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Automata, Logic and Semantics

Automata, Logic and Semantics

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59 articles

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This section of DMTCS is devoted to publishing original research from several domains covered by Volume B of the Handbook of Theoretical Computer Science (Elsevier Publisher). Our scope is suggested by the following list of keywords: automata theory, automata-theoretic complexity, automatic program verification, combinatorics of words, coding theory, concurrency, data bases, formal languages, functional programming, logic in computer science, logic programming, program specification, rewriting, semantics of programming languages, theorem proving.

Editors: Corina Cirstea, Henning Fernau, Juhani Eero Urho Karhumäki, Klaus-Jörn Lange, Andreas Maletti, Daniel Reidenbach, Hadas Shachnai, Howard Straubing, Val Tannen, Henning Urbat, Pascal Weil

Editors: Corina Cirstea, Henning Fernau, Juhani Eero Urho Karhumäki, Klaus-Jörn Lange, Andreas Maletti, Daniel Reidenbach, Hadas Shachnai, Howard Straubing, Val Tannen, Henning Urbat, Pascal Weil

59 articles

Article

Synchronization of strongly connected partial DFAs and prefix codes

Mikhail V. Berlinkov, Robert Ferens, Andrew Ryzhikov, Marek Szykuła

Abstract

We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a \emph{reset word}) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. The class of strongly connected partial DFAs is precisely the class of DFAs recognizing the Kleene star of prefix codes. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case, this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs to the same problems for complete DFAs. In particular, this includes the Černý and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other. Extended version of STACS 2021 paper

Published on June 21, 2026

Article

Sometimes Two Irrational Guards are Needed

Lucas Meijer, Tillmann Miltzow

Abstract

In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur. 24 pages, 13 figures

Published on March 16, 2026

Article

$\rm P$ has polynomial-time finite-state verifiers

M. Utkan Gezer, A. C. Cem Say

Abstract

Interactive proof systems whose verifiers are constant-space machines have interesting features that do not have counterparts in the better studied case where the verifiers operate under reasonably large space bounds. The language verification power of finite-state verifiers is known to be sensitive to the difference between private and public randomization. These machines also lack the capability of imposing worst-case superlinear bounds on their own runtime, and long interactions with untrustable provers can involve the risk of being fooled to loop forever. We analyze such verifiers under different bounds on the numbers of private and public random bits that they are allowed to use. This separate accounting for the private and public coin budgets as resource functions of the input length provides interesting characterizations of the collections of the associated languages. When the randomness bound is constant, the verifiable class is $\rm NL$ for private-coin machines, but equals just the regular languages when one uses public coins. Increasing the public coin budget while keeping the number of private coins constant augments the power: We show that the set of languages that are verifiable by such machines in expected polynomial time (with an arbitrarily small positive probability of looping) equals the complexity class $\rm P$. This hints that allowing a minuscule probability of looping may add significant power to polynomial-time finite-state automata, since it is still not known whether those machines can verify all of $\rm P$ when required to halt with probability 1, even with no bound on their private coin usage. We also show that logarithmic-space machines which hide a constant number of their coins are limited to verifying the languages in $\rm P$. 32 pages, of which 10 pages are appendix, substantially extended version of an earlier conference version that was published in the proceedings of ICTCS 2023

Published on December 15, 2025

Article

A Diamond Structure in the Transducer Hierarchy

Noah Kaufmann

Abstract

We answer an open question in the theory of transducer degrees on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite state transducer, which form an order based on which words can be transformed into other words. We provide a construction which proves the existence of a diamond structure, while also introducing a new function on streams which may be useful for proving more results about the transducer hierarchy.

Published on December 15, 2025

Article

Reduction for asynchronous Boolean networks: elimination of negatively autoregulated components

Robert Schwieger, Elisa Tonello

Abstract

To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph. Comment: 16 pages

Published on March 22, 2024

Article

Pseudoperiodic Words and a Question of Shevelev

Joseph Meleshko, Pascal Ochem, Jeffrey Shallit, Sonja Linghui Shan

Abstract

We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete.

