The No Endmarker Theorem for One-Way Probabilistic Pushdown Automata

TL;DR

This paper proves that for one-way probabilistic pushdown automata, endmarkers are unnecessary for language recognition, as they can be removed with a double exponential increase in stack-state complexity without affecting error probability.

Contribution

It establishes that endmarkers are always removable in probabilistic pushdown automata, extending known results from deterministic and nondeterministic models.

Findings

Endmarkers can be eliminated with double exponential complexity increase.

The removal preserves the automaton's error probability.

Provides an alternative proof for deterministic and nondeterministic cases.

Abstract

In various models of one-way pushdown automata, the explicit use of two designated endmarkers on a read-once input tape has proven to be extremely useful for making a conscious, final decision on the acceptance/rejection of each input word immediately after reading the right endmarker. With no endmarkers, by contrast, a machine must constantly stay in either accepting or rejecting states at any moment since it never notices the end of the input word. This situation, however, helps us analyze the behavior of the machine whose tape head makes the consecutive moves on all prefixes of a given extremely long input word. Since those two machine formulations have their own advantages, it is natural to ask whether the endmarkers are truly necessary to correctly recognize languages. In the deterministic and nondeterministic models, it is well-known that the endmarkers are removable without changing the acceptance criteria of each input word. This paper proves that, for a more general model of one-way probabilistic pushdown automata, the endmarkers are always removable. This is proven by employing probabilistic transformations from an "endmarker" machine to an equivalent "no-endmarker" machine at the cost of double exponential stack-state complexity without compromising its error probability. By setting this error probability appropriately, our proof also provides an alternative proof to both the deterministic and the nondeterministic models as well.