Verification power of rational-valued automata with deterministic and affine states
Zeyu Chen, Junde Wu
TL;DR
The paper investigates how rational-valued affine automata (ADfA) and two-way affine automata (2ADfA) can serve as verifiers in Arthur–Merlin interactive proofs, establishing strong verification capabilities under fully rational arithmetic. It provides a spectrum of results: real-time, perfect-completeness one-way verifiers for two nonregular languages; a weak AM protocol for every Turing-recognizable language using streaming configuration histories; a strengthened strong-verification protocol via a probabilistic continuation check that yields bounded-error verification for languages decidable in deterministic exponential space; and a reduction-based route via the Knapsack-game that achieves perfect completeness and PSPACE verification. Collectively, these findings show that two-way affine verifiers dramatically enlarge the verification landscape beyond classical PFAs/QFAs, while maintaining fully rational operations. The work further connects affine verification with alternation–space correspondences to place broad classes such as ATIME$(2^{O(n)})$ and PSPACE within AM_q frameworks, offering practical primitives like continuation checks and restart-on-accept to manage transcript length and halting. The results have implications for designing robust, low-memory verification protocols in theoretical computer science and demonstrate the versatility of affine computation in interactive settings.
Abstract
Previous research has shown that two-way automata with deterministic and affine states have strong verification capabilities, and that this power persists when all transition matrices are restricted to rational values. We investigate rational-valued affine automata as verifiers in Arthur--Merlin proof systems. For one-way verifiers, we give protocols with perfect completeness for two nonregular languages. For two-way verifiers, we first describe a weak protocol that verifies every Turing-recognizable language. We then strengthen this construction with a probabilistic continuation check to obtain strong verification with bounded error, establishing that every language decidable in deterministic exponential space is verifiable in Arthur--Merlin systems by rational-valued two-way affine automata. In a complementary, reduction-based route, we present a Knapsack-game verifier with perfect completeness, which implies that every language in PSPACE admits Arthur--Merlin verification by two-way affine automata with rational transitions. Taken together, these results illuminate the verification power of two-way affine automata while keeping arithmetic fully rational.