The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS '19] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We initiate a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractability within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18].

中文翻译：

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有限可处理的承诺约束满足问题

承诺约束满足问题 (PCSP) 是约束满足问题 (CSP) 的概括，包括可满足性和图着色问题的近似变体。Barto [LICS '19] 已经表明，特定的 PCSP，即为 1-in-3-SAT 实例找到有效的 Not-All-Equal 解决方案的问题，不是有限易处理的，因为它可以通过简单的归约来解决到一个易于处理的 CSP，但这样的 CSP 必然是在无限域上（除非 P=NP）。我们通过给出有限易处理性的一般必要条件并表征一类模板中的有限易处理性——对称布尔 PCSP 的二分定理中的“基本”易处理案例，允许 Brakensiek 和 Guruswami [SODA' 18]。