Mathematics for Computation (M4C)

The overall topic of the volume, Mathematics for Computation (M4C), is mathematics taking crucially into account the aspect of computation, investigating the interaction of mathematics with computation, bridging the gap between mathematics and computation wherever desirable and possible, and otherwise explaining why not.

Recently, abstract mathematics has proved to have more computational content than ever expected. Indeed, the axiomatic method, originally intended to do away with concrete computations, seems to suit surprisingly well the programs-from-proofs paradigm, with abstraction helping not only clarity but also efficiency.

Unlike computational mathematics, which rather focusses on objects of computational nature such as algorithms, the scope of M4C generally encompasses all the mathematics, including abstract concepts such as functions. The purpose of M4C actually is a strongly theory-based and therefore, is a more reliable and sustainable approach to actual computation, up to the systematic development of verified software.

While M4C is situated within mathematical logic and the related area of theoretical computer science, in principle it involves all branches of mathematics, especially those which prompt computational considerations. In traditional terms, the topics of M4C include proof theory, constructive mathematics, complexity theory, reverse mathematics, type theory, category theory and domain theory.

The aim of this volume is to provide a point of reference by presenting up-to-date contributions by some of the most active scholars in each field. A variety of approaches and techniques are represented to give as wide a view as possible and promote cross-fertilization between different styles and traditions.

Contents:

• Reflections on 50 Years of Constructive Research (Douglas S Bridges)
• Thoughts on Computational Mathematics (Fred Richman)
• Logic for Exact Real Arithmetic: Multiplication (Helmut Schwichtenberg)
• Information Systems with Witnesses: The Function Space Construction (Dieter Spreen)
• A Constructive Version of Carathéodory's Convexity Theorem (Josef Berger and Gregor Svindland)
• Varieties of the Weak Knig Lemma and the Disjunctive Dependent Choice (Josef Berger, Hajime Ishihara and Takako Nemoto)
• Intermediate Goodstein Principles (David Fernández-Duque, Oriola Gjetaj and Andreas Weiermann)
• Infinite Horizon Extensive Form Games, Coalgebraically (Matteo Capucci, Neil Ghani, Clemens Kupke, Jérémy Ledent and Fredrik Nordvall Forsberg)
• Concurrent Gaussian Elimination (Ulrich Berger, Monika Seisenberger, Dieter Spreen and Hideki Tsuiki)
• A Herbrandised Interpretation of Semi-Intuitionistic Second-Order Arithmetic with Function Variables (João Enes and Fernando Ferreira)
• More or Less Uniform Convergence (Henry Towsner)
• Constructive Theory of Ordinals (Thierry Coquand, Henri Lombardi and Stefan Neuwirth)
• No Speedup for Geometric Theories (Michael Rathjen)
• Domain Theory and Realisability over Scott's D in Constructive Set Theory (Eman Dihoum, Michael Rathjen and Avi Silterra)
• Proof Complexity of Quantified Boolean Logic A Survey (Olaf Beyersdorff)
• Subject Reduction in Multi-Universe Type Theories (Marco Benini)

Readership: Graduate students, researchers, and professors in Mathematics, Computer Science, and Philosophy.
Key Features:

• This book gathers recent contributions of distinguished experts
• It makes emerging fields accessible to a wider audience, appealing to a broad readership with diverse backgrounds
• It fills a gap between (under-)graduate level textbooks and state-of-the-art

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