A Boolean predicate $A$ is defined to be promise-useful if $PCSP(A,B)$ is tractable for some non-trivial $B$ and otherwise it is promise-useless. We initiate investigations of this notion and derive sufficient conditions for both promise-usefulness and promise-uselessness (assuming $\text{P} \ne \text{NP}$). While we do not obtain a complete characterization, our conditions are sufficient to classify all predicates of arity at most $4$ and almost all predicates of arity $5$. We also derive asymptotic results to show that for large arities a vast majority of all predicates are promise-useless.

Our results are primarily obtained by a thorough study of the ``Promise-SAT'' problem, in which we are given a $k$-SAT instance with the promise that there is a satisfying assignment for which the literal values of each clause satisfy some additional constraint.

The algorithmic results are based on the basic LP + affine IP algorithm of Brakensiek et al.~(SICOMP, 2020) while we use a number of novel criteria to establish NP-hardness.