Classical and quantum Merlin-Arthur automata
Abuzer Yakaryılmaz
TL;DR
This work studies Merlin-Arthur automata in which a certificate is supplied by a prover before the input is read, and analyzes how certificate length and postselection affect verification power across deterministic, probabilistic, and quantum finite-state verifiers. It defines MA-DFAs, MA-PFAs, MA-QFAs and their postselecting variants, and systematically maps certificate-size regimes—from sublinear to exponential to arbitrarily long—to the classes of languages they can verify on unary and binary inputs. A key finding is that MA-DFAs with constant certificates recognize exactly the regular languages (via equivalence with multi-entry DFAs), while longer certificates empower MA-PFAs and MA-QFAs to verify nonregular and even all decidable/universal binary languages under appropriate conditions; postselection further amplifies these capabilities, especially for quantum verifiers. Overall, the paper delineates a spectrum of certificate-size versus power in automata augmented with pre-input certificates, connecting classical automata, interactive proofs, and quantum verification concepts with concrete results for unary and binary languages.
Abstract
We introduce Merlin-Arthur (MA) automata where Merlin provides a certificate at the beginning of computation and it is scanned by Arthur before reading the input. We define Merlin-Arthur deterministic, probabilistic, and quantum finite state automata (resp., MA-DFAs, MA-PFAs, and MA-QFAs) and postselecting MA-PFAs and MA-QFAs (resp., MA-PostPFA and MA-PostQFA). We present several results using different certificate lengths. We show that MA-DFAs use constant length certificates, and they are equivalent to multi-entry DFAs. Thus, they recognize all and only regular languages, but they can be exponential and polynomial state efficient over binary and unary languages, respectively. With sublinear length certificates, MA-PFAs can recognize several nonstochastic unary languages with cutpoint 1/2. With linear length certificates, MA-PostPFAs can recognize these nonstochastic unary languages with bounded error. With arbitrarily long certificates, bounded-error MA-PostPFAs can verify every unary decidable language. With sublinear length certificates, bounded-error MA-PostQFAs can verify several nonstochastic unary languages. With linear length certificates, they can verify every unary language and some NP-complete binary languages. With exponential length certificates, they can verify every binary language.