The Complexity of the Max Word Problem

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.

[1]  Richard J. Lipton,et al.  On the complexity of space bounded interactive proofs , 1989, 30th Annual Symposium on Foundations of Computer Science.

[2]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[3]  John Gill,et al.  Computational Complexity of Probabilistic Turing Machines , 1977, SIAM J. Comput..

[4]  Azaria Paz,et al.  Introduction to Probabilistic Automata , 1971 .

[5]  Anne Condon.  Space-bounded probabilistic game automata , 1991, JACM.

[6]  Richard J. Lipton,et al.  Word Problems Solvable in Logspace , 1977, JACM.

[7]  Arto Salomaa,et al.  Automata-Theoretic Aspects of Formal Power Series , 1978, Texts and Monographs in Computer Science.

[8]  Jeffrey Shallit,et al.  The Ring of k-Regular Sequences , 1990, Theor. Comput. Sci..

[9]  J. Rotman.  The theory of groups : an introduction , 1966 .