# The complexity of the max word problem

and the power of one-way interactive proof systems

Complexity III

First Online:

## Abstract

We study the complexity of the *max word problem* for matrices, a variation of the well-known word problem for matrices. We show that the problem is *NP*-complete, and cannot be approximated within any constant factor, unless *P*=*NP*. We describe applications of this result to probabilistic finite state automata, rational series and *k*-regular sequences. Our proof is novel in that it employs the theory of interactive proof systems, rather than a standard reduction argument. As another consequence of our results, we characterize *NP* exactly in terms of *one-way* interactive proof systems.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1991