Lower Bounds for the Complexity of Monadic Second-Order Logic

Abstract

Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic (MSO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) can be decided in linear time on any class of graphs of bounded tree-width, or in other words, MSO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sup> n for some small c suffices) then MSO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -model checking is intractable on C (under a suitable assumption from complexity theory).