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

Markus Frick and Martin Grohe

To appear at Symposium on Logic in Computer Science (LICS'2002), Copenhagen, Denmark, July 22nd - 25th, 2002


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. 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 problem for first-order logic on structures of degree 2 is not solvable in time 2^{2^{o(k)}} p(n), for any polynomial p, again under an assumption from parameterized complexity theory. We match this lower bound by a corresponding upper bound.

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