Quantum Groups and Noncommutative Geometry

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of spacetime, in particular, amounts to a postulated new force or physical effect called cogravity.Comment: 72 pages, many figures; intended for wider theoretical physics community (special millenium volume of JMP

Synthetic Philosophy of Mathematics and Natural Sciences Conceptual analyses from a Grothendieckian Perspective

ISBN-13: 978-0692593974. Giuseppe Longo. Synthetic Philosophy of Mathematics and Natural Sciences, Conceptual analyses from a Grothendieckian Perspective, Reflections on “Synthetic Philosophy of Contemporary Mathematics” by F. Zalamea, Urbanomic (UK) and Sequence Press (USA), 2012. Invited Paper, in Speculations: Journal of Speculative Realism, Published: 12/12/2015, followed by an answer by F. Zalamea.International audienceZalamea’s book is as original as it is belated. It is indeed surprising, if we give it a moment’s thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions D’Espagnat’s work on quantum mechanics, but we could add several others who, in the last few decades, have elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy – since Kant’s crucial observations in the Critique of Judgment, at least – has often “run ahead” of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology..

Categorical structure of continuation passing style

Laboratory for Foundations of Computer ScienceThis thesis attempts to make precise the structure inherent in Continuation Passing Style (CPS). We emphasize that CPS translates lambda-calculus into a very basic calculus that does not have functions as primitive. We give an abstract categorical presentation of continuation semantics by taking the continuation type constructor (cont in Standard ML of New Jersey) as primitive. This constructor on types extends to a contravariant functor on terms which is adjoint to itself on the left; restricted to the subcategory of those programs that do not manipulate the current continuation, it is adjoint to itself on the right. The motivating example of such a category is built from (equivalence classes of typing judgements for) continuation passing style (CPS) terms. The categorical approach suggests a notion of effect-free term as well as some operators for manipulating continuations. We use these for writing programs that illustrate our categorical approach and refute some conjectures about control effects. A call-by-value lambda-calculus with the control operator callcc can be interpreted. Arrow types are broken down into continuation types for argument/result-continuations pairs, reflecting the fact that CPS compiles functions into a special case of continuations. Variant translations are possible, among them lazy call-by-name, which can be derived by way of argument thunking, and a genuinely call-by-name transform. Specialising the semantics to the CPS term model allows a rational reconstruction of various CPS transforms

A Dualities-Consolidating Framework to Support Systematic Programming Language Design