Published on October 16, 2023

Article

Dissecting power of intersection of two context-free languages

Josef Rukavicka

Abstract

We say that a language $L$ is \emph{constantly growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c+\vert u\vert$. We say that a language $L$ is \emph{geometrically growing} if there is a constant $c$ such that for every word $u\in L$ there is a word $v\in L$ with $\vert u\vert<\vert v\vert\leq c\vert u\vert$. Given two infinite languages $L_1,L_2$, we say that $L_1$ \emph{dissects} $L_2$ if $\vert L_2\setminus L_1\vert=\infty$ and $\vert L_1\cap L_2\vert=\infty$. In 2013, it was shown that for every constantly growing language $L$ there is a regular language $R$ such that $R$ dissects $L$. In the current article we show how to dissect a geometrically growing language by a homomorphic image of intersection of two context-free languages. Consider three alphabets $\Gamma$, $\Sigma$, and $\Theta$ such that $\vert \Sigma\vert=1$ and $\vert \Theta\vert=4$. We prove that there are context-free languages $M_1,M_2\subseteq \Theta^*$, an erasing alphabetical homomorphism $\pi:\Theta^*\rightarrow \Sigma^*$, and a nonerasing alphabetical homomorphism $\varphi : \Gamma^*\rightarrow \Sigma^*$ such that: If $L\subseteq \Gamma^*$ is a geometrically growing language then there is a regular language $R\subseteq \Theta^*$ such that $\varphi^{-1}\left(\pi\left(R\cap M_1\cap M_2\right)\right)$ dissects the language $L$.

Published on October 2, 2023

Article

Resynchronized Uniformization and Definability Problems for Rational Relations

Christof Löding, Sarah Winter

Abstract

Regular synchronization languages can be used to define rational relations of finite words, and to characterize subclasses of rational relations, like automatic or recognizable relations. We provide a systematic study of the decidability of uniformization and definability problems for subclasses of rational relations defined in terms of such synchronization languages. We rephrase known results in this setting and complete the picture by adding several new decidability and undecidability results.

Published on September 6, 2023

Article

Automatic sequences: from rational bases to trees

Michel Rigo, Manon Stipulanti

Abstract

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider those built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence. Comment: 26 pages, 16 figures; final version accepted for publication in Discrete Mathematics & Theoretical Computer Science

Published on July 19, 2022

Article

Freezing, Bounded-Change and Convergent Cellular Automata

Nicolas Ollinger, Guillaume Theyssier

Abstract

This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

Published on January 31, 2022

Article

Binary patterns in the Prouhet-Thue-Morse sequence

Jorge Almeida, Ondřej Klíma

Abstract

We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.

Published on August 30, 2021

Article

Crisp-determinization of weighted tree automata over strong bimonoids

Zoltán Fülöp, Dávid Kószó, Heiko Vogler

Abstract

We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.

Published on June 15, 2021

Article

Weak equivalence of higher-dimensional automata

Thomas Kahl

Abstract

This paper introduces a notion of equivalence for higher-dimensional automata, called weak equivalence. Weak equivalence focuses mainly on a traditional trace language and a new homology language, which captures the overall independence structure of an HDA. It is shown that weak equivalence is compatible with both the tensor product and the coproduct of HDAs and that, under certain conditions, HDAs may be reduced to weakly equivalent smaller ones by merging and collapsing cubes.

Published on May 18, 2021

Article

A B\"uchi-Elgot-Trakhtenbrot theorem for automata with MSO graph storage

Joost Engelfriet, Heiko Vogler

Abstract

We introduce MSO graph storage types, and call a storage type MSO-expressible if it is isomorphic to some MSO graph storage type. An MSO graph storage type has MSO-definable sets of graphs as storage configurations and as storage transformations. We consider sequential automata with MSO graph storage and associate with each such automaton a string language (in the usual way) and a graph language; a graph is accepted by the automaton if it represents a correct sequence of storage configurations for a given input string. For each MSO graph storage type, we define an MSO logic which is a subset of the usual MSO logic on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string case and the graph case. Moreover, we prove that (i) each MSO graph transduction can be used as storage transformation in an MSO graph storage type, (ii) every automatic storage type is MSO-expressible, and (iii) the pushdown operator on storage types preserves the property of MSO-expressibility. Thus, the iterated pushdown storage types are MSO-expressible.

