Quantum counter automata

The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.Comment: A revised version. 16 pages. A preliminary version of this paper appeared as A. C. Cem Say, Abuzer Yakary{\i}lmaz, and \c{S}efika Y\"{u}zsever. Quantum one-way one-counter automata. In R\={u}si\c{n}\v{s} Freivalds, editor, Randomized and quantum computation, pages 25--34, 2010 (Satellite workshop of MFCS and CSL 2010

Uncountable realtime probabilistic classes

We investigate the minimum cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On binary case, we follow the same result for double logarithmic space, which is tight. When replacing the worktape with some limited memories, we can follow uncountable results on unary languages for two counters.Comment: 12 pages. Accepted to DCFS201

Classical and quantum Merlin-Arthur automata

We introduce Merlin-Arthur (MA) automata as Merlin provides a single certificate 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, MA-QFAs) and postselecting MA-PFAs and MA-QFAs (resp., MA-PostPFA and MA-PostQFA). We obtain 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 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 recognize the same nonstochastic unary languages with bounded error. With arbitrarily long certificates, bounded-error MA-PostPFAs verify every unary decidable language. With sublinear length certificates, bounded-error MA-PostQFAs 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.Comment: 14 page

Finite state verifiers with constant randomness

We give a new characterization of $\mathsf{NL}$ as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio

State-efficient QFA Algorithm for Quantum Computers

The study of quantum finite automata (QFA's) is one of the possible approaches in exploring quantum computers with finite memory. Despite being one of the most restricted models, Moore-Crutchfield quantum finite automaton (MCQFA) is proven to be exponentially more succinct than classical finite automata models in recognizing certain languages such as $\mathtt{MOD}_{\rm p} = \{a^{j}~ |~ j \equiv 0 \mod p\}$, where $p$ is a prime number. In this paper, we present a modified MCQFA algorithm for the language $\mathtt{MOD}_{\rm p}$, the operators of which are selected based on the basis gates on the available real quantum computers. As a consequence, we obtain shorter quantum programs using less basis gates compared to the implementation of the original algorithm given in the literature