In the theory of programming languages, duality is increasingly recognized as being important for improving economy, offering the theoretical development for one of two dual concepts "for free". Two prevalent dualities are the extensibility duality, related to the Expression Problem, and the De Morgan duality, related to evaluation strategies and control flow; for instance, a language which is symmetric with respect to the extensibility duality has both a facility which allows for easy extension with new variants, similar to how classes implement an interface in certain object-oriented languages, and a dual facility which allows for easy extension with new operations, as in functional programming with algebraic data types. However, this theoretical knowledge arguably has yet to be made more accessible to the practician. In particular, the design of programming languages does not yet really benefit from it in a systematic way. As a step to improve this situation, building on these prior results, the present work presents a prototype of a, in the conceptual sense rather economical, foundational system, in which the extensibility duality and the De Morgan duality are consolidated. In particular, the system is inherently highly symmetric with respect to both dualities and their consolidation quite naturally allows to carve out the essence of the extensibility duality, thereby further optimizing the meta-level economy. As will be demonstrated, this system can serve as a framework in which various language features known from practical programming languages can be recovered (by local syntactic abstractions, a.k.a. macros) and systematically compared, including algebraic data types and function types as known from functional programming, classes and objects, and exception handling, in combination with the evaluation strategies employed by the respective languages. This is intended to facilitate a systematic analysis of programming language concepts which may aid in the design of parsimonious languages which are symmetric with respect to one or both of the mentioned dualities. For the more short-term perspective, the system may also serve as a cornerstone for the systematic development of tools which automatically semantically compare (and convert between) programs in different languages by means of analyzing the results of embedding them into the framework.In der theoretischen Betrachtung von Programmiersprachen wird Dualität als zunehmend wichtig für die Verbesserung der Ökonomie betrachtet, da diese ermöglicht, die Theorie-Entwicklung für eines von zwei dualen Konzepten "umsonst" zu erhalten. Zwei vorherrschende Dualitäten sind die Extensibilitäts-Dualität, die im Zusammenhang mit dem Expression Problem steht, und die De Morgan-Dualität, die im Zusammenhang mit Auswertungsstrategien und Kontrollfluss steht; zum Beispiel bietet eine Sprache, die symmetrisch in Bezug auf die Extensibilitäts-Dualität ist, sowohl ein Konstrukt, das die einfache Hinzufügung von neuen Varianten ermöglicht, ähnlich dazu wie in gewissen Objekt-Orientierten Sprachen Klassen ein Interface implementieren, als auch ein duales Konstrukt, das die einfache Hinzufügung von neuen Operationen ermöglicht, wie in der Funktionalen Programmierung mit algebraischen Datentypen. Dieses theoretische Wissen muss wohl allerdings dem Praktiker noch besser zugänglich gemacht werden. Insbesondere profitiert die Entwicklung von Programmiersprachen noch nicht wirklich auf eine systematische Weise davon. Als Schritt auf dem Weg dahin, diese Situation zu verbessern, präsentiert diese Arbeit, auf diesen bisherigen Resultaten aufbauend, ein grundlegendes, im konzeptuellen Sinne recht ökonomisches System, in dem die Extensibilitäts-Dualität und die De Morgan-Dualität miteinander vereinigt sind. Insbesondere ist dieses System inhärent höchst symmetrisch in Bezug auf beide Dualitäten und deren Vereinigung ermöglicht auf recht natürliche Weise die Essenz der Extensibilitäts-Dualität herauszuarbeiten, was die Ökonomie auf der Meta-Ebene weiter verbessert. Wie dargestellt werden wird, kann dieses System als Framework dienen, in dem sich verschiedene Sprach-Features aus in der Praxis relevanten Programmiersprachen darstellen lassen (durch lokale syntaktische Abstraktionen, auch bekannt als Macros) und in dem man diese vergleichen kann, wie etwa algebraische Datentypen und Funktionstypen, wie man sie aus der Funktionalen Programmierung kennt, Klassen und Objekte, sowie Exception-Handling, in Verbindung mit den Auswertungsstrategien die von den jeweiligen Sprachen verwendet werden. Dies soll dem Zweck dienen, eine systematische Analyse von Programmiersprachen-Konzepten zu ermöglichen, welche bei der Entwicklung von kompakten Sprachen helfen kann, die symmetrisch in Bezug auf eine oder beide der erwähnten Dualitäten sind. Für die kurzfristigere Perspektive bietet es das System auch als Grundstein für die systematische Entwicklung von Tools an, welche automatisch Programme in verschiedenen Sprache semantisch vergleichen (und ineinander umwandeln), indem sie die Ergebnisse von deren Einbettung in das Framework analysieren

An Analytic Propositional Proof System on Graphs

In this paper we present a proof system that operates on graphs instead of formulas. Starting from the well-known relationship between formulas and cographs, we drop the cograph-conditions and look at arbitrary undirected) graphs. This means that we lose the tree structure of the formulas corresponding to the cographs, and we can no longer use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalization of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic with mix, and we argue that our graphs form a notion of generalized connective

Session Types in Concurrent Calculi: Higher-Order Processes and Objects

This dissertation investigates different formalisms, in the form of programming language calculi, that are aimed at providing a theoretical foundation for structured concurrent programming based on session types. The structure of a session type is essentially a process-algebraic style description of the behaviour of a single program identifier serving as a communication medium (and usually referred to as a channel): the types incorporate typed inputs, outputs, and choices which can be composed to form larger protocol descriptions. The effectiveness of session typing can be attributed to the linear treatment of channels and session types, and to the use of tractable methods such as syntactic duality to decide if the types of two connected channels are compatible. Linearity is ensured when accumulating the uses of a channel into a composite type that describes also the order of those actions. Duality provides a tractable and intuitive method for deciding when two connected channels can interact and exchange values in a statically determined type-safe way. We present our contributions to the theory of sessions, distilled into two families of programming calculi, the first based on higher-order processes and the second based on objects. Our work unifies, improves and extends, in manifold ways, the session primitives and typing systems for the Lambda-calculus, the Pi-calculus, the Object-calculus, and their combinations in multi-paradigm languages. Of particular interest are: the treatment of infinite interactions expressed with recursive sessions; the capacity to encapsulate channels in higher-order structures which can be exchanged and kept suspended, i.e., the use of code as data; the integration of protocol structure directly into the description of objects, providing a powerful and uniformly extensible set of implementation abstractions; finally, the introduction of asynchronous subtyping, which enables controlled reordering of actions on either side of a session. Our work on higher-order processes and on object calculi for session-based concurrent programming provides a theoretical foundation for programming language design integrating functional, process, and object-oriented features