The quantifier structure of sentences that characterize nondeterministic time complexity

Abstract

For every nondeterministic Turing machineM of time complexityT(n), there is a second-order sentence σ of a very restricted form, whose set of finite models encodes the set of strings recognized byM. Specifically, σ has a relational symbol which is interpreted as addition restricted to finite segments of the natural numbers, and a prefix consisting of existentially quantified unary second-order variables followed by a universal-existential first-order part. Here, every input stringx is encoded by a model of sizeT(|x|). Using a closely related encoding of strings as models where the size of the model is the length of the string, a consequence is that ifT(n)=n d, then there is a sentence with a similar prefix but whose second-order variables ared-ary and whose finite models encode the strings accepted byM. Potential applications to low-level complexity are discussed.