## Paper: The complexity of first-order and monadic second-order logic revisited (at LICS 2002)

**Markus Frick Martin Grohe**

### Abstract

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.

### BibTeX

@InProceedings{FrickGrohe-Thecomplexityoffirs, author = {Markus Frick and Martin Grohe}, title = {The complexity of first-order and monadic second-order logic revisited}, booktitle = {Proceedings of the Seventeenth Annual IEEE Symposium on Logic in Computer Science (LICS 2002)}, year = {2002}, month = {July}, pages = {215--224}, location = {Copenhagen, Denmark}, publisher = {IEEE Computer Society Press} }