schemata. Special attention is paid to several equivalent axioms of finiteness, as well as to several equivalent ways of expressing regularity. The work also formalizes several facts about the independence of an axiom from a system of axioms by defining appropriate models.

cs.LO↗

arXiv2022

Binary codes that do not preserve primitivity

A code $X$ is not primitivity preserving if there is a primitive list ${\mathbf w} \in {\tt lists} X$ whose concatenation is imprimitive. We formalize a full characterization of such codes in the binary case in the proof assistant Isabelle/HOL. Part of the formalization, interesting on its own, is a description of $\{x,y\}$-interpretations of the square $xx$ if $|y| \leq |x|$. We also provide a formalized parametric solution of the related equation $x^jy^k = z^\ell$.

cs.FL↗

arXiv2020

The Intersection of $3$-Maximal Submonids

Very little is known about the structure of the intersection of two $k$-generated monoids of words, even for $k=3$. Here we investigate the case of $k$-maximal monoids, that is, monoids whose basis of cardinality $k$ cannot be non-trivially decomposed into at most $k$ words. We characterize the intersection in the case of two $3$-maximal monoids.

cs.FL↗

arXiv2020

Binary intersection formalized

We provide a reformulation and a formalization of the classical result by Juhani Karhumäki characterizing intersections of two languages of the form $\{x,y\}^*\cap \{u,v\}^*$. We use the terminology of morphisms which allows to formulate the result in a shorter and more transparent way, and we formalize the result in the proof assistant Isabelle/HOL.

cs.FL↗

arXiv2019

Maximal State Complexity and Generalized de Bruijn Words

We compute the exact maximum state complexity for the language consisting of $m$ words of length $N$, and characterize languages achieving the maximum. We also consider a special case, namely languages $C(w)$ consisting of the conjugates of a single word $w$. The words for which the maximum state complexity of $C(w)$ is achieved turn out to be a natural generalization of de Bruijn words. We show that generalized de Bruijn words exist for each length and consider the number of them.

cs.FL↗

arXiv2019

Pseudo-solutions of word equations

We present a framework which allows a uniform approach to the recently introduced concept of pseudo-repetitions on words in the morphic case. This framework is at the same time more general and simpler. We introduce the concept of a pseudo-solution and a pseudo-rank of an equation. In particular, this allows to prove that if a classical equation forces periodicity then it also forces pseudo-periodicity. Consequently, there is no need to investigate generalizations of important equations one by one.

cs.FL↗

arXiv2017

State spaces of convolutional codes, codings and encoders

In this paper we give a compact presentation of the theory of abstract spaces for convolutional codes and convolutional encoders, and show a connection between them that seems to be missing in the literature. We use it for a short proof of two facts: the size of a convolutional encoder of a polynomial matrix is at least its inner degree, and the minimal encoder has the size of the external degree if the matrix is reduced.

cs.IT↗

arXiv2017

On the Height of Towers of Subsequences and Prefixes

A tower is a sequence of words alternating between two languages in such a way that every word is a subsequence of the following word. The height of the tower is the number of words in the sequence. If there is no infinite tower (a tower of infinite height), then the height of all towers between the languages is bounded. We study upper and lower bounds on the height of maximal finite towers with respect to the size of the NFA (the DFA) representation of the languages. We show that the upper bound is polynomial in the number of states and exponential in the size of the alphabet, and that it is asymptotically tight if the size of the alphabet is fixed. If the alphabet may grow, then, using an alphabet of size approximately the number of states of the automata, the lower bound on the height of towers is exponential with respect to that number. In this case, there is a gap between the lower and upper bound, and the asymptotically optimal bound remains an open problem. Since, in many cases, the constructed towers are sequences of prefixes, we also study towers of prefixes.

cs.FL↗

arXiv2016

Prefix frequency of lost positions

The concept of "lost positions" is a recently introduced tool for counting the number of runs in words. We investigate the frequency of lost positions in prefixes of words. This leads to an algorithm that allows to show, using an extensive computer search, that the asymptotic density of runs in binary words is less than $183/193\approx 0.9482$.

cs.FL↗

arXiv2016

Fully bordered words

We characterize binary words that have exactly two unbordered conjugates and show that they can be expressed as a product of two palindromes.

cs.FL↗

arXiv2015

Alternating Towers and Piecewise Testable Separators

Two languages are separable by a piecewise testable language if and only if there exists no infinite tower between them. An infinite tower is an infinite sequence of strings alternating between the two languages such that every string is a subsequence (scattered substring) of all the strings that follow. For regular languages represented by nondeterministic finite automata, the existence of an infinite tower is decidable in polynomial time. In this paper, we investigate the complexity of a particular method to compute a piecewise testable separator. We show that it is closely related to the height of maximal finite towers, and provide the upper and lower bounds with respect to the size of the given nondeterministic automata. Specifically, we show that the upper bound is polynomial with respect to the number of states with the cardinality of the alphabet in the exponent. Concerning the lower bound, we show that towers of exponential height with respect to the cardinality of the alphabet exist. Since these towers mostly turn out to be sequences of prefixes, we also provide a comparison with towers of prefixes.

cs.FL↗

arXiv2015

Periods and borders of random words

We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length $k$ is a constant, depending only on $k$ and the alphabet size $\ell$. We give a recurrence that allows us to determine these constants with any required precision. This also allows us to evaluate the expected period of a random word. For the binary case, the expected period is asymptotically about $n-1.641$. We also give explicit formulas for the probability that a random word is unbordered or has maximum border length one.

cs.FL↗

arXiv2015

Beyond the Runs Theorem

Recently, a short and elegant proof was presented showing that a binary word of length $n$ contains at most $n-3$ runs. Here we show, using the same technique and a computer search, that the number of runs in a binary word of length $n$ is at most $\frac{22}{23}n<0.957n$.

cs.FL↗

arXiv2015

Equation $x^iy^jx^k=u^iv^ju^k$ in words

We will prove that the word $a^ib^ja^k$ is periodicity forcing if $j \geq 3$ and $i+k \geq 3$, where $i$ and $k$ are positive integers. Also we will give examples showing that both bounds are optimal.

cs.FL↗

arXiv2014

Algebraic properties of word equations

The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic properties of polynomials encoding the equations and their solutions. In this paper we develop further this approach and take into account other algebraic properties of polynomials, namely their factorization. This, in particular, allows to improve the bound for the number of independent equations with maximal rank from quadratic to linear.

cs.FL↗

arXiv2014

On Upper and Lower Bounds on the Length of Alternating Towers

A tower between two regular languages is a sequence of strings such that all strings on odd positions belong to one of the languages, all strings on even positions belong to the other language, and each string can be embedded into the next string in the sequence. It is known that if there are towers of any length, then there also exists an infinite tower. We investigate upper and lower bounds on the length of finite towers between two regular languages with respect to the size of the automata representing the languages in the case there is no infinite tower. This problem is relevant to the separation problem of regular languages by piecewise testable languages.

cs.FL↗

arXiv2013

Words with unbounded periodicity complexity

If an infinite non-periodic word is uniformly recurrent or is of bounded repetition, then the limit of its periodicity complexity is infinity. Moreover, there are uniformly recurrent words with the periodicity complexity arbitrarily high at infinitely many positions.

cs.FL↗

arXiv2013

On an algorithm for multiperiodic words

We consider an algorithm by Tijdeman and Zamboni constructing a word of a given length that has a given set of periods, and the richest possible alphabet. We show that this algorithm can be easily stated and its correctness briefly proved using the class equivalence approach.

cs.FL↗