  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].
