The complexity of first-order and monadic second-order logic revisited

The model-checking problem for a logic L on a class C of structures asks whether a given L-sentence holds in a given structure in C. In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic. We show that unless PTIME=NP, the model-checking problem for monadic second-order logic on finite words is not solvable in time f(k)·p(n), for any elementary function f and any polynomial p. Here k denotes the size of the input sentence and n the size of the input word. We prove the same result for first-order logic under a stronger complexity theoretic assumption from parameterized complexity theory. Furthermore, we prove that the model-checking problems for first-order logic on structures of degree 2 and of bounded degree d≥3 are not solvable in time 2(2^o(k))·p(n) (for degree 2) and 2(2^2o(k))·p(n) (for degree d), for any polynomial p, again under an assumption from parameterized complexity theory. We match these lower bounds by corresponding upper bounds.