Oracles and Quantifiers

We describe a general way of building logics with Lindstrom quantifiers, which capture regular complexity classes on ordered structures with polysize reductions. We then extend this method so as to accommodate complexity classes based on oracle Turing machines. Our main result shows an equivalence between enhancing a logic with a Lindstrom quantifier and enhancing a complexity class with an oracle such that, if K is a set of structures, Q K the associated Lindstrom quantifier and L a logic that captures a complexity class D, then the enhanced logic L[K] captures D K — the complexity class of machines in D using oracles for K. Our results are sensitive to the oracle computation model and hold in a natural modification of the unbounded model introduced by Buss [Bus88]. They do not hold in the, so called, space bounded oracle models or those that violate the ‘relativization thesis’ of Buss. Our results generalize and extend previous results of Stewart [Ste93a, Ste93b] and Makowsky and Pnueli [MP93].