arXiv subjects

Jarkko Peltomäki

Publications and source records attributed to Jarkko Peltomäki.

At least 19 recordsLinked to original sources

arXiv2025

The repetition threshold for ternary rich words

In 2017, Vesti proposed the problem of determining the repetition threshold for infinite rich words, i.e., for infinite words in which all factors of length $n$ contain $n$ distinct nonempty palindromic factors. In 2020, Currie, Mol, and Rampersad proved a conjecture of Baranwal and Shallit that the repetition threshold for binary rich words is $2 + \sqrt{2}/2$. In this paper, we prove a structure theorem for $16/7$-power-free ternary rich words. Using the structure theorem, we deduce that the repetition threshold for ternary rich words is $1 + 1/(3 - μ) \approx 2.25876324$, where $μ$ is the unique real root of the polynomial $x^3 - 2x^2 - 1$.

math.CO↗

arXiv2025

Requirement falsification for cyber-physical systems using generative models

We present the OGAN algorithm for automatic requirement falsification of cyber-physical systems. System inputs and outputs are represented as piecewise constant signals over time while requirements are expressed in signal temporal logic. OGAN can find inputs that are counterexamples for the correctness of a system revealing design, software, or hardware defects before the system is taken into operation. The OGAN algorithm works by training a generative machine learning model to produce such counterexamples. It executes tests offline and does not require any previous model of the system under test. We evaluate OGAN using the ARCH-COMP benchmark problems, and the experimental results show that generative models are a viable method for requirement falsification. OGAN can be applied to new systems with little effort, has few requirements for the system under test, and exhibits state-of-the-art CPS falsification efficiency and effectiveness.

cs.LG↗

arXiv2024

Learning test generators for cyber-physical systems

Black-box runtime verification methods for cyber-physical systems can be used to discover errors in systems whose inputs and outputs are expressed as signals over time and their correctness requirements are specified in a temporal logic. Existing methods, such as requirement falsification, often focus on finding a single input that is a counterexample to system correctness. In this paper, we study how to create test generators that can produce multiple and diverse counterexamples for a single requirement. Several counterexamples expose system failures in varying input conditions and support the root cause analysis of the faults. We present the WOGAN algorithm to create such test generators automatically. The algorithm works by training iteratively a Wasserstein generative adversarial network that models the target distribution of the uniform distribution on the set of counterexamples. WOGAN is an algorithm that trains generative models that act as test generators for runtime verification. The training is performed online without the need for a previous model or dataset. We also propose criteria to evaluate such test generators. We evaluate the trained generators on several well-known problems including the ARCH-COMP falsification benchmarks. Our experimental results indicate that generators trained by the WOGAN algorithm are as effective as state-of-the-art requirement falsification algorithms while producing tests that are as diverse as a sample from uniform random sampling. We conclude that WOGAN is a viable method to produce test generators automatically and that these test generators can generate multiple and diverse counterexamples for the runtime verification of cyber-physical systems.

cs.LG↗

arXiv2024

Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents

Regular episturmian words are episturmian words whose directive words have a regular and restricted form making them behave more like Sturmian words than general episturmian words. We present a method to evaluate the initial nonrepetitive complexity of regular episturmian words extending the work of Wojcik on Sturmian words. For this, we develop a theory of generalized Ostrowski numeration systems and show how to associate with each episturmian word a unique sequence of numbers written in this numeration system. The description of the initial nonrepetitive complexity allows us to obtain novel results on the Diophantine exponents of regular episturmian words. We prove that the Diophantine exponent of a regular episturmian word is finite if and only if its directive word has bounded partial quotients. Moreover, we prove that the Diophantine exponent of a regular episturmian word is strictly greater than $2$ if the sequence of partial quotients is eventually at least $3$. Given an infinite word $x$ over an integer alphabet, we may consider a real number $ξ_x$ having $x$ as a fractional part. The Diophantine exponent of $x$ is a lower bound for the irrationality exponent of $ξ_x$. Our results thus yield nontrivial lower bounds for the irrationality exponents of real numbers whose fractional parts are regular episturmian words. As a consequence, we identify a new uncountable class of transcendental numbers whose irrationality exponents are strictly greater than $2$. This class contains an uncountable subclass of Liouville numbers.

cs.FL↗

arXiv2022

Falsification of Multiple Requirements for Cyber-Physical Systems Using Online Generative Adversarial Networks and Multi-Armed Bandits

We consider the problem of falsifying safety requirements of Cyber-Physical Systems expressed in signal temporal logic (STL). This problem can be turned into an optimization problem via STL robustness functions. In this paper, our focus is in falsifying systems with multiple requirements. We propose to solve such conjunctive requirements using online generative adversarial networks (GANs) as test generators. Our main contribution is an algorithm which falsifies a conjunctive requirement $φ_1 \land \cdots \land φ_n$ by using a GAN for each requirement $φ_i$ separately. Using ideas from multi-armed bandit algorithms, our algorithm only trains a single GAN at every step, which saves resources. Our experiments indicate that, in addition to saving resources, this multi-armed bandit algorithm can falsify requirements with fewer number of executions on the system under test when compared to (i) an algorithm training a single GAN for the complete conjunctive requirement and (ii) an algorithm always training $n$ GANs at each step.

