Monotone Complexity of a Pair

Abstract

We define monotone complexity ${\textit{KM}}(x,y)$ of a pair of binary strings x,y in a natural way and show that ${\textit{KM}}(x,y)$ may exceed the sum of the lengths of x and y (and therefore the a priori complexity of a pair) by αlog(|x| + |y|) for every α< 1 (but not for α> 1).

We also show that decision complexity of a pair or triple of strings does not exceed the sum of its lengths.

The work was performed while visiting LIF Marseille (CNRS & Univ. Aix–Marseille); the visit was made possible by the CNRS France–Russia exchange program; preparation of the final text was supported also by NAFIT ANR 008-01 grant).