The No Endmarker Theorem for One-Way Probabilistic Pushdown Automata

TL;DR

This paper resolves a foundational question in one-way probabilistic pushdown automata by proving a No Endmarker Theorem: endmarkers on the input tape are not necessary for correct language recognition when the error probability is preserved. It provides a constructive, albeit complex, transformation from endmarker 1ppda to no-endmarker 1ppda with the same error bound, accompanied by explicit bounds on stack-state complexity that reveal a double-exponential blow-up. The authors introduce an ideal-shape normalization to better manage push and pop operations, and they establish a reversal-closure result for unbounded-error endmarker 1ppda using that form. The work also includes a systematic decomposition of the transformation into preparatory modifications and a rigorous sequence of remove-endmarker steps, culminating in a complete proof and open questions about potential tighter bounds and efficiency gains.

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.

Figures (4)

• Figure 1: (1) An endmarker 1ppda starts its computation by reading ${|}\!\!\mathrm{c}$. For each finite computation path, if the 1ppda does not halt before reading $\$$, then it reads $\$$, makes a (possible) final series of $\lambda$-moves, and enters a halting state. In this case, however, no blank cell outside of the scope between ${|}\!\!\mathrm{c}$ and $\$$ should be read. (2) A no-endmarker 1ppda starts reading the leftmost symbol of $x$ and continues reading all symbols of $x$. The final acceptance and rejection of the 1ppda is determined after a (possible) final series of $\lambda$-moves, however, no blank cell outside of the input $x$ should be read.
• Figure 2: Two completely opposite moving directions of the tape heads of $M$ and $N$.
• Figure 3: (1) An endmarker 1ppda starts with reading the left endmarker ${|}\!\!\mathrm{c}$ in the initial state $q_0$. The machine may make a series of consecutive $\lambda$-moves even after reading $\$$. (2) A no-left-endmarker 1ppda starts with reading the leftmost symbol of input $x$ in the initial state $q_0$. The machine may make a series of $\lambda$-moves after processing $\$$.
• Figure 4: (1) After reading $\$$, the underlying 1ppda makes a series of $\lambda$-moves to empty the stack. After the stack becomes empty, the 1ppda enters halting states. (2) The 1ppda enters halting states at reading $\$$, and thus the 1ppda makes no $\lambda$-move after reading $\$$.

Theorems & Definitions (9)

• Theorem 1.1
• Corollary 1.2
• Lemma 4.1
• Lemma 4.2
• Corollary 4.3
• Lemma 5.1
• Lemma 5.2
• Lemma 5.3
• Lemma 5.4