Symmetric two-counter machines

Krzysztof Jaros; Łukasz Kamiński; Maja Kądziołka; Sławomir Lasota

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arXiv · 2610.23495

Symmetric two-counter machines

Krzysztof Jaros·Łukasz Kamiński·Maja Kądziołka·Sławomir Lasota

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Abstract

Counter machines are a particularly simple, Turing-complete model of computation. We study the expressive power of nondeterministic 2-counter machines which are symmetric, in that their set of update transitions is invariant under swapping of counters. Our model is an extension of symmetric 2-VASS, in that we allow modifying both counters within a single transition. As the main result we demonstrate a dramatic impact of symmetry: we show that symmetric 2-counter machines are not Turing-complete, by proving decidability of their reachability problem. We conjecture that the decidability extends to machines with 3 or more counters that are symmetric, meaning the set of update transitions is invariant under any permutation of counters. We leave this problem open for machines with 3 and more counters, while identifying a frontier of the impact of symmetry: we prove undecidability for 3-counter machines invariant under the cyclic permutation group of order 3.

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BibTeX RIS

Krzysztof Jaros, Łukasz Kamiński, Maja Kądziołka, Sławomir Lasota. 2026-10-16. Symmetric two-counter machines. https://arxiv.org/abs/2610.23495

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