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Computer Science > Formal Languages and Automata Theory

arXiv:2212.13801 (cs)
[Submitted on 28 Dec 2022 (v1), last revised 17 Jul 2024 (this version, v2)]

Title:Classical and quantum Merlin-Arthur automata

Authors:Abuzer Yakaryılmaz
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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.
Comments: Major revision, extended content, 22 pages
Subjects: Formal Languages and Automata Theory (cs.FL); Computational Complexity (cs.CC); Quantum Physics (quant-ph)
Cite as: arXiv:2212.13801 [cs.FL]
  (or arXiv:2212.13801v2 [cs.FL] for this version)
  https://doi.org/10.48550/arXiv.2212.13801

Submission history

From: Abuzer Yakaryilmaz [view email]
[v1] Wed, 28 Dec 2022 12:46:18 UTC (24 KB)
[v2] Wed, 17 Jul 2024 22:14:08 UTC (22 KB)
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