Inductive definitions are the most natural means by which to represent many families of structures occurring in mathematics and computer science, and their corresponding induction / recursion principles provide the fundamental proof techniques by which to reason about such families. This thesis studies formal proof systems for inductive definitions, as needed, e.g., for inductive proof support in automated theorem proving tools. The systems are formulated as sequent calculi for classical first-order logic extended with a framework for (mutual) inductive definitions. The default approach to reasoning with inductive definitions is to formulate the induction principles of the inductively defined relations as suitable inference rules or axioms, which are incorporated into the reasoning framework of choice. Our first system LKID adopts this direct approach to inductive proof, with the induction rules formulated as rules for introducing atomic formulas involving inductively defined predicates on the left of sequents. We show this system to be sound and cut-free complete with respect to a natural class of Henkin models. As a corollary, we obtain cut-admissibility for LKID. The well-known method of infinite descent `a la Fermat, which exploits the fact that there are no infinite descending chains of elements of well-ordered sets, provides an alternative approach to reasoning with inductively defined relations. Our second proof system LKIDw formalises this approach. In this system, the left-introduction rules for formulas involving inductively defined predicates are not induction rules but simple case distinction rules, and an infinitary, global soundness condition on proof trees — formulated in terms of “traces” on infinite paths in the tree — is required to ensure soundness. This condition essentially ensures that, for every infinite branch in the proof, there is an inductive definition that is unfolded infinitely often along the branch. By an infinite descent argument based upon the well-foundedness of inductive definitions, the infinite branches of the proof can thus be disregarded, whence the remaining portion of proof is well-founded and hence sound. We show this system to be cutfree complete with respect to standard models, and again infer the admissibility of cut. The infinitary system LKIDw is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic” proof system, CLKIDw, is suitable for formal reasoning since proofs have finite representations and the soundness condition on proofs is thus decidable. We show how the formulation of our systems LKIDw and CLKIDw can be generalised to obtain soundness conditions for a general class of infinite proof systems and their corresponding cyclic restrictions. We provide machinery for manipulating and analysing the structure of proofs in these essentially arbitrary cyclic systems, based primarily on viewing them as generating regular infinite trees, and we show that any proof can be converted into an equivalent proof with a restricted cycle structure. For proofs in this “cycle normal form”, a finitary, localised soundness condition exists that is strictly stronger than the general, infinitary soundness condition, but provides more explicit information about the proof. Finally, returning to the specific setting of our systems for inductive definitions, we show that any LKID proof can be transformed into a CLKIDw proof (that, in fact, satisfies the finitary soundness condition). We conjecture that the two systems are in fact equivalent, i.e. that proof by induction is equivalent to regular proof by infinite descent.

This thesis studies three topics in quantum computation and information: The approximability of quantum problems, quantum proof systems, and non-classical correlations in quantum systems. Our first area of study concerns the approximability of computational problems which are complete for quantum complexity classes. In the classical setting, the study of approximation algorithms and hardness of approximation is one of the main research areas of theoretical computer science. Yet, little is known regarding approximability in the quantum setting. We first demonstrate a polynomial-time approximation algorithm for dense instances of the canonical QMA-complete quantum constraint satisfaction problem, the local Hamiltonian problem. We next go in the opposite direction by first introducing a quantum generalization of the polynomial-time hierarchy. We then introduce problems which are not only complete for the second level of this hierarchy, but are in fact hard to approximate. Our second area of study concerns quantum proof systems. Here, an interesting question which remains open despite much effort is whether a proof system with multiple unentangled quantum provers is equal in expressive power to a proof system with a single quantum prover (i.e. is QMA(poly) equal to QMA?). Our results here study variants of this question, and include a proof that the class BellQMA(poly) collapses to QMA. We also give an alternate proof that SepQMA(m) admits perfect parallel repetition. This proof is novel in that it utilizes cone programming duality. Our final area of study concerns non-classical correlations in quantum systems. Specifically, there exist genuinely quantum correlations beyond entanglement in mixed quantum states which may prove useful from a computing and information theoretic perspective. We first explore the presence of such correlations in the locking of classical correlations and the DQC1 model of mixed-state quantum computing. Our second result introduces a novel scheme for quantifying non-classical correlations using local unitary operations. Our third result introduces a protocol through which non-classical correlations in a starting system can be “activated”' into distillable entanglement with an ancilla system. Our last result determines when the entanglement generated in the activation protocol above can be mapped back onto the starting state via entanglement swapping.

