# Automata, Logic and Semantics

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.

### Resynchronized Uniformization and Definability Problems for Rational Relations

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.

### Automatic sequences: from rational bases to trees

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.

### Freezing, Bounded-Change and Convergent Cellular Automata

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 […]

### Binary patterns in the Prouhet-Thue-Morse sequence

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.

### Crisp-determinization of weighted tree automata over strong bimonoids

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 […]

### Weak equivalence of higher-dimensional automata

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.

### A Büchi-Elgot-Trakhtenbrot theorem for automata with MSO graph storage

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 Büchi-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.

### A Type System Describing Unboundedness

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.

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

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.

### Factoriality and the Pin-Reutenauer procedure

We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.

### Some undecidable problems about the trace-subshift associated to a Turing machine

We consider three problems related to dynamics of one-tape Turing machines: Existence of blocking configurations, surjectivity in the trace, and entropy positiveness. In order to address them, a reversible two-counter machine is simulated by a reversible Turing machine on the right side of its tape. By completing the machine in different ways, we prove that none of the former problems is decidable. In particular, the problems about blocking configurations and entropy are shown to be undecidable for the class of reversible Turing machines.

### Classical Automata on Promise Problems

Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and one-way deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.

### On the Hausdorff measure of regular ω-languages in Cantor space

This paper deals with the calculation of the Hausdorff measure of regular ω-languages, that is, subsets of the Cantor space definable by finite automata. Using methods for decomposing regular ω-languages into disjoint unions of parts of simple structure we derive two sufficient conditions under which ω-languages with a closure definable by a finite automaton have the same Hausdorff measure as this closure. The first of these condition is related to the homogeneity of the local behaviour of the Hausdorff dimension of the underlying set, and the other with a certain topological density of the set in its closure.

### A note on a recent attempt to improve the Pin-Frankl bound

We provide a counterexample to a lemma used in a recent tentative improvement of the Pin-Frankl bound for synchronizing automata. This example naturally leads us to formulate an open question, whose answer could fix the line of the proof, and improve the bound.

### Parameterized complexity of synchronization and road coloring

First, we close the multi-parameter analysis of a canonical problem concerning short reset words (SYN) initiated by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multi-parameter analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum length of a reset word.

### An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n+1) - p(n) ≤ 2

An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n + 1) − p(n) ≤ 2 S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is S-adic with Card(S) ≤ 3 27. In this paper, we improve this result by giving an S-adic characterization of these subshifts with a set S of 5 morphisms, solving by this way the S-adic conjecture for this particular case.

The implicit signature κ consists of the multiplication and the (ω-1)-power. We describe a procedure to transform each κ-term over a finite alphabet A into a certain canonical form and show that different canonical forms have different interpretations over some finite semigroup. The procedure of construction of the canonical forms, which is inspired in McCammond\textquoterights normal form algorithm for ω-terms interpreted over the pseudovariety A of all finite aperiodic semigroups, consists in applying elementary changes determined by an elementary set Σ of pseudoidentities. As an application, we deduce that the variety of κ-semigroups generated by the pseudovariety S of all finite semigroups is defined by the set Σ and that the free κ-semigroup generated by the alphabet A in that variety has decidable word problem. Furthermore, we show that each ω-term has a unique ω-term in canonical form with the same value over A. In particular, the canonical forms provide new, simpler, representatives for ω-terms interpreted over that pseudovariety.

### The Cerný conjecture for automata respecting intervals of a directed graph

The Cerný's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n - 1)2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors.

### Surjective cellular automata far from the Garden of Eden

One of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure.

### Derivatives of approximate regular expressions

Our aim is to construct a finite automaton recognizing the set of words that are at a bounded distance from some word of a given regular language. We define new regular operators, the similarity operators, based on a generalization of the notion of distance and we introduce the family of regular expressions extended to similarity operators, that we call AREs (Approximate Regular Expressions). We set formulae to compute the Brzozowski derivatives and the Antimirov derivatives of an ARE, which allows us to give a solution to the ARE membership problem and to provide the construction of two recognizers for the language denoted by an ARE. As far as we know, the family of approximative regular expressions is introduced for the first time in this paper. Classical approximate regular expression matching algorithms are approximate matching algorithms on regular expressions. Our approach is rather to process an exact matching on approximate regular expressions.

