Computational power of quantum and probabilistic automata

Elektroniskā versija nesatur pielikumusAnot¹acija ņSis darbs apvieno p¹et¹³jumus par diviem autom¹atu veidiem: varb¹utiskajiem apgrieņzamajiem autom¹atiem (PRA), kas ir saist¹³ti ar kvantu gal¹³gajiem auto- m¹atiem (QFA), un vienvirziena kvantu autom¹atiem ar skait¹³t¹aju (Q1CA), kas ir l»oti ierobeņzots kvantu autom¹atu modelis, kam atbilstoņsa kvantu sist¹ema nav gal¹³ga. Darba m¹erk»is ir aprakst¹³t valodu klases, ko paz¹³st ņsie autom¹ati, un sal¹³dzin¹at kvantu un varb¹utiskos autom¹atus. M¹es pied¹av¹ajam varb¹utisk¹a apgrieņzama autom¹ata modeli. M¹es p¹et¹am vienvirziena PRA gan ar klasisko (C-PRA) v¹ardu akcept¹eņsanu, gan ar ap- st¹adin¹aņsanu (DH-PRA). M¹es par¹ad¹am valodu klases a¤1 a¤2 : : : a¤n paz¹³ņsanu ar PRA. M¹es par¹ad¹am vai valodu klase, ko paz¹³st PRA, ir sl¹egta pret B¹ula oper¹acij¹am. M¹es par¹ad¹am visp¹ar¹³gas valodu klases, ko C-PRA un DH-PRA nepaz¹³st. M¹es apskat¹am v¹ajas apgrieņzam¹³bas de¯n¹³ciju un par¹ad¹am atņsk»ir¹³bu no apgrieņzam¹³bas. M¹es pied¹av¹ajam vispar¹³gu kvantu vienvirziena autom¹ata modeli ar skai- t¹³t¹aju (Q1CA). M¹es pier¹ad¹am ka ņsis modelis apmierina transform¹acijas uni- tarit¹ates principu. Tiek pied¹av¹ats speci¹als Q1CA veids - vienk¹arņsais Q1CA, kas l»auj konstru¹et autom¹atu piem¹erus konkr¹et¹am valod¹am. M¹es par¹ad¹am vair¹aku kontekstatkar¹³go valodu paz¹³ņsanu ar Q1CA. M¹es pier¹ad¹am ka past¹av valodas, ko paz¹³st Q1CA, bet ko nepaz¹³st varb¹utiskais autom¹ats ar skait¹³t¹aju.The thesis assembles research on two models of automata - probabilistic reversible (PRA) that appear very similar to 1-way quantum ¯nite automata (1-QFA) and quantum one-way one counter automata (Q1CA), that is the most restricted model of non-¯nite space quantum automata. The objective of the research is to describe classes of languages recognizable by these models and compare related quantum and probabilistic automata. We propose the model of probabilistic reversible automata. We study both one-way PRA with classical (1-C-PRA) and decide and halt (1-DH- PRA) acceptance. We show recognition of general class of languages Ln = a¤1a¤2 : : : a¤n with probability 1 ¡ ". We show whether the classes of languages they recognize are closed under boolean operations and describe general class of languages not recognizable by these automata in terms of \forbidden con- structions" for the minimal deterministic automaton of the language. We also consider \weak" reversibility as equivalent de¯nition for 1-way automata and show the di®erence from ordinary reversibility in 1.5-way case. We propose the general notion of quantum one-way one counter au- tomata(Q1CA). We describe well-formedness conditions for the Q1CA that ensure unitarity of its evolution. A special kind of Q1CA, called simple, that satis¯es the well-formedness conditions is introduced. We show recognition of several non context free languages by Q1CA. We show that there is a lan- guage that can be recognized by quantum one-way one counter automaton, but not by the probabilistic one counter automaton

Quantum computation with devices whose contents are never read

In classical computation, a "write-only memory" (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented with WOM can solve problems that neither a classical computer with WOM nor a quantum computer without WOM can solve, when all other resource bounds are equal. We focus on realtime quantum finite automata, and examine the increase in their power effected by the addition of WOMs with different access modes and capacities. Some problems that are unsolvable by two-way probabilistic Turing machines using sublogarithmic amounts of read/write memory are shown to be solvable by these enhanced automata.Comment: 32 pages, a preliminary version of this work was presented in the 9th International Conference on Unconventional Computation (UC2010

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

New results on classical and quantum counter automata

We show that one-way quantum one-counter automaton with zero-error is more powerful than its probabilistic counterpart on promise problems. Then, we obtain a similar separation result between Las Vegas one-way probabilistic one-counter automaton and one-way deterministic one-counter automaton. We also obtain new results on classical counter automata regarding language recognition. It was conjectured that one-way probabilistic one blind-counter automata cannot recognize Kleene closure of equality language [A. Yakaryilmaz: Superiority of one-way and realtime quantum machines. RAIRO - Theor. Inf. and Applic. 46(4): 615-641 (2012)]. We show that this conjecture is false, and also show several separation results for blind/non-blind counter automata.Comment: 21 page

Applying causality principles to the axiomatization of probabilistic cellular automata

Cellular automata (CA) consist of an array of identical cells, each of which may take one of a finite number of possible states. The entire array evolves in discrete time steps by iterating a global evolution G. Further, this global evolution G is required to be shift-invariant (it acts the same everywhere) and causal (information cannot be transmitted faster than some fixed number of cells per time step). At least in the classical, reversible and quantum cases, these two top-down axiomatic conditions are sufficient to entail more bottom-up, operational descriptions of G. We investigate whether the same is true in the probabilistic case. Keywords: Characterization, noise, Markov process, stochastic Einstein locality, screening-off, common cause principle, non-signalling, Multi-party non-local box.Comment: 13 pages, 6 figures, LaTeX, v2: refs adde

Superiority of one-way and realtime quantum machines and new directions

In automata theory, the quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to the QFAs augmented with counters or stacks. Moreover, to our knowledge, there is no result related to QFAs having more than one input head. In this paper, we focus on such generalizations of QFAs whose input head(s) operate(s) in one-way or realtime mode and present many superiority of them to their classical counterparts. Furthermore, we propose some open problems and conjectures in order to investigate the power of quantumness better. We also give some new results on classical computation.Comment: A revised edition with some correction