Affine Automata Achieve Real-Time Verification of Non-Regular Languages with Tunable Bounded Error

Rational-valued automata, complex computational models extending traditional finite state machines, now demonstrate a surprising capacity for verifying mathematical proofs, according to new research. Zeyu Chen from Zhejiang University, Abuzer Yakaryılmaz from the University of Latvia, and Junde Wu, among others, reveal that these automata can verify languages, patterns within sequences, that challenge simpler verification methods. The team’s work establishes a concrete advantage over existing probabilistic and finite-state verifiers for specific, non-regular languages, and importantly, places complex computational classes, such as exponential time, within the realm of affine Arthur, Merlin verification. This advancement signifies a powerful step towards more robust and efficient methods for automated theorem proving and program verification, potentially impacting fields from computer science to mathematics.

The research presents real-time protocols with perfect completeness and tunable bounded error for one-way verifiers, demonstrated using the balanced-middle language and the centered-palindrome language. This illustrates a concrete advantage over existing probabilistic and quantum finite-state verifiers, showcasing improved performance on complex linguistic structures. For two-way verifiers, the team initially designs a protocol capable of verifying every Turing-recognizable language by streaming and checking a configuration history, establishing a foundational verification method. This protocol is then strengthened with a probabilistic continuation check, which bounds the prover’s transcript length and ensures halting with high probability, ultimately yielding strong verification capabilities.

Two-Way Affine Automata Verify Every Language

This research paper explores the power of affine automata as verifiers in interactive proof systems. The core result demonstrates that every language can be verified by a two-way affine automaton, establishing a strong connection between affine computation and verification power. The authors prove that two-way affine automata are as powerful as any other verifier, meaning they can verify any language. The paper presents two distinct approaches: a reduction-based approach, which reduces a language to a Knapsack Game instance and uses a two-way affine automaton for the game to verify the language with bounded error, and a direct construction, which demonstrates that every language can be verified directly by a two-way affine automaton without relying on reductions.

The research establishes that languages decidable in space 2 O(n) can be verified by a two-way affine automaton in an interactive proof system, and the same holds for quantum space. The team employed a two-state restart-on-accept register to convert exact algebraic checks into bounded-error, halting protocols, and a probabilistic continuation check to control the expected running time of the verification process. This work significantly expands our understanding of the computational power of affine automata, demonstrating their ability to serve as universal verifiers, connects affine computation to interactive proof systems, and could have practical applications in areas such as verification of complex systems and development of secure protocols. In essence, this research demonstrates that affine automata are surprisingly powerful computational devices capable of verifying any language, solidifying their place as a fundamental model of computation and interactive proof systems.

Affine Automata Enhance Computational Verification Power

Scientists have demonstrated a significant advancement in computational verification using affine automata within the framework of Arthur-Merlin proof systems. Their research establishes a powerful connection between these automata and the ability to verify complex computational problems, surpassing the capabilities of previously established models like probabilistic and finite-state verifiers. The team achieved protocols for the balanced-middle language and the centered-palindrome language, demonstrating a clear advantage in verification power. For two-way verification, researchers initially designed a protocol capable of verifying any Turing-recognizable language.

To address the initial limitation of unbounded message length, they then developed a sophisticated probabilistic continuation check, which effectively bounds the prover’s communication and ensures termination with high probability. This breakthrough yields strong verification capabilities, with expected running time proportional to the product of the machine’s space and time, scaled by the input length and a factor related to the desired error margin. Furthermore, the team developed an alternative verification route based on a reduction to the Knapsack-Game language, restoring perfect completeness while maintaining strong verification guarantees and rational-valued transitions. This complementary approach provides flexibility and is particularly useful when perfect completeness is a priority. Experiments confirm that rational-valued two-way affine automata can strongly verify any language accepted by an alternating Turing machine running in exponential time relative to the input length, showcasing a substantial improvement in efficiency compared to existing models like two-way quantum finite automata. These findings have significant implications for the development of more robust and efficient verification systems, with potential applications in areas such as secure computation and formal verification of software and hardware.

Affine Automata Verify Languages Efficiently

This research investigates the capabilities of rational-valued affine automata within Arthur, Merlin proof systems, a framework for verifying computational problems. The study demonstrates that these automata can verify certain non-regular languages, such as balanced-middle and centered-palindrome languages, more efficiently than traditional probabilistic or quantum finite-state machines under comparable constraints. This represents a concrete advantage for verifying these specific types of problems. Furthermore, the researchers developed protocols for two-way affine automata that can verify any Turing-recognizable language, bounding the prover’s transcript length and ensuring halting with high probability.

This work establishes a connection between alternating exponential time and deterministic exponential space, and demonstrates that the complexity class PSPACE can be verified using these automata with bounded error. The protocols rely on two key techniques: a probabilistic continuation check to control running time and a restart-on-accept register to convert exact algebraic checks into bounded-error procedures. While the current work focuses on specific automata and proof systems, future research could explore extending these techniques to other interactive settings and refining the trade-offs between state and register usage.