Lower Bounds for the Complexity of Monadic Second-Order Logic

Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic (MSO
₂
) can be decided in linear time on any class of graphs of bounded tree-width, or in other words, MSO
₂
is fixed-parameter 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 MSO
₂
-model 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 second-order logic. In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log
^c
n for some small c suffices) then MSO
₂
-model checking is intractable on C (under a suitable assumption from complexity theory).