Containment Problem for Deterministic Multicounter Machine Models

There are many types of automata and grammar models that have been studied in the literature, and for these models, it is common to determine whether certain problems are decidable. One problem that has been difficult to answer throughout the history of automata and formal language theory is to decide whether a given system $M$ accepts a bounded language (whether there exist words $w_1, \ldots,w_k$ such that $L(M) \subseteq w_1^* \cdots w_k^*$?). Boundedness was only known to be decidable for regular and context-free languages until recently when it was shown to also be decidable for finite automata and pushdown automata augmented with reversal-bounded counters, and for vector addition systems with states. However, decidability of this problem has still gone unanswered for the majority of automata/grammar models with a decidable emptiness problem that have been studied in the literature. In this paper, we develop new techniques to show that the boundedness problem is decidable for larger classes of one-way nondeterministic automata and grammar models by reducing the problem to the decidability of boundedness for simpler classes of automata. One technique involves characterizing the models in terms of multi-tape automata. We give new characterizations of finite-turn Turing machines, finite-turn Turing machines augmented with various storage structures (like a pushdown, multiple reversal-bounded counters, partially-blind counters, etc.), and simple matrix grammars. The characterizations are then used to show that the boundedness problem for these models is decidable. Another technique uses the concept of the store language of an automaton. This is used to show that the boundedness problem is decidable for pushdown automata that can "flip" their pushdown a bounded number of times. Boundedness remains decidable even if we augment this device with additional stores.