Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2601.21862

Cellular Automaton Reducibility as a Measure of Complexity for Infinite Words

Markel Zubia·Herman Geuvers
View original source ↗Read source document ↗Export metadata ↓

Abstract

Infinite words, also known as streams, hold significant interest in computer science and mathematics, raising the natural question of how their complexity should be measured. We introduce cellular automaton reducibility as a measure of stream complexity: σ is at least as complex as τ when there exists a cellular automaton mapping σ to τ. This enables the categorization of streams into degrees of complexity, analogous to Turing degrees in computability theory. We investigate the algebraic properties of the hierarchy that emerges from the partial ordering of degrees, showing that it is not well-founded and not dense, that ultimately periodic streams are ordered by divisibility of their period, that sparse streams are atoms, that maximal streams have maximal subword complexity, and that suprema of sets of streams do not generally exist. We also provide a pseudo-algorithm for classifying streams up to this reducibility.

Explore related subjects

cs.FL

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Markel Zubia, Herman Geuvers. 2026-01-29. Cellular Automaton Reducibility as a Measure of Complexity for Infinite Words. https://arxiv.org/abs/2601.21862

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↗