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

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

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

Computação quântica : autômatos, jogos e complexidade

Orientador: Arnaldo Vieira MouraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Desde seu surgimento, Teoria da Computação tem lidado com modelos computacionais de maneira matemática e abstrata. A noção de computação eficiente foi investigada usando esses modelos sem procurar entender as capacidades e limitações inerentes ao mundo físico. A Computação Quântica representa uma ruptura com esse paradigma. Enraizada nos postulados da Mecânica Quântica, ela é capaz de atribuir um sentido físico preciso à computação segundo nosso melhor entendimento da natureza. Esses postulados dão origem a propriedades fundamentalmente diferentes, uma em especial, chamada emaranhamento, é de importância central para computação e processamento de informação. O emaranhamento captura uma noção de correlação que é única a modelos quânticos. Essas correlações quânticas podem ser mais fortes do que qualquer correlação clássica estando dessa forma no coração de algumas capacidades quânticas que vão além do clássico. Nessa dissertação, nós investigamos o emaranhamento da perspectiva da complexidade computacional quântica. Mais precisamente, nós estudamos uma classe bem conhecida, definida em termos de verificação de provas, em que um verificador tem acesso à múltiplas provas não emaranhadas (QMA(k)). Assumir que as provas não contêm correlações quânticas parece ser uma hipótese não trivial, potencialmente fazendo com que essa classe seja maior do que aquela em que há apenas uma prova. Contudo, encontrar cotas de complexidade justas para QMA(k) permanece uma questão central sem resposta por mais de uma década. Nesse contexto, nossa contribuição é tripla. Primeiramente, estudamos classes relacionadas mostrando como alguns recursos computacionais podem afetar seu poder de forma a melhorar a compreensão a respeito da própria classe QMA(k). Em seguida, estabelecemos uma relação entre Probabilistically Checkable Proofs (PCP) clássicos e QMA(k). Isso nos permite recuperar resultados conhecidos de maneira unificada e simplificada. Para finalizar essa parte, mostramos que alguns caminhos para responder essa questão em aberto estão obstruídos por dificuldades computacionais. Em um segundo momento, voltamos nossa atenção para modelos restritos de computação quântica, mais especificamente, autômatos quânticos finitos. Um modelo conhecido como Two-way Quantum Classical Finite Automaton (2QCFA) é o objeto principal de nossa pesquisa. Seu estudo tem o intuito de revelar o poder computacional provido por memória quântica de dimensão finita. Nos estendemos esse autômato com a capacidade de colocar um número finito de marcadores na fita de entrada. Para qualquer número de marcadores, mostramos que essa extensão é mais poderosa do que seus análogos clássicos determinístico e probabilístico. Além de trazer avanços em duas linhas complementares de pesquisa, essa dissertação provê uma vasta exposição a ambos os campos: complexidade computacional e autômatosAbstract: Since its inception, Theoretical Computer Science has dealt with models of computation primarily in a very abstract and mathematical way. The notion of efficient computation was investigated using these models mainly without seeking to understand the inherent capabilities and limitations of the actual physical world. In this regard, Quantum Computing represents a rupture with respect to this paradigm. Rooted on the postulates of Quantum Mechanics, it is able to attribute a precise physical notion to computation as far as our understanding of nature goes. These postulates give rise to fundamentally different properties one of which, namely entanglement, is of central importance to computation and information processing tasks. Entanglement captures a notion of correlation unique to quantum models. This quantum correlation can be stronger than any classical one, thus being at the heart of some quantum super-classical capabilities. In this thesis, we investigate entanglement from the perspective of quantum computational complexity. More precisely, we study a well known complexity class, defined in terms of proof verification, in which a verifier has access to multiple unentangled quantum proofs (QMA(k)). Assuming the proofs do not exhibit quantum correlations seems to be a non-trivial hypothesis, potentially making this class larger than the one in which only a single proof is given. Notwithstanding, finding tight complexity bounds for QMA(k) has been a central open question in quantum complexity for over a decade. In this context, our contributions are threefold. Firstly, we study closely related classes showing how computational resources may affect its power in order to shed some light on \QMA(k) itself. Secondly, we establish a relationship between classical Probabilistically Checkable Proofs and QMA(k) allowing us to recover known results in unified and simplified way, besides exposing the interplay between them. Thirdly, we show that some paths to settle this open question are obstructed by computational hardness. In a second moment, we turn our attention to restricted models of quantum computation, more specifically, quantum finite automata. A model known as Two-way Quantum Classical Finite Automaton (2QCFA) is the main object of our inquiry. Its study is intended to reveal the computational power provided by finite dimensional quantum memory. We extend this automaton with the capability of placing a finite number of markers in the input tape. For any number of markers, we show that this extension is more powerful than its classical deterministic and probabilistic analogues. Besides bringing advances to these two complementary lines of inquiry, this thesis also provides a vast exposition to both subjects: computational complexity and automata theoryMestradoCiência da ComputaçãoMestre em Ciência da Computaçã

Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

Homo Aleator: a sociological study of gambling in western society

The subject of this thesis is the nature and form of gambling in western society. Unlike other academic studies, which approach the subject in piecemeal fashion and treat it as essentially problematic, this research aims to provide a comprehensive analysis of the activity by situating the phenomenological experience of modern gambling within a formal and historical framework, and examining it from a variety of theoretical perspectives. This thesis is divided into five Chapters. The first describes the historical development of the concept of chance and argues that, after a lengthy period in which it existed first as a sacred and later as an epistemological category, in the twentieth century chance was secularised and ascribed ontological status as an explanatory feature of the modern world. In Chapter Two, a study of the historical development of gambling forms, the social affiliation of the various groups associated with them complements the outline of chance given in Chapter One. Here it is argued that the nature of these games reflected wider social attitudes towards the perception of randomness, and also the general configuration of the society in which they were played. The development of games from their genesis in divination ritual to their formation into a recognisably modern gambling economy, is also traced, and the historical specificity of their various forms examined. The formal parameters and phenomenological experience of the gambling economy - the 'phenomenological sites' - is the subject of Chapter Three. It is argued that commercial organisation of games of chance in the late twentieth century is beginning to overcome the traditional stratification that for years has determined the social formation of gambling forms