From Specifications to Programs: A Fork-Algebraic Approach to Bridge the Gap

The development of programs from first-order specifications has as its main difficulty that of dealing with universal quantifiers. This work is focused in that point, i.e., in the construction of programs whose specifications involve universal quantifiers. This task is performed within a relational calculus based on fork algebras. The fact that first-order theories can be translated into equational theories in abstract fork algebras suggests that such work can be accomplished in a satisfactory way. Furthermore, the fact that these abstract algebras are representable guarantees that all properties valid in the standard models are captured by the axiomatization given for them, allowing the reasoning formalism to be shifted back and forth between any model and the abstract algebra. In order to cope with universal quantifiers, a new algebraic operation — relational implication — is introduced. This operation is shown to have deep significance in the relational statement of first-order expressions involving universal quantifiers. Several algebraic properties of the relational implication are stated showing its usefulness in program calculation. Finally, a non-trivial example of derivation is given to asses the merits of the relational implication as an specification tool, and also in calculation steps, where its algebraic properties are clearly appropriate as transformation rules.Laboratorio de Investigación y Formación en Informática Avanzad

The Subpower Membership Problem for Finite Algebras with Cube Terms

The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP we are given a set of tuples in a direct product of algebras from a fixed finite set $\mathcal{K}$ of finite algebras, and are asked whether or not a given tuple belongs to the subalgebra of the direct product generated by a given set. Our main result is that the subpower membership problem SMP($\mathcal{K}$) is in P if $\mathcal{K}$ is a finite set of finite algebras with a cube term, provided $\mathcal{K}$ is contained in a residually small variety. We also prove that for any finite set of finite algebras $\mathcal{K}$ in a variety with a cube term, each one of the problems SMP($\mathcal{K}$), SMP($\mathbb{HS} \mathcal{K}$), and finding compact representations for subpowers in $\mathcal{K}$, is polynomial time reducible to any of the others, and the first two lie in NP

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

A logic of hypothetical conjunction

This work was supported by the UK Engineering and Physical Sciences Research Council under the research grants EP/K033042/1 and EP/P011829/1.Peer reviewedPostprin

An estimation theoretic approach to quantum realizability problems

This thesis seeks to develop a general method for solving so-called quantum realizability problems, which are questions of the following form: under which conditions does there exist a quantum state exhibiting a given collection of properties? The approach adopted by this thesis is to utilize mathematical techniques previously developed for the related problem of property estimation which is concerned with learning or estimating the properties of an unknown quantum state. Our primary result is to recognize a correspondence between (i) property values which are realized by some quantum state, and (ii) property values which are occasionally produced as estimates of a generic quantum state. In Chapter 3, we review the concepts of stability and norm minimization from geometric invariant theory and non-commutative optimization theory for the purposes of characterizing the flow of a quantum state under the action of a reductive group. In particular, we discover that most properties of quantum states are related to the gradient of this flow, also known as the moment map. Afterwards, Chapter 4 demonstrates how to estimate the value of the moment map of a quantum state by performing a covariant quantum measurement on a large number of identical copies of the quantum state. These measurement schemes for estimating the moment map of a quantum state arise naturally from the decomposition of a large tensor-power representation into its irreducible sub-representations. Then, in Chapter 5, we prove an exact correspondence between the realizability of a moment map value on one hand and the asymptotic likelihood it is produced as an estimate on the other hand. In particular, by composing these estimation schemes, we derive necessary and sufficient conditions for the existence of a quantum state jointly realizing any finite collection of moment maps. Finally, in Chapter 6 we apply these techniques to the quantum marginals problem which aims to characterize precisely the relationships between the marginal density operators describing the various subsystems of a composite quantum system. We make progress toward an analytic solution to the quantum marginals problem by deriving a complete hierarchy of necessary inequality constraints

Topology of Arrangements and Representation Stability

The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research

The definable content of homological invariants II: \v{C}ech cohomology and homotopy classification

