Two-Way Counter Machines and Diophantine Equations

Abstract

Let Q be the class of determmistlc two-way l-counter machines accepting only bounded languages Each machine m Q has the property that m every accepting computation, the counter makes at most a fixed number of reversals It is shown that the emptiness problem for Q is decidable. When the counter is unrestricted or the machine is prowded with two reversal-bounded counters, the emptiness problem becomes undecidable. The decidability of the emptmess problem for Q is useful in proving the solvabdity of some number-theoreuc problems It can also be used to prove that the language L = {u11u121 ≥ 0} cannot be accepted by any machme in Q (u1 and u2 are symbols). The proof techmque is new m that it does not employ the usual “pumpmg,“ “counting,“ or “thagonal“ argument. Note that L can be accepted by a deterministic two-way machine with two counters, each of which makes exactly one reversal. © 1982, ACM. All rights reserved.