Causality in the Semantics of Esterel: Revisited

We re-examine the challenges concerning causality in the semantics of Esterel and show that they pertain to the known issues in the semantics of Structured Operational Semantics with negative premises. We show that the solutions offered for the semantics of SOS also provide answers to the semantic challenges of Esterel and that they satisfy the intuitive requirements set by the language designers

Explaining Gabriel-Zisman localization to the computer

This explains a computer formulation of Gabriel-Zisman localization of categories in the proof assistant Coq. It includes both the general localization construction with the proof of GZ's Lemma 1.2, as well as the construction using calculus of fractions. The proof files are bundled with the other preprint "Files for GZ localization" posted simultaneously

On the concept of (homo)morphism : a key notion in the learning of abstract algebra

This article is dedicated to the investigation of difficulties involved in the understanding of the homomorphism concept. It doesn't restrict to group-theory but on the contrary raises the issue of developing teaching strategies aiming at gaining access to structuralist thinking. Emphasis is put on epistemological analysis and its interaction with didactics in an attempt to make Abstract Algebra more accessible

Set Theory or Higher Order Logic to Represent Auction Concepts in Isabelle?

When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL system, and we have produced verified code for combinatorial Vickrey auctions. A fundamental question in this was how to represent some basic concepts: since set theory is available inside Isabelle/HOL, when introducing new definitions there is often the issue of balancing the amount of set-theoretical objects and of objects expressed using entities which are more typical of higher order logic such as functions or lists. Likewise, a user has often to answer the question whether to use a constructive or a non-constructive definition. Such decisions have consequences for the proof development and the usability of the formalization. For instance, sets are usually closer to the representation that economists would use and recognize, while the other objects are closer to the extraction of computational content. In this paper we give examples of the advantages and disadvantages for these approaches and their relationships. In addition, we present the corresponding Isabelle library of definitions and theorems, most prominently those dealing with relations and quotients.Comment: Preprint of a paper accepted for the forthcoming CICM 2014 conference (cicm-conference.org/2014): S.M. Watt et al. (Eds.): CICM 2014, LNAI 8543, Springer International Publishing Switzerland 2014. 16 pages, 1 figur

Free Fermi and Bose Fields in TQFT and GBF

We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case

Non-reformist reform for Haskell Modularity

In this thesis, I present Backpack, a new language for building separately-typecheckable packages on top of a weak module system like Haskell’s. The design of Backpack is the first to bring the rich world of type systems to the practical world of packages via mixin modules. It’s inspired by the MixML module calculus of Rossberg and Dreyer but by choosing practicality over expressivity Backpack both simplifies that semantics and supports a flexible notion of applicative instantiation. Moreover, this design is motivated less by foundational concerns and more by the practical concern of integration into Haskell. The result is a new approach to writing modular software at the scale of packages.Modulsysteme wie die in Haskell erlauben nur eine weiche Art der Modularität, in dem Modulimplementierungen direkt von anderen Implementierungen abhängen und in dieser Abhängigkeitsreihenfolge verarbeitet werden müssen. Modulsysteme wie die in ML andererseits erlauben eine kräftige Art der Modularität, in dem explizite Schnittstellen Vermutungen über Abhängigkeiten ausdrücken und jeder Modultyp überprüft und unabhängig ergründet werden kann. In dieser Dissertation präsentiere ich Backpack, eine neue Sprache zur Entwicklung separattypenüberprüfbarer Pakete über einem weichen Modulsystem wie Haskells. Das Design von Backpack überführt erstmalig die reichhaltige Welt der Typsysteme in die praktische Welt der Pakete durch Mixin-Module. Es wird von der MixML-Kalkulation von Rossberg und Dreyer angeregt. Backpack vereinfacht allerdings diese Semantik durch die Auswahl von Anwendbarkeit statt Expressivität und fördert eine flexible Art von geeigneter Applicative- Instantiierung. Zudem wird dieses Design weniger von grundlegenden Anliegen als von dem praktischen Anliegen der Eingliederung in Haskell begründet. Die Semantik von Backpack wird durch die Ausarbeitung in Mengen von Haskell-Modulen und „binary interface files“ definiert, und zeigt so, wie Backpack Interoperabilität mit Haskell erhält, während Backpack es mit Schnittstellen nachrüstet. In meiner Formalisierung Backpacks präsentiere ich ein neuartiges Typsystem für Haskellmodule und überprüfe einen entscheidenen Korrektheitssatz, um die Semantik von Backpack zu validieren.Max Planck Institute for Software Systems (MPI-SWS

Certified Symbolic Manipulation: Bivariate Simplicial Polynomials

Certified symbolic manipulation is an emerging new field where programs are accompanied by certificates that, suitably interpreted, ensure the correctness of the algorithms. In this paper, we focus on algebraic algorithms implemented in the proof assistant ACL2, which allows us to verify correctness in the same programming environment. The case study is that of bivariate simplicial polynomials, a data structure used to help the proof of properties in Simplicial Topology. Simplicial polynomials can be computationally interpreted in two ways. As symbolic expressions, they can be handled algorithmically, increasing the automation in ACL2 proofs. As representations of functional operators, they help proving properties of categorical morphisms. As an application of this second view, we present the definition in ACL2 of some morphisms involved in the Eilenberg-Zilber reduction, a central part of the Kenzo computer algebra system. We have proved the ACL2 implementations are correct and tested that they get the same results as Kenzo does.Ministerio de Ciencia e Innovación MTM2009-13842Unión Europea nr. 243847 (ForMath