cs.SE↗

arXiv2022

Wasserstein Generative Adversarial Networks for Online Test Generation for Cyber Physical Systems

We propose a novel online test generation algorithm WOGAN based on Wasserstein Generative Adversarial Networks. WOGAN is a general-purpose black-box test generator applicable to any system under test having a fitness function for determining failing tests. As a proof of concept, we evaluate WOGAN by generating roads such that a lane assistance system of a car fails to stay on the designated lane. We find that our algorithm has a competitive performance respect to previously published algorithms.

cs.LG↗

arXiv2022

WOGAN at the SBST 2022 CPS Tool Competition

WOGAN is an online test generation algorithm based on Wasserstein generative adversarial networks. In this note, we present how WOGAN works and summarize its performance in the SBST 2022 CPS tool competition concerning the AI of a self-driving car.

cs.RO↗

arXiv2022

Automatic winning shifts

To each one-dimensional subshift $X$, we may associate a winning shift $W(X)$ which arises from a combinatorial game played on the language of $X$. Previously it has been studied what properties of $X$ does $W(X)$ inherit. For example, $X$ and $W(X)$ have the same factor complexity and if $X$ is a sofic subshift, then $W(X)$ is also sofic. In this paper, we develop a notion of automaticity for $W(X)$, that is, we propose what it means that a vector representation of $W(X)$ is accepted by a finite automaton. Let $S$ be an abstract numeration system such that addition with respect to $S$ is a rational relation. Let $X$ be a subshift generated by an $S$-automatic word. We prove that as long as there is a bound on the number of nonzero symbols in configurations of $W(X)$ (which follows from $X$ having sublinear factor complexity), then $W(X)$ is accepted by a finite automaton, which can be effectively constructed from the description of $X$. We provide an explicit automaton when $X$ is generated by certain automatic words such as the Thue-Morse word.

cs.FL↗

arXiv2021

On prefix palindromic length of automatic words

The prefix palindromic length $\mathrm{PPL}_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. Since 2013, it is still unknown if $\mathrm{PPL}_{\mathbf{u}}(n)$ is unbounded for every aperiodic infinite word $\mathbf{u}$, even though this has been proven for almost all aperiodic words. At the same time, the only well-known nontrivial infinite word for which the function $\mathrm{PPL}_{\mathbf{u}}(n)$ has been precisely computed is the Thue-Morse word $\mathbf{t}$. This word is $2$-automatic and, predictably, its function $\mathrm{PPL}_{\mathbf{t}}(n)$ is $2$-regular, but is this the case for all automatic words? In this paper, we prove that this function is $k$-regular for every $k$-automatic word containing only a finite number of palindromes. For two such words, namely the paperfolding word and the Rudin-Shapiro word, we derive a formula for this function. Our computational experiments suggest that generally this is not true: for the period-doubling word, the prefix palindromic length does not look $2$-regular, and for the Fibonacci word, it does not look Fibonacci-regular. If proven, these results would give rare (if not first) examples of a natural function of an automatic word which is not regular.

cs.FL↗

arXiv2020

Standard words and solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$

We consider solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$ such that the squares $X_i^2$ are minimal squares found in optimal squareful infinite words. We apply a method developed by the second author for studying word equations and prove that there are exactly two families of solutions: reversed standard words and words obtained from reversed standard words by a simple substitution scheme. A particular and remarkable consequence is that a word $w$ is a standard word if and only if its reversal is a solution to the word equation and $\gcd(|w|, |w|_1) = 1$. This result can be interpreted as a yet another characterization for standard Sturmian words. We apply our results to the symbolic square root map $\sqrt{\cdot}$ studied by the first author and M. A. Whiteland. We prove that if the language of a minimal subshift $Ω$ contains infinitely many solutions to the word equation, then either $Ω$ is Sturmian and $\sqrt{\cdot}$-invariant or $Ω$ is a so-called SL-subshift and not $\sqrt{\cdot}$-invariant. This result is progress towards proving the conjecture that a minimal and $\sqrt{\cdot}$-invariant subshift is necessarily Sturmian.

cs.FL↗

arXiv2020

Avoiding abelian powers cyclically

We study a new notion of cyclic avoidance of abelian powers. A finite word $w$ avoids abelian $N$-powers cyclically if for each abelian $N$-power of period $m$ occurring in the infinite word $w^ω$, we have $m \geq |w|$. Let $\mathcal{A}(k)$ be the least integer $N$ such that for all $n$ there exists a word of length $n$ over a $k$-letter alphabet that avoids abelian $N$-powers cyclically. Let $\mathcal{A}_\infty(k)$ be the least integer $N$ such that there exist arbitrarily long words over a $k$-letter alphabet that avoid abelian $N$-powers cyclically. We prove that $5 \leq \mathcal{A}(2) \leq 8$, $3 \leq \mathcal{A}(3) \leq 4$, $2 \leq \mathcal{A}(4) \leq 3$, and $\mathcal{A}(k) = 2$ for $k \geq 5$. Moreover, we show that $\mathcal{A}_\infty(2) = 4$, $\mathcal{A}_\infty(3) = 3$, and $\mathcal{A}_\infty(4) = 2$.

