Lower Bounds for the Complexity of Monadic SecondOrder Logic
Abstract
Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth, or in other words, MSO is fixedparameter tractable in linear time on any such class of graphs. From a logical perspective, Courcelle's theorem establishes a sufficient condition, or an upper bound, for tractability of MSOmodel checking. Whereas such upper bounds on the complexity of logics have received significant attention in the literature, almost nothing is known about corresponding lower bounds. In this paper we establish a strong lower bound for the complexity of monadic secondorder logic. In particular, we show that if C is any class of graphs which is closed under taking subgraphs and whose treewidth is not bounded by a polylogarithmic function (in fact, $\log^c n$ for some small c suffices) then MSOmodel checking is intractable on C (under a suitable assumption from complexity theory).
 Publication:

arXiv eprints
 Pub Date:
 January 2010
 arXiv:
 arXiv:1001.5019
 Bibcode:
 2010arXiv1001.5019K
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms;
 F.4.1;
 F.2.2;
 G.2.2
 EPrint:
 Preliminary version appeared in proceedings of the 25th IEEE symposium on Logic in Computer Science (LICS'10), Edinburgh, Scotland, UK, pp. 189198, 2010