R 69 - 18 Multitape One - Way

A multitape one-way nonwriting automaton (MONA) is a finite state machine with a finite number of one-way input tapes which are advanced independently. This paper summarizes closure properties and decision problems of both deterministic and nondeterministic varieties. The family of n-ary word relations defined by deterministic (nondeterministic) n-tape MONA's is called D5(NV). Clearly D1 = NJ =regular sets and requires no further comment. As is often the case, the results for deterministic machines are more difficult and less general than those for nondeterministic machines. Indeed, if one attaches multiple input tapes to any AFA [11 or closed class of one-way nondeterministic balloon automata [3], one obtains at once a family L. of n-ary relations with most of the properties of Nn. Defining operations componentwise in the obvious manner, it turns out that any such L. is a full AFL [1] (closed under union, concatenation, star, homomorphism, inverse homomorphism, and intersection with "regular" relations of the form R1X ... XRn, each Ri in N,). Any projection of a member of L. on k coordinates clearly belongs to Lk. Since each L. must contain N,, the Rabin and Scott proof of the undecidability of the disjointness problem for two-tape deterministic MONA yields the undecidability of the universe problem for N. and so L, and hence all the usual undecidability results follow by appropriate generalization of methods of [2] including the fact that if the deterministic subfamily is properly contained in L, then it is undecidable whether a member of L. can be defined by a deterministic machine. In view of the connection between Na and linear context-free languages established by Rosenberg [5], it is not surprising to find that the results on closure properties and decision problems for N2 (and N.) echo those for the context-free languages. Most of the results for P2 are parallel to those for deterministic context-free languages (the connection is [5] does not extend in a natural way to the deterministic case); the equivalence problem for D2 remains open. An interesting example of the pathology present even in as simple a family as N2 is the table demonstrating the independence of the three properties: 1) L in D2, 2) Lk in D2for all k> 1, and 3) L* in D2. S. A. GREIBACH Harvard University Cambridge, Mass.