Two characterizations of the context-sensitive languages

An n-dimensional bug-automation is generalization of a finite state acceptor to n-dimensions. With each bug B, we associate the language L(B) which is the set of top rows of the n-dimensional rectangular arrays accepted by B. One-dimensional bugs define trivially the regular sets. Twodimensional bugs define precisely the context-sensitive languages, while bugs of dimension 3 or greater define all the recursively enumerable sets. We consider also finite state acceptors with n two-way non-writing input tapes. For each such machine M, let domain (M) be the set of all strings which are the first component of some n-tuple of tapes accepted by M. For any n ≥ l, the domains of n-tape two-way finite state acceptors are precisely the same as the languages definable by n-dimensional bugs, so as a corollary, the domains of two-tape two-way finite state acceptors are precisely the context-sensitive languages.