On the log-space reducibility among array languages /preliminary version/

Abstract

In this paper we consider a 2-dimensional array acceptor, called Turing array machine /TAM/, which is a generalization of k-tape Turing machine for string languages. By means of this automaton the complexity classes of array languages are defined. For 2-dimension array languages a generalization of log-space reducibility relation is introduced so that every language which is NL-complete is also complete for the class of array languages accepted by TAMs in log space. It is also shown that there exists array language over 1-letter alphabet which is complete for the class of array languages accepted by nondeterministic TAMs in log space. It turned out that when proving equality or inequality of the classes NL and L we face the same difficulties when proving this property for array counterparts of the above classes. At the end we introduce an array language, called projection accessibility problem /PAP/, over 1-letter alphabet, which is log-space complete and is accepted by some nondeterministic finite automaton. It follows, that if there exists any deterministic automaton with a finite number of pebbles which accepts PAP then NL=L.