From Block Orthogonality to Decidability in Complex-Weighted Counting CSP

[2608.14845] From Block Orthogonality to Decidability in Complex-Weighted Counting CSP Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Computational Complexity arXiv:2608.14845 (cs) [Submitted on 14 Aug 2026] Title: From Block Orthogonality to Decidability in Complex-Weighted Counting CSP Authors: Chenghua Liu , Boning Meng View a PDF of the paper titled From Block Orthogonality to Decidability in Complex-Weighted Counting CSP, by Chenghua Liu and Boning Meng View PDF HTML (experimental) Abstract: In a landmark JACM paper recognized with the 2021 G{ö}del Prize, Cai and Chen established a complete complexity dichotomy for counting CSPs over arbitrary finite domains with algebraic complex weights. Its polynomial-time side is characterized by three conditions---Block Orthogonality, Type Partition, and preservation by a common Mal'tsev operation---quantified over the countably infinite family $W_{\mathcal{F}}$ generated from arbitrary $\#\mathrm{CSP}(\mathcal{F})$ instances by partial summation. They asked whether these infinitary conditions are decidable from the finite language $\mathcal{F}$ alone---equivalently, whether the polynomial-time side of this complete fixed-language classification is uniformly recognizable. We settle this problem by giving, for every nonempty finite domain $D$ and every finite exactly encoded algebraic-complex language $\mathcal{F}$, a total exact algorithm that decides all three conditions on the full unbounded family $W_{\mathcal{F}}$. Beyond decidability, we prove that Block Orthogonality alone forces both Type Partition and the existence of a single Mal'tsev operation preserving all generated support and row-equivalence relations. Thus the three-condition characterization collapses to Block Orthogonality, and the finite input $(D,\mathcal{F})$ determines which side of the dichotomy applies. The same framework decides the corresponding conditions in the dichotomy theorem for degree-multiple counting CSP proved by Lin. Subjects: Computational Complexity (cs.CC) Cite as: arXiv:2608.14845 [cs.CC] (or arXiv:2608.14845v1 [cs.CC] for this version) https://doi.org/10.48550/arXiv.2608.14845 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Boning Meng [ view email ] [v1] Fri, 14 Aug 2026 19:37:56 UTC (56 KB) Full-text links: Access Paper: View a PDF of the paper titled From Block Orthogonality to Decidability in Complex-Weighted Counting CSP, by Chenghua Liu and Boning Meng View PDF HTML (experimental) TeX Source view license Current browse context: cs.CC < prev | next > new | recent | 2026-08 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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