On the universe, disjointness, and containment problems for simple machines

consider counter machines, which are nondeterministic one-way finite automata augmented with one counter, where at each step the counter can be incremented by 1, decremented by 1, or left unchanged, and can be tested for zero. Acceptance is by accepting state. We show that there is a fixed nondeterministic real-time (i.e., no ϵ-move) 1-reversal counter machine A (once the counter decrements it can no longer increment) such that it is undecidable to determine, given an arbitrary positive integer d, whether Ad (which is A with initial counter value d) accepts all strings.then strengthen the above result by showing that it also holds when A’s counter is partially blind in that it cannot be tested for zero (the machine aborts when there is an attempt to decrement the counter when it is zero), and the only condition for acceptance is for the counter to be zero when the input head falls off the right end of the input. We assume that on every input x for which d⩾|x|, the counter is initially set to |x|. (We make this assumption since, otherwise, Ad cannot be both real-time and satisfy the zero-counter requirement for acceptance.) We also prove a similar result for partially blind 2-head nondeterministic finite automata.ally, we sharpen the well-known results that the disjointness and containment problems for deterministic pushdown automata are undecidable.