cs.FL↗

arXiv2020

Abelian periods of factors of Sturmian words

We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle $α$ with continued fraction expansion $[0; a_1, a_2, \ldots]$ is either $tq_k$ with $1 \leq t \leq a_{k+1}$ (a multiple of a denominator $q_k$ of a convergent of $α$) or $q_{k,\ell}$ (a denominator $q_{k,\ell}$ of a semiconvergent of $α$). This result generalizes a result of Fici et. al stating that the abelian period set of the Fibonacci word is the set of Fibonacci numbers. A characterization of the Fibonacci word in terms of its abelian period set is obtained as a corollary.

cs.FL↗

arXiv2019

On $k$-abelian Equivalence and Generalized Lagrange Spectra

We study the set of $k$-abelian critical exponents of all Sturmian words. It has been proven that in the case $k = 1$ this set coincides with the Lagrange spectrum. Thus the sets obtained when $k > 1$ can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when $k > 1$ the spectrum is a dense non-closed set. This is in contrast with the case $k = 1$, where the spectrum is a closed set containing a discrete part and a half-line. We describe explicitly the least accumulation points of the generalized spectra. Our geometric approach allows the study of $k$-abelian powers in Sturmian words by means of continued fractions.

cs.DM↗

arXiv2019

Every nonnegative real number is an abelian critical exponent

The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number $θ$ there exists an infinite word having abelian critical exponent $θ$. We also extend this result to the $k$-abelian setting.

cs.FL↗

arXiv2018

Automatic sequences based on Parry or Bertrand numeration systems

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical $k$-automatic sequences and Pisot-automatic sequences. We show that, like $k$-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for $k$-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is $U$-automatic with respect to a positional numeration system $U$ with regular language of numeration if and only if its $U$-kernel is finite.

cs.FL↗

arXiv2018

On winning shifts of marked uniform substitutions

The second author introduced with I. Törmä a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has a predetermined (possibly finite) choice sequence $α_1$, $α_2$, $\ldots$ of integers such that on round $n$ the player $A$ chooses a subset $S_n$ of size $α_n$ of some fixed finite alphabet and the player $B$ picks a letter from the set $S_n$. The outcome is determined by whether the word obtained by concatenating the letters $B$ picked lies in a prescribed target set $X$ (a win for player $A$) or not (a win for player $B$). Typically, we consider $X$ to be a subshift. The winning shift $W(X)$ of a subshift $X$ is defined as the set of choice sequences for which $A$ has a winning strategy when the target set is the language of $X$. The winning shift $W(X)$ mirrors some properties of $X$. For instance, $W(X)$ and $X$ have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that $W(X)$ and $X$ have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.

cs.FL↗

arXiv2018

More on the dynamics of the symbolic square root map

In our earlier paper [A square root map on Sturmian words, Electron. J. Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal squareful infinite word $s$ contains exactly six minimal squares and can be written as a product of these squares: $s = X_1^2 X_2^2 \cdots$. The square root $\sqrt{s}$ of $s$ is the infinite word $X_1 X_2 \cdots$ obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.

cs.FL↗

arXiv2016

Abelian Powers and Repetitions in Sturmian Words

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $α$. vAs an analogue of the critical exponent, we introduce the abelian critical exponent $A(s_α)$ of a Sturmian word $s_α$ of angle $α$ as the quantity $A(s_α) = limsup\ k_{m}/m=limsup\ k'_{m}/m$, where $k_{m}$ (resp. $k'_{m}$) denotes the maximum exponent of an abelian power (resp.~of an abelian repetition) of abelian period $m$ (the superior limits coincide for Sturmian words). We show that $A(s_α)$ equals the Lagrange constant of the number $α$. This yields a formula for computing $A(s_α)$ in terms of the partial quotients of the continued fraction expansion of $α$. Using this formula, we prove that $A(s_α) \geq \sqrt{5}$ and that the equality holds for the Fibonacci word. We further prove that $A(s_α)$ is finite if and only if $α$ has bounded partial quotients, that is, if and only if $s_α$ is $β$-power-free for some real number $β$. Concerning the infinite Fibonacci word, we prove that: i) The longest prefix that is an abelian repetition of period $F_j$, $j>1$, has length $F_j( F_{j+1}+F_{j-1} +1)-2$ if $j$ is even or $F_j( F_{j+1}+F_{j-1} )-2$ if $j$ is odd, where $F_{j}$ is the $j$th Fibonacci number; ii) The minimum abelian period of any factor is a Fibonacci number. Further, we derive a formula for the minimum abelian periods of the finite Fibonacci words

math.CO↗