Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1504.05159

Complexity of Suffix-Free Regular Languages

Janusz Brzozowski·Marek Szykuła
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study various complexity properties of suffix-free regular languages. The quotient complexity of a regular language $L$ is the number of left quotients of $L$; this is the same as the state complexity of $L$. A regular language $L'$ is a dialect of a regular language $L$ if it differs only slightly from $L$. The quotient complexity of an operation on regular languages is the maximal quotient complexity of the result of the operation expressed as a function of the quotient complexities of the operands. A sequence $(L_k,L_{k+1},\dots)$ of regular languages in some class ${\mathcal C}$, where $n$ is the quotient complexity of $L_n$, is called a stream. A stream is most complex in class ${\mathcal C}$ if its languages $L_n$ meet the complexity upper bounds for all basic measures. It is known that there exist such most complex streams in the class of regular languages, in the class of prefix-free languages, and also in the classes of right, left, and two-sided ideals. In contrast to this, we prove that there does not exist a most complex stream in the class of suffix-free regular languages. However, we do exhibit one ternary suffix-free stream that meets the bound for product and whose restrictions to binary alphabets meet the bounds for star and boolean operations. We also exhibit a quinary stream that meets the bounds for boolean operations, reversal, size of syntactic semigroup, and atom complexities. Moreover, we solve an open problem about the bound for the product of two languages of quotient complexities $m$ and $n$ in the binary case by showing that it can be met for infinitely many $m$ and $n$.

Explore related subjects

cs.FL

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Janusz Brzozowski, Marek Szykuła. 2016-12-12. Complexity of Suffix-Free Regular Languages. https://arxiv.org/abs/1504.05159

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adding Reconfiguration to Zielonka's Asynchronous Automata

We study an extension of Zielonka's (fixed) asynchronous automata called reconfigurable asynchronous automata where processes can dynamically change who they communicate with. We show that reconfigurable asynchronous automata are not more expressive than fixed asynchronous automata by giving translations from one to the other. However, going from reconfigurable to fixed comes at the cost of disseminating communication (and knowledge) to all processes in the system. We then show that this is unavoidable by describing a language accepted by a reconfigurable automaton such that in every equivalent fixed automaton, every process must either be aware of all communication or be irrelevant.

cs.FL↗
arXiv2026

Certificates for short extending words in a finite automaton

Let $\mathcal A$ be a complete deterministic finite automaton on a state set $Q$ of size $n$ with $k$ letters, and for a proper nonempty subset $S$ of $Q$ let $\mathrm{minext}(S)$ be the length of a shortest word $u$ with $|Su^{-1}|>|S|$, where $Su^{-1}=\{q: q\cdot u\in S\}$. To each state $q$ attach the integer $β^{\ast}_q=\sum_{t=1}^{n-1}k^{\,n-1-t}(\mathrm{indeg}_t(q)-k^{t})$, where $\mathrm{indeg}_t(q)$ counts the pairs $(p,u)$ with $|u|=t$ and $p\cdot u=q$, and let $B(S)=\sum_{q\in S}β^{\ast}_q$. On every synchronizing automaton, $B(S)\ge0$ implies $\mathrm{minext}(S)\le n-1$, so, as $B(Q)=0$, one of $S$ and $Q\setminus S$ extends within $n-1$; when $B(S)>0$ no hypothesis is needed. Kari's Eulerian extension lemma is the case $β^{\ast}=0$, and $β^{\ast}$, like every member of the family $\sum_{t=1}^{n-1}c_tσ_t$, $c_t>0$, vanishes identically if and only if the automaton is Eulerian, where $σ_t(S)=\sum_{q\in S}(\mathrm{indeg}_t(q)-k^{t})$. On strongly connected automata $σ_t(S)/k^{t}$ has Cesàro limit $n\,e(S)/e(Q)-|S|$ for Friedman's weight $e$; that limit certifies singletons but no larger subset in general. The hypothesis $B(S)\ge0$ cannot be relaxed by one integer unit, nor can the constant $n-1$ be improved. A second-moment test on the sizes $|Su^{-1}|$ certifies 60 to 95 percent of the subsets with $B(S)<0$ at $n\le7$. Along non-Eulerian automata whose words of length $n-1$ merge a fraction of the state pairs bounded below, with $\max_q\mathrm{indeg}_{n-1}(q)=o(nk^{n-1})$, it certifies all but a vanishing share of them. The functional $B$ certifies half of the subsets outside $\{B=0\}$. At each subset size coprime to $n$ ($n\ge4$) some synchronizing Eulerian binary automaton attains the constant $n-1$; whether only there is open. No reset bound follows: Černý's automata have subsets not extending within $n-1$.

cs.FL↗
arXiv2026

Quadratic Word Equations with a Linear Side: Polynomial Nielsen Graph Diameter and NP-Completeness

The satisfiability problem for word equations asks whether variables can be replaced by words so that the two sides become equal. For regular word equations, in which each variable occurs at most once on each side, satisfiability is NP-complete. For general quadratic word equations, in which each variable occurs at most twice in total, satisfiability is NP-hard, but its membership in NP remains open. We consider an intermediate class: quadratic word equations with a linear side, where each variable occurs at most once on one designated side. We show that the Nielsen graph of an equation $U=V$ in this class, with total length $N=|U|+|V|$, has diameter $O(N^{12})$, measured over reachable pairs of vertices. Together with the known NP-hardness for regular word equations, this result establishes NP-completeness of satisfiability for this class.

cs.FL↗