Matt Valeriote, McMaster University
I will discuss a question dealing with the solutions of primitive positive (pp) formulas over finite structures. The question is concerned with the query complexity of algorithms for making decisions about the correctness of a proposed solution to a pp-formula over a finite structure. In general, in order to determine with high probability whether some hidden proposed solution to a pp-formula is close to being an actual solution, one will need to check a large fraction of its values. It turns out that for some (special) structures, there are algorithms that only need to check (or query) a constant number of values of a proposed solution to a pp-formula (no matter how many free variables the formula has) in order to conclude, with high probability, whether or not it is close to a solution to the formula over the structure.
With Chen and Yoshida, we characterize, for finite structures A, the query complexity of testing solutions of pp-formulas over A in terms of algebraic conditions on A. In particular, we characterize those structures A for which solution testing can be carried out by an algorithm that only needs to make a constant number of queries of a proposed solution to a pp-formula, independent of the number of free variables the formula has.