Characterising Computational Devices with Logical Systems

In this thesis we shall present and develop the concept of a theory machine. Theory machines describe computation via logical systems, providing an overarching formalism for characterising computational systems such as Turing machines, type-2 machines, quantum computers, infinite time Turing machines, and various physical computation devices. Notably we prove that the class of finite problems that are computable by a finite theory machine acting in first-order logic is equal to the class Turing machine computable problems. Whereas the class infinite problems that are computable by a finite first-order theory machine is equal to the class type-2 machine computable problems. A key property of a theory machine computation is that it does not have to occur in a causally ordered manner. A consequence of this fact is that the class of problems that are computable by finite first-order theory machine in polynomial resources is equal to $NP \cap co-NP$. Since there are problems which appear to lie in $NP \cap co-NP \setminus P$ that are efficiently solvable by a quantum computer (such as the factorisation problem), this gives weight to the argument that there is an atemporal/non-causal component to the apparent speed-up offered by quantum computers

Strengths and Weaknesses of Quantum Computing

Recently a great deal of attention has focused on quantum computation following a sequence of results suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of NP can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turing machine in time $o(2^{n/2})$. We also show that relative to a permutation oracle chosen uniformly at random, with probability 1, the class $NP \cap coNP$ cannot be solved on a quantum Turing machine in time $o(2^{n/3})$. The former bound is tight since recent work of Grover shows how to accept the class NP relative to any oracle on a quantum computer in time $O(2^{n/2})$.Comment: 18 pages, latex, no figures, to appear in SIAM Journal on Computing (special issue on quantum computing

Cell-like and Tissue-like Membrane Systems as Recognizer Devices

Most of the variants of membrane systems found in the literature are generally thought as generating devices. In this paper recognizer computational devices (cell–like and tissue–like) are presented in the framework of Membrane Computing, using the biological membranes arranged hierarchically, inspired from the structure of the cell, and using the biological membranes placed in the nodes of a graph, inspired from the cell inter–communication in tissues. In this context, polynomial complexity classes of recognizer membrane systems are introduced. The paper also addresses the P versus NP problem, and the (efficient) solvability of computationally hard problems, in the framework of these new complexity classes.Ministerio de Educación y Ciencia TIN2005-09345-C04-0

On security analysis of periodic systems: expressiveness and complexity

Development of automated technological systems has seen the increase in interconnectivity among its components. This includes Internet of Things (IoT) and Industry 4.0 (I4.0) and the underlying communication between sensors and controllers. This paper is a step toward a formal framework for specifying such systems and analyzing underlying properties including safety and security. We introduce automata systems (AS) motivated by I4.0 applications. We identify various subclasses of AS that reflect different types of requirements on I4.0. We investigate the complexity of the problem of functional correctness of these systems as well as their vulnerability to attacks. We model the presence of various levels of threats to the system by proposing a range of intruder models, based on the number of actions intruders can use

On the computational complexity of ethics: moral tractability for minds and machines

Why should moral philosophers, moral psychologists, and machine ethicists care about computational complexity? Debates on whether artificial intelligence (AI) can or should be used to solve problems in ethical domains have mainly been driven by what AI can or cannot do in terms of human capacities. In this paper, we tackle the problem from the other end by exploring what kind of moral machines are possible based on what computational systems can or cannot do. To do so, we analyze normative ethics through the lens of computational complexity. First, we introduce computational complexity for the uninitiated reader and discuss how the complexity of ethical problems can be framed within Marr’s three levels of analysis. We then study a range of ethical problems based on consequentialism, deontology, and virtue ethics, with the aim of elucidating the complexity associated with the problems themselves (e.g., due to combinatorics, uncertainty, strategic dynamics), the computational methods employed (e.g., probability, logic, learning), and the available resources (e.g., time, knowledge, learning). The results indicate that most problems the normative frameworks pose lead to tractability issues in every category analyzed. Our investigation also provides several insights about the computational nature of normative ethics, including the differences between rule- and outcome-based moral strategies, and the implementation-variance with regard to moral resources. We then discuss the consequences complexity results have for the prospect of moral machines in virtue of the trade-off between optimality and efficiency. Finally, we elucidate how computational complexity can be used to inform both philosophical and cognitive-psychological research on human morality by advancing the moral tractability thesis

