The class of languages recognized by two-way probabilistic finite automata with one-counter (2P1CAs) with bounded error is known to be contained within the complexity class BPL (bounded-error probabilistic log-space).

BPL consists of languages decidable by probabilistic Turing machines using O(log n) space with bounded error probability (less than 1/2). Two-way probabilistic finite automata ...

Short answer: They are contained in BPL, and hence in DSPACE(log^{3/2} n) and in Σ2^L ∩ Π2^L (in particular, in P). No substantially tighter, model-specific upper bound is known.

Reasoning:

• If a two-way 1-counter automaton halts on a given input, then every computation path halts in O(n^2) steps (the Duris–Galil bound). A probabilistic machine’s computation paths are just deterministic runs wi...

Let M be a two–way probabilistic one–counter automaton (2P1CA) that works with bounded error, say it accepts every word w that is in the language with probability ≥ 2/3 and rejects every other word with probability ≥ 2/3.
Fix an input word w of length n.
During the run of M on w a configuration consists of

• the finite control state of M (|Q| possibilities)
• the current head position i (0 ...

Upper Bounds for Two-Way Probabilistic Finite Automata with One-Counter (2P1CAs)

The question is asking about the best known upper bound for languages recognized by two-way probabilistic finite automata with one counter (2P1CAs) with bounded error.

From what I understand of the research literature, the best known upper bound for 2P1CAs with bounded error is likely BPL (Bounded-error Probab...

Excellent question. It touches on a classic but non-trivial result in automata theory and computational complexity.

Summary

The best known upper bound for the class of languages recognized by two-way probabilistic finite automata with one-counter (2P1CA) with bounded error is P (polynomial time).

This result is due to Ioan I. Macarie and Mitsunori Ogihara in their 1998 paper:

Macar...

The question asks about the computational power of two-way probabilistic finite automata with one counter (2P1CAs) with bounded error, specifically what upper bounds are known for this model.

To answer this comprehensively:

Known Upper Bounds

The best known upper bound for 2P1CAs with bounded error is BPL (bounded-error probabilistic logspace). This can be shown through the following rea...

Below is what is known about the upper‐bound of the 2‐way probabilistic one–counter machines (2P1CAs) with bounded error. In all cases the proofs are just “cook‐ups’’ of a finite (but large) absorbing Markov chain over the set of configurations, and then solving for the absorption probabilities.

1. Configuration‐graph size.
   – A single configuration is determined by
   • the contro...

<think> We are given a question about the best known upper bound for two-way probabilistic finite automata with one-counter (2P1CAs) with bounded error.

First, recall that a two-way probabilistic finite automaton with one-counter (2P1CA) is a two-way automaton that has a finite state control, a read-only input tape with endmarkers, and a counter that can store an integer and be incremented, decr...

Let

2P1CA = the class of languages that can be recognised with two–sided bounded error
( say ≤ 1/3 ) by two–way probabilistic finite automata equipped with one counter and that halt with probability 1 on every input.

The best upper bound that is currently known for 2P1CA is the bounded-error probabilistic log–space class BPL (and therefore also P).

Why 2P1CA ⊆ ...