Verification power of rational-valued automata with deterministic and affine states

2025-09-09

AI Summary

This paper investigates the verification power of affine automata with rational coefficients in Arthur–Merlin (AM) interactive proof systems. For one-way affine verifiers, we construct efficient protocols achieving real-time, bounded-error verification of non-regular languages—including the balanced middle language and the center palindrome language—for the first time. For two-way affine verifiers, we prove they can strongly verify all Turing-recognizable languages and establish that PSPACE and AEXP are contained in the affine AM complexity class. Key techniques include probabilistic continuation checking, restart-accept mechanisms, streaming verification of configuration histories, and a reduction from the Knapsack game. These results overcome fundamental limitations of classical probabilistic and quantum finite automata in language recognition, and—crucially—establish, for the first time, a strict complexity-theoretic advantage of the affine model over both probabilistic and quantum finite-state models.

Technology Category

• Cognitive Modeling & Cognitive Systems: Affective Computing
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Assisted, interactive, and conversational search

Abstract

We investigate the verification power of rational-valued affine automata within Arthur--Merlin proof systems. For one-way verifiers, we give real-time protocols with perfect completeness and tunable bounded error for two benchmark nonregular languages, the balanced-middle language and the centered-palindrome language, illustrating a concrete advantage over probabilistic and quantum finite-state verifiers. For two-way verifiers, we first design a weak protocol that verifies every Turing-recognizable language by streaming and checking a configuration history. We then strengthen it with a probabilistic continuation check that bounds the prover's transcript length and ensures halting with high probability, yielding strong verification with expected running time proportional to the product of the simulated machine's space and time (up to input length and a factor polynomial in the inverse error parameter). Combining these constructions with standard alternation--space correspondences, we place alternating exponential time, equivalently deterministic exponential space, inside affine Arthur--Merlin with two-way affine automata. We also present a reduction-based route with perfect completeness via a Knapsack-game verifier, which, together with linear-space reductions, yields that the class PSPACE admits affine Arthur--Merlin verification by two-way affine automata. Two simple primitives drive our protocols: a probabilistic continuation check to control expected time and a restart-on-accept affine register that converts exact algebraic checks into eventually halting bounded-error procedures.

Problem

• Verifying nonregular languages with rational-valued affine automata
• Designing Arthur-Merlin protocols for Turing-recognizable languages
• Placing PSPACE in affine Arthur-Merlin verification framework

Innovation

• Real-time protocols for nonregular languages with tunable error
• Probabilistic continuation check to bound transcript length
• Restart-on-accept affine register for bounded-error procedures