Automatic constraint satisfaction problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915947281907712 |
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| author | Bulatov, Andrei Gong, Xiaoyang Khoussainov, Bakh Wang, Xinyao |
| author_facet | Bulatov, Andrei Gong, Xiaoyang Khoussainov, Bakh Wang, Xinyao |
| contents | We study constraint satisfaction problems (CSPs) where the constraint languages are defined by finite automata, giving rise to automata-based CSPs. The key notion is the concept of Automatic Constraint Satisfaction Problem ($AutCSP$), where constraint languages and instances are specified by finite automata. The $AutCSP$ captures infinite yet finitely describable sets of relations, enabling concise representations of complex constraints. Studying the complexity of the $AutCSP$s illustrates the interplay between classical CSPs, automata, and logic, sharpening the boundary between tractable and intractable constraints. We show that checking whether an operation is a polymorphism of such a language can be done in polynomial time. Building on this, we establish several complexity classification results for the $AutCSP$. In particular, we prove that Schaefer's Dichotomy Theorem extends to the $AutCSP$ over the Boolean domain, and we provide algorithms that decide tractability of some classes of $AutCSP$s over arbitrary finite domains via automatic polymorphisms. An important part of our work is that our polynomial-time algorithms run on $AutCSP$ instances that can be exponentially more succinct than their standard CSP counterparts. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_19266 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Automatic constraint satisfaction problem Bulatov, Andrei Gong, Xiaoyang Khoussainov, Bakh Wang, Xinyao Logic in Computer Science Formal Languages and Automata Theory We study constraint satisfaction problems (CSPs) where the constraint languages are defined by finite automata, giving rise to automata-based CSPs. The key notion is the concept of Automatic Constraint Satisfaction Problem ($AutCSP$), where constraint languages and instances are specified by finite automata. The $AutCSP$ captures infinite yet finitely describable sets of relations, enabling concise representations of complex constraints. Studying the complexity of the $AutCSP$s illustrates the interplay between classical CSPs, automata, and logic, sharpening the boundary between tractable and intractable constraints. We show that checking whether an operation is a polymorphism of such a language can be done in polynomial time. Building on this, we establish several complexity classification results for the $AutCSP$. In particular, we prove that Schaefer's Dichotomy Theorem extends to the $AutCSP$ over the Boolean domain, and we provide algorithms that decide tractability of some classes of $AutCSP$s over arbitrary finite domains via automatic polymorphisms. An important part of our work is that our polynomial-time algorithms run on $AutCSP$ instances that can be exponentially more succinct than their standard CSP counterparts. |
| title | Automatic constraint satisfaction problem |
| topic | Logic in Computer Science Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2604.19266 |