Generalized Quantifiers and Logical Reducibilities

We consider extensions of first order logic (FO) and least fixed point logic (LFP) with generalized quantifiers in the sense of Lindström [Lin66]. We show that adding a finite set of such quantifiers to LFP fails to capture all polynomial time properties of structures, even over a fixed signature. We show that this strengthens results in [Hel92] and [KV92a]. We also consider certain regular infinite sets of Lindström quantifiers, which correspond to a natural notion of logical reducibility. We show that if there is any recursively enumerable set of quantifiers that can be added to FO (or LFP) to capture P, then there is one with strong uniformity conditions. This is established through a general result, linking the existence of complete problems for complexity classes with respect to the first order translations of [Imm87] or the elementary reductions of [LG77] with the existence of recursive index sets for these classes.