Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory

In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige Schlüsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenössischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen Vollständigkeits-, Unentscheidbarkeits- und Unvollständigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jüngeren Untersuchungen, eine zunehmende Bedeutung als Maßstab für Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgeführt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermöglicht neue Einsichten in den konstruktiven Gehalt von Vollständigkeit, einen synthetischen Ansatz für Unentscheidbarkeit und Unvollständigkeit, der großteils den berüchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natürliche Repräsentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen für Nutzung und Beiträge externer Anwender ist

An Algorithmic Interpretation of Quantum Probability

The Everett (or relative-state, or many-worlds) interpretation of quantum mechanics has come under fire for inadequately dealing with the Born rule (the formula for calculating quantum probabilities). Numerous attempts have been made to derive this rule from the perspective of observers within the quantum wavefunction. These are not really analytic proofs, but are rather attempts to derive the Born rule as a synthetic a priori necessity, given the nature of human observers (a fact not fully appreciated even by all of those who have attempted such proofs). I show why existing attempts are unsuccessful or only partly successful, and postulate that Solomonoff's algorithmic approach to the interpretation of probability theory could clarify the problems with these approaches. The Sleeping Beauty probability puzzle is used as a springboard from which to deduce an objectivist, yet synthetic a priori framework for quantum probabilities, that properly frames the role of self-location and self-selection (anthropic) principles in probability theory. I call this framework "algorithmic synthetic unity" (or ASU). I offer no new formal proof of the Born rule, largely because I feel that existing proofs (particularly that of Gleason) are already adequate, and as close to being a formal proof as one should expect or want. Gleason's one unjustified assumption--known as noncontextuality--is, I will argue, completely benign when considered within the algorithmic framework that I propose. I will also argue that, to the extent the Born rule can be derived within ASU, there is no reason to suppose that we could not also derive all the other fundamental postulates of quantum theory, as well. There is nothing special here about the Born rule, and I suggest that a completely successful Born rule proof might only be possible once all the other postulates become part of the derivation. As a start towards this end, I show how we can already derive the essential content of the fundamental postulates of quantum mechanics, at least in outline, and especially if we allow some educated and well-motivated guesswork along the way. The result is some steps towards a coherent and consistent algorithmic interpretation of quantum mechanics

Implementation of Web Query Languages Reconsidered

Visions of the next generation Web such as the "Semantic Web" or the "Web 2.0" have triggered the emergence of a multitude of data formats. These formats have different characteristics as far as the shape of data is concerned (for example tree- vs. graph-shaped). They are accompanied by a puzzlingly large number of query languages each limited to one data format. Thus, a key feature of the Web, namely to make it possible to access anything published by anyone, is compromised. This thesis is devoted to versatile query languages capable of accessing data in a variety of Web formats. The issue is addressed from three angles: language design, common, yet uniform semantics, and common, yet uniform evaluation. % Thus it is divided in three parts: First, we consider the query language Xcerpt as an example of the advocated class of versatile Web query languages. Using this concrete exemplar allows us to clarify and discuss the vision of versatility in detail. Second, a number of query languages, XPath, XQuery, SPARQL, and Xcerpt, are translated into a common intermediary language, CIQLog. This language has a purely logical semantics, which makes it easily amenable to optimizations. As a side effect, this provides the, to the best of our knowledge, first logical semantics for XQuery and SPARQL. It is a very useful tool for understanding the commonalities and differences of the considered languages. Third, the intermediate logical language is translated into a query algebra, CIQCAG. The core feature of CIQCAG is that it scales from tree- to graph-shaped data and queries without efficiency losses when tree-data and -queries are considered: it is shown that, in these cases, optimal complexities are achieved. CIQCAG is also shown to evaluate each of the aforementioned query languages with a complexity at least as good as the best known evaluation methods so far. For example, navigational XPath is evaluated with space complexity O(q d) and time complexity O(q n) where q is the query size, n the data size, and d the depth of the (tree-shaped) data. CIQCAG is further shown to provide linear time and space evaluation of tree-shaped queries for a larger class of graph-shaped data than any method previously proposed. This larger class of graph-shaped data, called continuous-image graphs, short CIGs, is introduced for the first time in this thesis. A (directed) graph is a CIG if its nodes can be totally ordered in such a manner that, for this order, the children of any node form a continuous interval. CIQCAG achieves these properties by employing a novel data structure, called sequence map, that allows an efficient evaluation of tree-shaped queries, or of tree-shaped cores of graph-shaped queries on any graph-shaped data. While being ideally suited to trees and CIGs, the data structure gracefully degrades to unrestricted graphs. It yields a remarkably efficient evaluation on graph-shaped data that only a few edges prevent from being trees or CIGs

Natural Communication

In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science

Certificates for Incremental Type Checking

The central topic of this thesis is the study of algorithms for type checking, both from the programming language and from the proof-theoretic point of view. A type checking algorithm takes a program or a proof, represented as a syntactical object, and checks its validity with respect to a specification or a statement. It is a central piece of compilers and proof assistants. We postulate that since type checkers are at the interface between proof theory and program theory, their study can let these two fields mutually enrich each other. We argue by two main instances: first, starting from the problem of proof reuse, we develop an incremental type checker; secondly, starting from a type checking program, we evidence a novel correspondence between natural deduction and the sequent calculus

Introduction to Discrete Mathematics: An OER for MA-471

The first objective of this book is to define and discuss the meaning of truth in mathematics. We explore logics, both propositional and first-order , and the construction of proofs, both formally and human-targeted. Using the proof tools, this book then explores some very fundamental definitions of mathematics through set theory. This theory is then put in practice in several applications. The particular (but quite widespread) case of equivalence and order relations is studied with detail. Then we introduces sequences and proofs by induction, followed by number theory. Finally, a small introduction to combinatorics is given