The CSP of a first-order theory $T$ is the problem of deciding for a given finite set $S$ of atomic formulas whether $T \cup S$ is satisfiable. Let $T_1$ and $T_2$ be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomial-time decidability) of $\mathrm{CSP}(T_1 \cup T_2)$ under the assumption that $\mathrm{CSP}(T_1)$ and $\mathrm{CSP}(T_2)$ are decidable (or polynomial-time decidable). We show that for a large class of $\omega$-categorical theories $T_1, T_2$ the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of $\mathrm{CSP}(T_1 \cup T_2)$ (unless P=NP).

Source : oai:arXiv.org:1801.05965

Volume: Volume 16, Issue 1

Published on: February 20, 2020

Submitted on: May 20, 2019

Keywords: Mathematics - Logic,Computer Science - Computational Complexity,Computer Science - Logic in Computer Science,03C98, 03D15, 08A70,F.4.1,F.2.2,G.2.1

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