This thesis studies three topics in quantum computation and information: The approximability of quantum problems, quantum proof systems, and non-classical correlations in quantum systems. Our first area of study concerns the approximability of computational problems which are complete for quantum complexity classes. In the classical setting, the study of approximation algorithms and hardness of approximation is one of the main research areas of theoretical computer science. Yet, little is known regarding approximability in the quantum setting. We first demonstrate a polynomial-time approximation algorithm for dense instances of the canonical QMA-complete quantum constraint satisfaction problem, the local Hamiltonian problem. We next go in the opposite direction by first introducing a quantum generalization of the polynomial-time hierarchy. We then introduce problems which are not only complete for the second level of this hierarchy, but are in fact hard to approximate. Our second area of study concerns quantum proof systems. Here, an interesting question which remains open despite much effort is whether a proof system with multiple unentangled quantum provers is equal in expressive power to a proof system with a single quantum prover (i.e. is QMA(poly) equal to QMA?). Our results here study variants of this question, and include a proof that the class BellQMA(poly) collapses to QMA. We also give an alternate proof that SepQMA(m) admits perfect parallel repetition. This proof is novel in that it utilizes cone programming duality. Our final area of study concerns non-classical correlations in quantum systems. Specifically, there exist genuinely quantum correlations beyond entanglement in mixed quantum states which may prove useful from a computing and information theoretic perspective. We first explore the presence of such correlations in the locking of classical correlations and the DQC1 model of mixed-state quantum computing. Our second result introduces a novel scheme for quantifying non-classical correlations using local unitary operations. Our third result introduces a protocol through which non-classical correlations in a starting system can be “activated”' into distillable entanglement with an ancilla system. Our last result determines when the entanglement generated in the activation protocol above can be mapped back onto the starting state via entanglement swapping.

No description available.

We consider the concept of local sets of inference rules. Locality is a syntactic condition on rule sets which guarantees that the inference relation defined by those rules is polynomial time decidable. Unfortunately, determining whether a given rule set is local can be difficult. In this paper we define inductive locality, a strengthening of locality. We also give a procedure which can automatically recognize the locality of any inductively local rule set. Inductive locality seems to be more useful that the earlier concept of strong locality. We show that locality, as a property of rule sets, is undecidable in general.

Dedukti est un cadre logique résultant de la combinaison du typage dépendant et de la réécriture. Il permet d'encoder de nombreux systèmes logiques au moyen de plongements superficiels qui préservent la notion de réduction. Ces traductions de systèmes logiques dans un format commun sont une première étape nécessaire à l'échange de preuves entre ces systèmes. Cet objectif d'interopérabilité des systèmes de preuve est la motivation principale de cette thèse. Pour y parvenir, nous nous inspirons du monde des langages de programmation et plus particulièrement des langages orientés-objet parce qu'ils mettent en œuvre des mécanismes avancés d'encapsulation, de modularité et de définitions par défaut. Pour cette raison, nous commençons par une traduction superficielle d'un calcul orienté-objet en Dedukti. L'aspect le plus intéressant de cette traduction est le traitement du sous-typage. Malheureusement, ce calcul orienté-objet ne semble pas adapté à l'incorporation de traits logiques. Afin de continuer, nous devons restreindre les mécanismes orientés-objet à des mécanismes statiques, plus faciles à combiner avec la logique et apparemment suffisant pour notre objectif d'interopérabilité. Une telle combinaison de mécanismes orientés-objet et de logique est présente dans l'environnement FoCaLiZe donc nous proposons un encodage superficiel de FoCaLiZe dans Dedukti. Les difficultés principales proviennent de l'intégration de Zenon, le prouveur automatique de théorèmes sur lequel FoCaLiZe repose, et de la traduction du langage d'implantation fonctionnel de FoCaLiZe qui présente deux constructions qui n'ont pas de correspondance simple en Dedukti : le filtrage de motif local et la récursivité. Nous démontrons finalement comment notre encodage de FoCaLiZe dans Dedukti peut servir en pratique à l'interopérabilité entre des systèmes de preuve à l'aide de FoCaLiZe, Zenon et Dedukti. Pour éviter de trop renforcer la théorie dans laquelle la preuve finale est obtenue, nous proposons d'utiliser Dedukti en tant que méta-langage pour éliminer des axiomes superflus. / Dedukti is a Logical Framework resulting from the combination ofdependent typing and rewriting. It can be used to encode many logical systems using shallow embeddings preserving their notion of reduction. These translations of logical systems in a common format are a necessary first step for exchanging proofs between systems. This objective of interoperability of proof systems is the main motivation of this thesis.To achieve it, we take inspiration from the world of programming languages and more specifically from object-oriented languages because they feature advanced mechanisms for encapsulation, modularity, and default definitions. For this reason we start by a shallow translation of an object calculus to Dedukti. The most interesting point in this translation is the treatment of subtyping. Unfortunately, it seems very hard to incorporate logic in this object calculus. To proceed, object-oriented mechanisms should be restricted to static ones which seem enough for interoperability. Such a combination of static object-oriented mechanisms and logic is already present in the FoCaLiZe environment so we propose a shallow embedding of FoCaLiZe in Dedukti. The main difficulties arise from the integration of FoCaLiZe automatic theorem prover Zenon and from the translation of FoCaLiZe functional implementation language featuring two constructs which have no simple counterparts in Dedukti: local pattern matching and recursion. We then demonstrate how this embedding of FoCaLiZe to Dedukti can be used in practice for achieving interoperability of proof systems through FoCaLiZe, Zenon, and Dedukti. In order to avoid strengthening to much the theory in which the final proof is expressed, we use Dedukti as a meta-language for eliminating unnecessary axioms.

