Algebraic Properties for P-Selectivity

Karp and Lipton, in their seminal 1980 paper, introduced the notion of advice (nonuniform) complexity, which since has been of central importance in complexity theory. Nonetheless, much remains unknown about the optimal advice complexity of classes having polynomial advice complexity.In particular, let P-sel denote the class of all P-selective sets [23] For the nondeterministic advice complexity of P-sel, linear upper and lower bounds are known [10]. However, for the deterministic advice complexity of P-sel, the best known upper bound is quadratic [13], and the best known lower bound is the linear lower bound inherited from the nondeterministic case. This paper establishes an algebraic sufficient condition for P-sel to have a linear upper bound: If all P-selective sets are associatively P-selective then the deterministic advice complexity of P-sel is linear. (The weakest previously known sufficient condition was P = NP.) Relatedly, we prove that every associatively P-selective set is commutatively, associatively P-selective.