Complexity classes and theories of finite models

Abstract

LetL $$ \subseteq $$ Σ* be accepted in timef(n) by a nondeterministic Turing machine. Then there is a monadic existential second-order sentence σ in the language of + such that for everyx∈Σ*,x∈L if and only if a certain structureU of cardinalityf(|x|) satisfies σ. It follows that ifL is accepted in nondeterministic timen d, d a natural number, then there is a sentence whose relational symbols ared-ary or less, whose finite spectrum isL.