This book comprises 10 years of research by Matthias Baaz and Alexander Leitsch on the topic of cut-elimination. The aim of this research was to consider computational aspects of cut-elimination, the most important method for analyzing formal first-order proofs. During this period a new method of cut-elimination, cut-elimination by resolution (CERES), has been developed which is based on the refutations of formulas characterizing the cut-structure of the proofs. This new method connects automated theorem proving with classical proof theory, allowing the development of new methods and more efficient implementations; moreover, CERES opens a new view on cut-elimination in general. This field of research is evolving quite fast and we expect further results in the near future (in particular concerning cut-elimination in higher-order logic and in nonclassical logics).
We thank Daniele Mundici for his encouragement to write a book on this topic and for his steady interest in our research during the last 15 years. We also are grateful to the Austrian Science Fund for supporting the research on cut-elimination by funding the projects P16264, P17995, and P19875.
The research on this topic began with the authors and Alessandra Carbone during the time of her Lise Meitner fellowship. In the course of the following FWF research projects P16264, P17995, and P19875, the Ph.D. students Stefan Hetzl, Clemens Richter, Hendrik Spohr, Daniel Weller, and Bruno Woltzenlogel-Paleo contributed substantially to the theoretic and, especially, to the practical development of the CERES method. The extension of the method to Gödel logic has been carried out together with Agata Ciabattoni and Chris Fermüller.
Our special thanks go to Tomer Libal, Daniel Weller and Bruno Woltzenlogel for their careful and critical reading of the text. Their comments and suggestions have been integrated in the text and have resulted in a substantial improvement of the book.
We are very grateful to the reviewer for his numerous critical comments and suggestions for improvements which had a substantial impact on the final version of text.