Word Equations Where a Power Equals a Product of Powers

We solve a long-standing open problem on word equations by proving that if the words x_0, ..., x_n satisfy the equation x_0^k = x_1^k ... x_n^k for three positive values of k, then the words commute. One of our methods is to assign numerical values for the letters, and then study the sums of the letters of words and their prefixes. We also give a geometric interpretation of our methods

Hardness Results for Constant-Free Pattern Languages and Word Equations

We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism ? such that ?(xy) ? ?(yx) for given variables x and y

Word Equations and Related Topics. Independence, Decidability and Characterizations

The three main topics of this work are independent systems and chains of word equations, parametric solutions of word equations on three unknowns, and unique decipherability in the monoid of regular languages. The most important result about independent systems is a new method giving an upper bound for their sizes in the case of three unknowns. The bound depends on the length of the shortest equation. This result has generalizations for decreasing chains and for more than three unknowns. The method also leads to shorter proofs and generalizations of some old results. Hmelevksii’s theorem states that every word equation on three unknowns has a parametric solution. We give a significantly simplified proof for this theorem. As a new result we estimate the lengths of parametric solutions and get a bound for the length of the minimal nontrivial solution and for the complexity of deciding whether such a solution exists. The unique decipherability problem asks whether given elements of some monoid form a code, that is, whether they satisfy a nontrivial equation. We give characterizations for when a collection of unary regular languages is a code. We also prove that it is undecidable whether a collection of binary regular languages is a code.Siirretty Doriast

On a generalization of Abelian equivalence and complexity of infinite words

In this paper we introduce and study a family of complexity functions of infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+ \cup {+\infty} and $A$ be a finite non-empty set. Two finite words $u$ and $v$ in $A^*$ are said to be $k$-Abelian equivalent if for all $x\in A^*$ of length less than or equal to $k,$ the number of occurrences of $x$ in $u$ is equal to the number of occurrences of $x$ in $v.$ This defines a family of equivalence relations $\thicksim_k$ on $A^*,$ bridging the gap between the usual notion of Abelian equivalence (when $k=1$) and equality (when $k=+\infty).$ We show that the number of $k$-Abelian equivalence classes of words of length $n$ grows polynomially, although the degree is exponential in $k.$ Given an infinite word \omega \in A^\nats, we consider the associated complexity function \mathcal {P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of $k$-Abelian equivalence classes of factors of $\omega$ of length $n.$ We show that the complexity function $\mathcal {P}^{(k)}$ is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if $\mathcal {P}^{(k)}_{\omega}(n)<q^k(n)$ for some k \in \ints ^+ \cup {+\infty} and $n\geq 0,$ the $\omega$ is ultimately periodic. Moreover if $\omega$ is aperiodic, then $\mathcal {P}^{(k)}_{\omega}(n)=q^k(n)$ if and only if $\omega$ is Sturmian. We also study $k$-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if $\omega$ has bounded $k$-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer $N,$ there exists a $k$-Abelian $N$ power occurring in $\omega$ at some position $j\in D.

An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences

We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open

47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism α such that α(xy) ≠ α(yx) for given variables x and y. </p