From Block Orthogonality to Decidability in Complex-Weighted Counting CSP

2026-08-14

AI Summary

This study addresses the decidability and unified identification of polynomial-time side infinite conditions for weighted counting CSPs. Integrating algebraic complexity theory with block orthogonality analysis, we design an exact decision algorithm. The core innovation lies in proving that block orthogonality implies both type partitioning and the existence of Mal'tsev operations, thereby collapsing the original three-condition characterization into a single criterion. This result establishes the centrality of block orthogonality and achieves unified identification for complexity dichotomies. Furthermore, we present a fully exact algorithm for determining infinite family conditions, completely resolving the decidability problem associated with classification theorems in this domain.

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.

Problem

• Counting CSP
• Decidability
• Complexity Dichotomy
• Block Orthogonality
• Algebraic Complex Weights

Innovation

• Counting CSP
• Block Orthogonality
• Decidability
• Complexity Dichotomy
• Mal'tsev Operation