Published on August 27, 2020

Article

A Type System Describing Unboundedness

Paweł Parys

Abstract

We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP.

Published on August 18, 2020

Article

Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence

Łukasz Merta

Abstract

A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue{Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin{Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of strings of consecutive identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences. Comment: 20 pages

Published on May 25, 2020

Article

Complexity of Leading Digit Sequences

Xinwei He, A. J. Hildebrand, Yuchen Li, Yunyi Zhang

Abstract

Let $S_{a,b}$ denote the sequence of leading digits of $a^n$ in base $b$. It is well known that if $a$ is not a rational power of $b$, then the sequence $S_{a,b}$ satisfies Benford's Law; that is, digit $d$ occurs in $S_{a,b}$ with frequency $\log_{b}(1+1/d)$, for $d=1,2,\dots,b-1$. In this paper, we investigate the \emph{complexity} of such sequences. We focus mainly on the \emph{block complexity}, $p_{a,b}(n)$, defined as the number of distinct blocks of length $n$ appearing in $S_{a,b}$. In our main result we determine $p_{a,b}(n)$ for all squarefree bases $b\ge 5$ and all rational numbers $a>0$ that are not integral powers of $b$. In particular, we show that, for all such pairs $(a,b)$, the complexity function $p_{a,b}(n)$ is \emph{affine}, i.e., satisfies $p_{a,b}(n)=c_{a,b} n + d_{a,b}$ for all $n\ge1$, with coefficients $c_{a,b}\ge1$ and $d_{a,b}\ge0$, given explicitly in terms of $a$ and $b$. We also show that the requirement that $b$ be squarefree cannot be dropped: If $b$ is not squarefree, then there exist integers $a$ with $1<a<b$ for which $p_{a,b}(n)$ is not of the above form. We use this result to obtain sharp upper and lower bounds for $p_{a,b}(n)$, and to determine the asymptotic behavior of this function as $b\to\infty$ through squarefree values. We also consider the question which linear functions $p(n)=cn+d$ arise as the complexity function $p_{a,b}(n)$ of some leading digit sequence $S_{a,b}$. We conclude with a discussion of other complexity measures for the sequences $S_{a,b}$ and some open problems.

Published on April 30, 2020

Article

The minimal probabilistic and quantum finite automata recognizing uncountably many languages with fixed cutpoints

Aleksejs Naumovs, Maksims Dimitrijevs, Abuzer Yakaryılmaz

Abstract

It is known that 2-state binary and 3-state unary probabilistic finite automata and 2-state unary quantum finite automata recognize uncountably many languages with cutpoints. These results have been obtained by associating each recognized language with a cutpoint and then by using the fact that there are uncountably many cutpoints. In this note, we prove the same results for fixed cutpoints: each recognized language is associated with an automaton (i.e., algorithm), and the proofs use the fact that there are uncountably many automata. For each case, we present a new construction. Comment: 12 pages, minor revisions, changing the format to "dmtcs-episciences" style

Published on April 30, 2020

Article

New tools for state complexity

Pascal Caron, Edwin Hamel-De le court, Jean-Gabriel Luque, Bruno Patrou

Abstract

A monster is an automaton in which every function from states to states is represented by at least one letter. A modifier is a set of functions allowing one to transform a set of automata into one automaton. We revisit some language transformation algorithms in terms of modifier and monster. These new theoretical concepts allow one to find easily some state complexities. We illustrate this by retrieving the state complexity of the Star of Intersection and the one of the Square root operation. Comment: 18 pages

Published on March 16, 2020

Article

A Characterization of Morphic Words with Polynomial Growth

Tim Smith

Abstract

A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each $n \geq 1$ in one of three ways: it grows forward [$t(1)\ t(2)\ \dotsm\ t(n)]$, backward [$t(n)\ \dotsm\ t(2)\ t(1)$], or just occurs once [$t$]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth $\Theta(n^k)$ iff it is a zigzag word of depth $k$. As corollaries, we obtain that the morphic words with growth $O(n)$ are exactly the ultimately periodic words, and the morphic words with growth $O(n^2)$ are exactly the multilinear words.

Published on February 6, 2020

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