[Paper Review] One-counter verifiers for decidable languages

Abuzer Yakaryılmaz|arXiv (Cornell University)|Jul 17, 2012

semigroups and automata theory12 references3 citations

TL;DR

This paper introduces one-counter verifiers—two-way probabilistic and quantum one-counter automata—as efficient interactive proof system (IPS) verifiers for decidable languages, demonstrating that bounded-error quantum counter automata can recognize languages beyond probabilistic counterparts. It proves that quantum counters enable public-coin (Arthur-Merlin) protocols, and presents a new programming technique enabling 2qcca to recognize a language not decidable by any bounded-error 2pca.

ABSTRACT

Condon and Lipton (FOCS 1989) showed that the class of languages having a space-bounded interactive proof system (IPS) is a proper subset of decidable languages, where the verifier is a probabilistic Turing machine. In this paper, we show that if we use architecturally restricted verifiers instead of restricting the working memory, i.e. replacing the working tape(s) with a single counter, we can define some IPS's for each decidable language. Such verifiers are called two-way probabilistic one-counter automata (2pca's). Then, we show that by adding a fixed-size quantum memory to a 2pca, called a two-way one-counter automaton with quantum and classical states (2qcca), the protocol can be space efficient. As a further result, if the 2qcca can use a quantum counter instead of a classical one, then the protocol can even be public, also known as Arthur-Merlin games. We also investigate the computational power of 2pca's and 2qcca's as language recognizers. We show that bounded-error 2pca's can be more powerful than their deterministic counterparts by giving a bounded-error simulation of their nondeterministic counterparts. Then, we present a new programming technique for bounded-error 2qcca's and show that they can recognize a language which seems not to be recognized by any bounded-error 2pca. We also obtain some interesting results for bounded-error 1-pebble quantum finite automata based on this new technique. Lastly, we prove a conjecture posed by Ravikumar (FSTTCS 1992) regarding 1-pebble probabilistic finite automata, i.e. they can recognize some nonstochastic languages with bounded error.

Motivation & Objective

• To construct interactive proof systems (IPS) for all decidable languages using architecturally restricted verifiers instead of space-bounded Turing machines.
• To investigate the computational power of two-way probabilistic one-counter automata (2pca) and two-way one-counter automata with quantum and classical states (2qcca) as language recognizers.
• To prove Ravikumar's conjecture that 1-pebble probabilistic finite automata can recognize nonstochastic languages with bounded error.
• To develop a new programming technique for bounded-error 2qcca that enables recognition of languages not decidable by 2pca.

Proposed method

• Uses two-way probabilistic one-counter automata (2pca) as verifiers in interactive proof systems for decidable languages, with the verifier using a single counter instead of a work tape.
• Introduces two-way one-counter automata with quantum and classical states (2qcca), incorporating a fixed-size quantum register to enhance verification efficiency.
• Employs quantum counters (2qca) to transform private-coin protocols into public-coin (Arthur-Merlin) protocols, enabling full verifier information disclosure to the prover.
• Applies superoperators and amplitude encoding techniques to encode input substrings into quantum state amplitudes, enabling comparison via unitary evolution and measurement.
• Uses a novel programming technique involving superoperators with multiple Kraus operators to encode and compare strings $u_1$ and $u_2$ in the TWIN language.
• Applies bounded-error simulation techniques to show that 2pca can simulate nondeterministic one-counter automata with bounded error, increasing their expressive power.

Experimental results

Research questions

• RQ1 Can one-counter verifiers (2pca) be used to construct interactive proof systems for all decidable languages?
• RQ2 Can adding a fixed-size quantum register to a 2pca (forming 2qcca) enable space-efficient verification of decidable languages?
• RQ3 Can replacing the classical counter with a quantum counter (2qca) transform a private-coin protocol into a public-coin (Arthur-Merlin) protocol?
• RQ4 Can bounded-error 2qcca recognize languages not decidable by bounded-error 2pca?
• RQ5 Do 1-pebble probabilistic finite automata recognize nonstochastic languages with bounded error, as conjectured by Ravikumar?

Key findings

• Bounded-error 2pca can simulate two-way nondeterministic one-counter automata, proving that 2pca is strictly more powerful than 2dca.
• The 2qcca-based protocol recognizes the TWIN language with bounded error: accepted with probability 1 if $x \in \mathtt{TWIN}$, rejected with probability at least $4/5$ if $x \notin \mathtt{TWIN}$.
• The new programming technique for 2qcca enables recognition of a language that appears not decidable by any bounded-error 2pca, demonstrating increased computational power.
• The protocol using a quantum counter (2qca) becomes public-coin (Arthur-Merlin), meaning the prover has full knowledge of the verifier’s state, a key result in interactive proof theory.
• The paper proves Ravikumar’s conjecture: 1-pebble probabilistic finite automata can recognize nonstochastic languages with bounded error.
• The results represent the first nontrivial findings on two-way quantum one-counter automata (2qca), which were previously unexplored in this context.