The quantifier structure of sentences that characterize nondeterministic time complexity

Abstract

The well-known model theoretic characterization of nondeterministic time complexity is studied. It is shown that for every nondeterministic Turing machine M of time complexity T(n), there is an existential second-order sentence sigma of a very restricted form, whose set of finite models corresponds to the set of strings recognized by M. Specifically, sigma is in the theory of addition restricted to finite segments of the natural numbers. Also, for every string x accepted by M, the length of the encoding of the model corresponding to x is O(T( mod x mod )). A consequence is that if T(n)=n/sup d/, then an upper bound is obtained for the degree of the second-order variables of sigma . Potential applications to low-level complexity are discussed. >