In classical propositional logic (CPL) logical reasoning is formalised as logical entailment and can be computed by means of tableau and resolution proof procedures. Unfortunately CPL is not expressive enough and using first order logic (FOL) does not solve the problem either since proof procedures for these logics are not decidable. Modal propositional logics (MPL) on the other hand are both decidable and more expressive than CPL. It therefore seems reasonable to apply tableau and resolution proof systems to MPL in order to compute logical entailment in MPL. Although some of the principles in CPL are present in MPL, there are complexities in MPL that are not present in CPL. Tableau and resolution proof systems which address these issues and others will be surveyed here. In particular the work of Abadi & Manna (1986), Chan (1987), del Cerro & Herzig (1988), Fitting (1983, 1990) and Gore (1995) will be reviewed. / Computing / M. Sc. (Computer Science)

In classical propositional logic (CPL) logical reasoning is formalised as logical entailment and can be computed by means of tableau and resolution proof procedures. Unfortunately CPL is not expressive enough and using first order logic (FOL) does not solve the problem either since proof procedures for these logics are not decidable. Modal propositional logics (MPL) on the other hand are both decidable and more expressive than CPL. It therefore seems reasonable to apply tableau and resolution proof systems to MPL in order to compute logical entailment in MPL. Although some of the principles in CPL are present in MPL, there are complexities in MPL that are not present in CPL. Tableau and resolution proof systems which address these issues and others will be surveyed here. In particular the work of Abadi & Manna (1986), Chan (1987), del Cerro & Herzig (1988), Fitting (1983, 1990) and Gore (1995) will be reviewed. / Computing / M. Sc. (Computer Science)

Die Theorie disjunkter NP-Paare, die auf natürliche Weise statt einzelner Sprachen Paare von NP-Mengen zum Objekt ihres Studiums macht, ist vor allem wegen ihrer Anwendungen in der Kryptografie und Beweistheorie interessant. Im Zentrum dieser Dissertation steht die Analyse der Beziehung zwischen disjunkten NP-Paaren und aussagenlogischen Beweissystemen. Haben die Anwendungen der NP-Paare in der Beweistheorie maßgeblich das Verständnis aussagenlogischer Beweissysteme gefördert, so beschreiten wir in dieser Arbeit gewissermaßen den umgekehrten Weg, indem wir Methoden der Beweistheorie zur genaueren Untersuchung des Verbands disjunkter NP-Paare heranziehen. Insbesondere ordnen wir jedem Beweissystem P eine Klasse DNPP(P) von NP-Paaren zu, deren Disjunktheit in dem Beweissystem P mit polynomiell langen Beweisen gezeigt werden kann. Zu diesen Klassen DNPP(P) zeigen wir eine Reihe von Resultaten, die illustrieren, dass robust definierten Beweissystemen sinnvolle Komplexitätsklassen DNPP(P) entsprechen. Als wichtiges Hilfsmittel zur Untersuchung aussagenlogischer Beweissysteme und der daraus abgeleiteten Klassen von NP-Paaren benutzen wir die Korrespondenz starker Beweissysteme zu erststufigen arithmetischen Theorien, die gemeinhin unter dem Schlagwort beschränkte Arithmetik zusammengefasst werden. In der Praxis trifft man statt auf zwei häufig auf eine größere Zahl konkurrierender Bedingungen. Daher widmen wir uns der Erweiterung der Theorie disjunkter NP-Paare auf disjunkte Tupel von NP-Mengen. Unser Hauptergebnis in diesem Bereich besteht in der Charakterisierung der Fragen nach der Existenz optimaler Beweissysteme und vollständiger NP-Paare mit Hilfe disjunkter Tupel. / Disjoint NP-pairs are an interesting complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this dissertation we explore the connection between disjoint NP-pairs and propositional proof complexity. This connection is fruitful for both fields. Various disjoint NP-pairs have been associated with propositional proof systems which characterize important properties of these systems, yielding applications to areas such as automated theorem proving. Further, conditional and unconditional lower bounds for the separation of disjoint NP-pairs can be translated to results on lower bounds to the length of propositional proofs. In this way disjoint NP-pairs have substantially contributed to the understanding of propositional proof systems. Conversely, this dissertation aims to transfer proof-theoretic knowledge to the theory of NP-pairs to gain a more detailed understanding of the structure of the class of disjoint NP-pairs and in particular of the NP-pairs defined from propositional proof systems. For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P). Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist. As an important tool for our investigation we use the connection of propositional proof systems and disjoint NP-pairs to theories of bounded arithmetic.