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the \v{C}ech cohomology functors $\check{\mathrm{H}}^n$ on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in the category $\mathsf{GPC}$ of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from $\mathsf{GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors $\check{\mathrm{H}}^n_{\mathrm{def}}$ to provide strong solutions to higher-dimensional and equivariant generalizations of a seminal problem in the development of algebraic topology, namely Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslash\Sigma$ to the $2$-sphere. In the course of this work, we record Borel definable versions of a number of classical results bearing on both the combinatorial and homotopical formulations of \v{C}ech cohomology; in aggregate, this work may be regarded as laying foundations for the descriptive set theoretic study of the homotopy relation on the space of maps from a locally compact Polish space to a polyhedron, a relation which embodies a substantial variety of classification problems arising throughout mathematics.Comment: 62 page

Cartesian closed bicategories: type theory and coherence

In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. However, calculations in such bicategories quickly fall into a quagmire of coherence data. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the difficulty of calculating---bring the data of cartesian closed bicategories down to the familiar level of cartesian closed categories. In fact, I prove this result in two ways. The first argument is closely related to Power's coherence theorem for bicategories with flexible bilimits. For the second, which is the central preoccupation of this thesis, the proof strategy has two parts: the construction of a type theory, and the proof that it satisfies a form of normalisation I call local coherence. I synthesise the type theory from algebraic principles using a novel generalisation of the (multisorted) abstract clones of universal algebra, called biclones. The result brings together two extensions of the simply-typed lambda calculus: a 2-dimensional type theory in the style of Hilken, which encodes the 2-dimensional nature of a bicategory, and a version of explicit substitution, which encodes a composition operation that is only associative and unital up to isomorphism. For products and exponentials I develop the theory of cartesian and cartesian closed biclones and pursue a connection with the representable multicategories of Hermida. Unlike preceding 2-categorical type theories, in which products and exponentials are encoded by postulating a unit and counit satisfying the triangle laws, the universal properties for products and exponentials are encoded using T. Fiore's biuniversal arrows. Because the type theory is extracted from the construction of a free biclone, its syntactic model satisfies a suitable 2-dimensional freeness universal property generalising the classical Curry--Howard--Lambek correspondence. One may therefore describe the type theory as an `internal language'. The relationship with the classical situation is made precise by a result establishing that the type theory I construct is the simply-typed lambda calculus up to isomorphism. This relationship is exploited for the proof of local coherence. It is has been known for some time that one may use the normalisation-by-evaluation strategy to prove the simply-typed lambda calculus is strongly normalising. Using a bicategorical treatment of M. Fiore's categorical analysis of normalisation-by-evaluation, I prove a normalisation result which entails the coherence theorem for cartesian closed bicategories. In contrast to previous coherence results for bicategories, the argument does not rely on the theory of rewriting or strictify using the Yoneda embedding. I prove bicategorical generalisations of a series of well-established category-theoretic results, present a notion of glueing of bicategories, and bicategorify the folklore result providing sufficient conditions for a glueing category to be cartesian closed. Once these prerequisites have been met, the argument is remarkably similar to that in the categorical setting

Non-Archimedean Geometry and Applications

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields, in particular to number theory and algebraic geometry. These applications included Mirror Symmetry, the Langlands program, p-adic Hodge theory, tropical geometry, resolution of singularities and the geometry of moduli spaces. Much emphasis was put on making the list of talks to reflect this diversity, thereby fostering the mutual inspiration which comes from such interactions

Congruences on lattices (with application to amalgamation)

Bibliography: pages 124-128.We present some aspects of congruences on lattices. An overview of general results on congruence distributive algebras is given in Chapter 1 and in Chapter 2 we examine weak projections; including Dilworth's characterization of congruences on lattices and a finite basis theorem for lattices. The outstanding problem of whether congruence lattices of lattices characterize distributive algebraic lattices is discussed in Chapter 3 and we look at some of the partial results known to date. The last chapter (Chapter 6) characterizes the amalgamation class of a variety B generated by a B-lattice, B, as the intersection of sub direct products of B, 2-congruence extendible members of B and 2-chain limited members of B. To this end we consider 2-congruence extendibility in Chapter 4 and n-chain limited lattices in Chapter 5. Included in Chapter 4 is the result that in certain lattice varieties the amalgamation class is contained in the class of 2-congruence extendible members of the variety. A final theorem in Chapter 6 states that the amalgamation class of a B-lattice variety is a Horn class