On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure

Undecidability of a weak version of MSO+U

We prove the undecidability of MSO on ω-words extended with the second-order predicate U1(X) which says that the distance between consecutive positions in a set X⊆N is unbounded. This is achieved by showing that adding U1 to MSO gives a logic with the same expressive power as MSO+U, a logic on ω-words with undecidable satisfiability. As a corollary, we prove that MSO on ω-words becomes undecidable if allowing to quantify over sets of positions that are ultimately periodic, i.e., sets X such that for some positive integer p, ultimately either both or none of positions x and x+p belong to X

Linear Distances between Markov Chains

We introduce a general class of distances (metrics) between Markov chains, which are based on linear behaviour. This class encompasses distances given topologically (such as the total variation distance or trace distance) as well as by temporal logics or automata. We investigate which of the distances can be approximated by observing the systems, i.e. by black-box testing or simulation, and we provide both negative and positive results

Separation Property for wB- and wS-regular Languages

In this paper we show that {\omega}B- and {\omega}S-regular languages satisfy the following separation-type theorem If L1,L2 are disjoint languages of {\omega}-words both recognised by {\omega}B- (resp. {\omega}S)-automata then there exists an {\omega}-regular language Lsep that contains L1, and whose complement contains L2. In particular, if a language and its complement are recognised by {\omega}B- (resp. {\omega}S)-automata then the language is {\omega}-regular. The result is especially interesting because, as shown by Boja\'nczyk and Colcombet, {\omega}B-regular languages are complements of {\omega}S-regular languages. Therefore, the above theorem shows that these are two mutually dual classes that both have the separation property. Usually (e.g. in descriptive set theory or recursion theory) exactly one class from a pair C, Cc has the separation property. The proof technique reduces the separation property for {\omega}-word languages to profinite languages using Ramsey's theorem and topological methods. After that reduction, the analysis of the separation property in the profinite monoid is relatively simple. The whole construction is technically not complicated, moreover it seems to be quite extensible. The paper uses a framework for the analysis of B- and S-regular languages in the context of the profinite monoid that was proposed by Toru\'nczyk

Cyclic Complexity of Words

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equal to that of $x.$ Since $c_x(n)=1$ for some $n\geq 1$ implies $x$ is periodic, it is natural to consider the quantity $\liminf_{n\rightarrow \infty} c_x(n).$ We show that if $x$ is a Sturmian word, then $\liminf_{n\rightarrow \infty} c_x(n)=2.$ We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue-Morse word $t$, $\liminf_{n\rightarrow \infty} c_t(n)=+\infty.$Comment: To appear in Journal of Combinatorial Theory, Series