Resolution and the binary encoding of combinatorial principles.

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn for m > n. Using the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn requires proofs of size 2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2 p n log n there are quasipolynomial size proofs of Bin-PHPmn in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form and with no finite model

Unificação das interpretações funcionais: via lógica linear intuicionista

Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2014A presente dissertação em Lógica Matemática, enquadrada no âmbito da Teoria da Demonstração, centra-se no conceito de “interpretação funcional”. Em termos informais, uma interpretação funcional consiste numa interpretação (de fórmulas de um sistema para fórmulas de outro sistema), que pode ser encarada como uma correspondência entre fórmulas e jogos, e num teorema que, intuitivamente, faz corresponder provas a estratégias vencedoras. A terminologia “interpretação funcional” parece ser relativamente recente, mas o conceito (ou, no mínimo, exemplos concretos do mesmo) data pelo menos da década de 50. Com efeito, foi em 1958 que Gödel propôs a famosa interpretação Dialéctica, e apenas um ano mais tarde surge a Realizabilidade Modificada de Kreisel. É sobre estas duas interpretações funcionais, juntamente com a variante da Dialéctica mais tarde proposta por Diller e Nahm, que nos debruçamos ao longo deste texto. O nosso principal objectivo é demonstrar que é possível apresentar as três interpretações funcionais do parágrafo anterior como “variantes” de uma única interpretação funcional, parametrizada. Para tal, introduziremos e motivaremos o contexto da Lógica Linear, criada por J.–Y. Girard e apresentada pela primeira vez em [17]. Estaremos em particular interessados na Lógica Linear Intuicionista, uma vez que é nesta lógica que será feita a unificação. A unificação será obtida apresentando uma interpretação linear “básica”, que representará a parte comum à realizabilidade modificada, à Dialéctica e à variante de Diller–Nahm, juntamente com três instanciações de um parâmetro, que corresponderá ao que as diferencia.We present a dissertation in Mathematical Logic, specifically within the framework of Proof Theory, centered on the concept of functional interpretation. Informally, a functional interpretation consists of an interpretation, which maps formulas of a system into formulas of another system and which can be thought of as a correspondence between formulas and games, and a theorem, which, intuitively, associates proofs with winning strategies. The terminology “functional interpretation” seems to be relatively recent, but the concept it represents (or, at least, specific examples of it) is known and studied since the fifties. It was indeed in 1958 that Gödel presented his famous Dialectica interpretation, and barely a year after that Kreisel introduced his Modified Realizability. These two functional interpretations, together with the Dialectica variant presented later by Diller and Nahm, will be the object of our study throughout the text. Our main goal will be to show that it is possible to present the three functional interpretations mentioned above as “variants” of one parametrized functional interpretation. In order to do so, we will motivate and introduce the context of Linear Logic, first introduced by J.–Y. Girard in [17]. We will be specifically interested in Intuitionistic Linear Logic, since the unification will be obtained within this context. The unification, as we will see, will be done by means of presenting a “basic” functional interpretation, representing the common part to modified realizability, Dialectica and Diller–Nahm’s variant, together with three instantiations of a parameter, corresponding to what differentiates these interpretations

From Quantum Metalanguage to the Logic of Qubits

The main aim of this thesis is to look for a logical deductive calculus (we will adopt sequent calculus, originally introduced in Gentzen, 1935), which could describe quantum information and its properties. More precisely, we intended to describe in logical terms the formation of the qubit (the unit of quantum information) which is a particular linear superposition of the two classical bits 0 and 1. To do so, we had to introduce the new connective "quantum superposition", in the logic of one qubit, Lq, as the classical conjunction cannot describe this quantum link.Comment: 138 pages, PhD thesis in Mathematic

Model checking quantum protocols

This thesis describes model checking techniques for protocols arising in quantum information theory and quantum cryptography. We discuss the theory and implementation of a practical model checker, QMC, for quantum protocols. In our framework, we assume that the quantum operations performed in a protocol are restricted to those within the stabilizer formalism; while this particular set of operations is not universal for quantum computation, it allows us to develop models of several useful protocols as well as of systems involving both classical and quantum information processing. We detail the syntax, semantics and type system of QMC’s modelling language, the logic QCTL which is used for verification, and the verification algorithms that have been implemented in the tool. We demonstrate our techniques with applications to a number of case studies.EThOS - Electronic Theses Online ServiceUniversity of Warwick. Dept. of Computer ScienceEngineering and Physical Sciences Research Council (Great Britain) (EPSRC) (GR/S34090/01, EP/E006833/2, GR/S86037/01)Sixth Framework Programme (European Commission) (SFP)Fundação para a Ciência ea Tecnologia (FCT) (POCI/MAT/55796/2004)Conselho de Reitores das Universidades Portuguesas (CRUP)GBUnited Kingdo

Saturation-based decision procedures for extensions of the guarded fragment

We apply the framework of Bachmair and Ganzinger for saturation-based theorem proving to derive a range of decision procedures for logical formalisms, starting with a simple terminological language EL, which allows for conjunction and existential restrictions only, and ending with extensions of the guarded fragment with equality, constants, functionality, number restrictions and compositional axioms of form S ◦ T ⊆ H. Our procedures are derived in a uniform way using standard saturation-based calculi enhanced with simplification rules based on the general notion of redundancy. We argue that such decision procedures can be applied for reasoning in expressive description logics, where they have certain advantages over traditionally used tableau procedures, such as optimal worst-case complexity and direct correctness proofs.Wir wenden das Framework von Bachmair und Ganzinger für saturierungsbasiertes Theorembeweisen an, um eine Reihe von Entscheidungsverfahren für logische Formalismen abzuleiten, angefangen von einer simplen terminologischen Sprache EL, die nur Konjunktionen und existentielle Restriktionen erlaubt, bis zu Erweiterungen des Guarded Fragment mit Gleichheit, Konstanten, Funktionalität, Zahlenrestriktionen und Kompositionsaxiomen der Form S ◦ T ⊆ H. Unsere Verfahren sind einheitlich abgeleitet unter Benutzung herkömmlicher saturierungsbasierter Kalküle, verbessert durch Simplifikationsregeln, die auf dem Konzept der Redundanz basieren. Wir argumentieren, daß solche Entscheidungsprozeduren für das Beweisen in ausdrucksvollen Beschreibungslogiken angewendet werden können, wo sie gewisse Vorteile gegenüber traditionell benutzten Tableauverfahren besitzen, wie z.B. optimale worst-case Komplexität und direkte Korrektheitsbeweise