Two-way affine automata can verify every language
Zeyu Chen, Abuzer Yakaryılmaz
TL;DR
The paper proves that two-way affine automata can verify membership in every language with bounded error inside Arthur–Merlin proof systems, achieving single-exponential expected time. It introduces two verifier protocols: a weak one employing a single affine register read in one pass, and a strong one using two registers with a probabilistic termination step, both leveraging a real-number encoding of language membership via $\alpha_L$. The key innovation lies in encoding membership digits of $\alpha_L$ and using affine-state updates to probabilistically distinguish members from non-members, yielding exponential-time verification for binary and $r$-ary languages. This establishes a powerful verification capability for AfAs and suggests further exploration of rational-transition variants and other interactive proof configurations.
Abstract
When used as verifiers in Arthur-Merlin systems, two-way quantum finite automata can verify membership in all languages with bounded error with double-exponential expected running time, which cannot be achieved by their classical counterparts. We obtain the same result for affine automata with single-exponential expected time. We show that every binary (and r-ary) language is verified by some two-way affine finite automata verifiers by presenting two protocols: A weak verification protocol uses a single affine register and the input is read once; and, a strong verification protocol uses two affine registers. These results reflects the remarkable verification capabilities of affine finite automata.