### A stronger recognizability condition for two-dimensional languages

The paper presents a condition necessarily satisfied by (tiling system) recognizable two-dimensional languages. The new recognizability condition is compared with all the other ones known in the literature (namely three conditions), once they are put in a uniform setting: they are stated as bounds on the growth of some complexity functions defined for two-dimensional languages. The gaps between such functions are analyzed and examples are shown that asymptotically separate them. Finally the new recognizability condition results to be the strongest one, while the remaining ones are its particular cases. The problem of deciding whether a two-dimensional language is recognizable is here related to the one of estimating the minimal size of finite automata recognizing a sequence of (one-dimensional) string languages.

### Automaticity of primitive words and irreducible polynomials

If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.

### Digraph complexity measures and applications in formal language theory

We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.

### The Join of the Varieties of R-trivial and L-trivial Monoids via Combinatorics on Words

The join of two varieties is the smallest variety containing both. In finite semigroup theory, the varieties of R-trivial and L-trivial monoids are two of the most prominent classes of finite monoids. Their join is known to be decidable due to a result of Almeida and Azevedo. In this paper, we give a new proof for Almeida and Azevedo's effective characterization of the join of R-trivial and L-trivial monoids. This characterization is a single identity of omega-terms using three variables.

### Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law

Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.

### Negative bases and automata

We study expansions in non-integer negative base -beta introduced by Ito and Sadahiro. Using countable automata associated with (-beta)-expansions, we characterize the case where the (-beta)-shift is a system of finite type. We prove that, if beta is a Pisot number, then the (-beta)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base beta to negative base -beta. When beta is a Pisot number, the conversion can be realized by a finite on-line transducer.

### Deciding whether the ordering is necessary in a Presburger formula

We characterize the relations which are first-order definable in the model of the group of integers with the constant 1. This allows us to show that given a relation defined by a first-order formula in this model enriched with the usual ordering, it is recursively decidable whether or not it is first-order definable without the ordering.

### On the asymptotic enumeration of accessible automata

We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.

### Cubefree words with many squares

We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.

### On the length of shortest 2-collapsing words

Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.

### Directed figure codes are decidable

Two-dimensional structures of various kinds can be viewed as generalizations of words. Codicity verification and the defect effect, important properties related to word codes, are studied also in this context. Unfortunately, both are lost in the case of two common structures, polyominoes and figures. We consider directed figures defined as labelled polyominoes with designated start and end points, equipped with catenation operation that uses a merging function to resolve possible conflicts. We prove that in this setting verification whether a given finite set of directed figures is a code is decidable and we give a constructive algorithm. We also clarify the status of the defect effect for directed figures.

### Multidimensional cellular automata and generalization of Fekete's lemma

Fekete's lemma is a well-known combinatorial result on number sequences: we extend it to functions defined on d-tuples of integers. As an application of the new variant, we show that nonsurjective d-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index.

### Leftmost derivations of propagating scattered context grammars: a new proof

In 1973, V. Virkkunen proved that propagating scattered context grammars which use leftmost derivations are as powerful as context-sensitive grammars. This paper brings a significantly simplified proof of this result.

### Interaction properties of relational periods

We consider relational periods where the relation is a compatibility relation on words induced by a relation on letters. We introduce three types of periods, namely global, external and local relational periods, and we compare their properties by proving variants of the theorem of Fine and Wilf for these periods.

### Shifts with decidable language and non-computable entropy

We consider subshifts of the full shift of all binary bi-infinite sequences. On the one hand, the topological entropy of any subshift with computably co-enumerable language is a right-computable real number between 0 and 1. We show that, on the other hand, any right-computable real number between 0 and 1, whether computable or not, is the entropy of some subshift with even polynomial time decidable language. In addition, we show that computability of the entropy of a subshift does not imply any kind of computability of the language of the subshift

### On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the critical exponent, i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.

### Latin square Thue-Morse sequences are overlap-free

We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences, generalizing results of Allouche - Shallit and Frid.