No weakly factor-universal cellular automaton
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908911831875584 |
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| author | Gwozdz, Maja |
| author_facet | Gwozdz, Maja |
| contents | Hochman asked whether there exists a cellular automaton $F$ such that every cellular automaton is a factor of $F$ in the dynamical sense. In particular, we do not require the factor map to commute with the spatial shifts. We show that no such cellular automaton exists. More generally, if $F$ weakly factors onto the radius-zero $q$-clock automaton $C_q^{(k)}$, then every periodic point of $F$ has period divisible by $q$. For a cellular automaton $F:A^{\mathbb Z^d}\to A^{\mathbb Z^d}$, define $φ_F:A\to A$ by $F(\underline a)=\underline{φ_F(a)}$, and let $g_F$ be the greatest common divisor of the cycle lengths of $φ_F$. We prove that if $C_q^{(k)}$ is a weak factor of $F$, then $q\mid g_F$ holds. It follows that the action of $F$ on constant configurations yields an explicit divisibility obstruction to clock weak factors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23570 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | No weakly factor-universal cellular automaton Gwozdz, Maja Dynamical Systems Formal Languages and Automata Theory 37B15, 37B50, 68Q80 Hochman asked whether there exists a cellular automaton $F$ such that every cellular automaton is a factor of $F$ in the dynamical sense. In particular, we do not require the factor map to commute with the spatial shifts. We show that no such cellular automaton exists. More generally, if $F$ weakly factors onto the radius-zero $q$-clock automaton $C_q^{(k)}$, then every periodic point of $F$ has period divisible by $q$. For a cellular automaton $F:A^{\mathbb Z^d}\to A^{\mathbb Z^d}$, define $φ_F:A\to A$ by $F(\underline a)=\underline{φ_F(a)}$, and let $g_F$ be the greatest common divisor of the cycle lengths of $φ_F$. We prove that if $C_q^{(k)}$ is a weak factor of $F$, then $q\mid g_F$ holds. It follows that the action of $F$ on constant configurations yields an explicit divisibility obstruction to clock weak factors. |
| title | No weakly factor-universal cellular automaton |
| topic | Dynamical Systems Formal Languages and Automata Theory 37B15, 37B50, 68Q80 |
| url | https://arxiv.org/abs/2603.23570 |