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.

About the Author

Marco Benini is an Assistant Professor for Mathematical Logic at the University of Insubria. After a doctorate in computer science at the University of Milano, he became Assistant Professor for Computer Science at the University of Insubria and then, he won a Marie Curie Fellowship in Mathematical Logic at the University of Leeds. Apart from constructive mathematics in general, his principal research interests are about point-free semantics and the computational content of formal theories.

Olaf Beyersdorff is a Professor of Theoretical Computer Science at Friedrich-Schiller University Jena. He obtained his PhD from Humboldt University Berlin and his Habilitation from Leibniz University Hanover and was visiting professor at Sapienza University Rome. His principal research interests are in computational logic, computational complexity, and particularly in proof complexity.

Michael Rathjen is a Professor of Pure Mathematics at the University of Leeds. He has worked in general proof theory and constructivism for around 20 years, particularly on ordinal analysis of strong theories, models and extensions of Martin-L�f type theory, and constructive set theories. From 2002 to 2005 he was Professor of Mathematics at Ohio State University, having previously held a Heisenberg Fellowship and appointments at Leeds, Stanford, Ohio State University and M�nster.

Peter Schuster is a Professor for Mathematical Logic at the University of Verona. After both doctorate and Habilitation in mathematics at the University of Munich, he was Lecturer at the University of Leeds and member of the Leeds Logic Group. Apart from constructive mathematics at large, his principal research interests are about the computational content of classical proofs in abstract algebra.

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