Monadic logical definability of nondeterministic linear time

Abstract.

We answer a question asked by J. F. Lynch by proving that existential monadic second-order logic with addition captures not only the class NTIME(n) but also the class NLIN (i.e., linear time on nondeterministic RAMs), so enlarging considerably the set of natural problems expressible in this logic, since most combinatorial NP-complete problems belong to NLIN. Moreover, our result still holds if the first-order part of the formulas is required to be $\forall^{\ast} \exists ^{\ast}$, so improving the recent similar result by J. F. Lynch about NTIME(n).¶In addition, we explicitly state that a graph problem is recognizable in nondeterministic linear time O(n + e) (where n and e are the numbers of vertices and edges, respectively) if and only if it can be defined in existential second-order logic with unary functions and only one variable on the vertices-edges domain.