A model-theoretical approach to classical Turing computability

Orientador: Walter Alexandre CarnielliTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências HumanasResumo: Esta tese propõe uma nova abordagem da computabilidade de Turing clássica, denominada abordagem modelo-teórica. De acordo com essa abordagem, estruturas e teorias são associadas às máquinas de Turing a fim de investigar as características de suas computações. Uma abordagem modelo-teórica da computabilidade de Turing através da lógica de primeira ordem é desenvolvida, e resultados de correspondência, correção, representação e completude entre máquinas, estruturas e teorias de Turing são demonstrados. Nessa direção, os resultados obtidos a respeito de propriedades tais como estabilidade, absoluticidade, universalidade e logicidade enfatizam as potencialidades da computabilidade modelo-teórica de primeira ordem. Demonstra-se que a lógica subjacente às teorias de Turing é uma lógica minimal intuicio-nista, sendo capaz, inclusive, de internalizar um operador de negação clássico. As técnicas formuladas nesta tese permitem, sobretudo, investigar a computabilidade de Turing em modelos não-padrão da aritmética. Nesse contexto, uma nova perspectiva acerca do fenômeno de Tennenbaum e uma avaliação crítica da abordagem de Dershowitz e Gurevich da tese de Church-Turing sào apresentadas. Como conseqüência, postula-se um princípio de interna-lidade aritmética na computabilidade, segundo o qual o próprio conceito de computação é relativo ao modelo aritmético em que as máquinas de Turing operam. Assim, a tese unifica as caracterizações modelo-aritméticas do problema P versus NP existentes na literatura, revelando, por fim, uma barreira modelo-aritmética para a possibilidade de solução desse problema central em complexidade computacional no que diz respeito a certos métodos. Em sua totalidade, a tese sustenta que características cruciais do conceito de computação podem ser vislumbradas a partir da dualidade entre finitude e infinitude presente na distinção entre números naturais padrão e não-padrãoAbstract: This PhD thesis proposes a new approach to classical Turing computability, called a model-theoretic approach. In that approach, structures and theories are associated to Turing machines in order to study the characteristics of their computations. A model-theoretic approach to Turing computability through first-order logic is developed, and first results about correspondence, soundness, representation and completeness among Turing machines, structures and theories are proved. In this line, the results about properties as stability, absoluteness, universality and logicality emphasize the importance of the model-theoretic standpoint. It is shown that the underlying logic of Turing theories is a minimal intuicionistic logic, being able to internalize a classical negation operator. The techniques obtained in the present dissertation permit us to examine the Turing computability over nonstandard models of arithmetic as well. In this context, a new perspective about Tennenbaum's phenomenon and a critical evaluation of Dershowitz and Gurevich's account on Church-Turing's thesis are given. As a consequence, an arithmetic internality principle is postulated, according to which the concept of computation itself is relative to the arithmetic model that Turing machines operate. In this way, the dissertation unifies the existing model-arithmetic characterizations of the P versus NP problem, leading, as a by-product, to a model-arithmetic barrier to the solvability of that central problem in computational complexity with respect to certain techniques. As a whole, the dissertation sustains that crucial characteristics of the concept of computation may be understood from the duality between finiteness and infiniteness inherent within the distinction between standard and nonstandard natural numbersDoutoradoFilosofiaDoutor em Filosofi

Temporal Query Answering in DL-Lite with Negation

Ontology-based query answering augments classical query answering in databases by adopting the open-world assumption and by including domain knowledge provided by an ontology. We investigate temporal query answering w.r.t. ontologies formulated in DL-Lite, a family of description logics that captures the conceptual features of relational databases and was tailored for efficient query answering. We consider a recently proposed temporal query language that combines conjunctive queries with the operators of propositional linear temporal logic (LTL). In particular, we consider negation in the ontology and query language, and study both data and combined complexity of query entailment