Schemata are formal tools for describing inductive reasoning. They opened a new area in the analysis of inductive proofs.
The book introduces schemata for first-order terms, first-order formulas and first-order inference systems. Based on general first-order schemata, the cut-elimination-by-resolution (CERES) methoddeveloped around the year 2000is extended to schematic proofs. This extension requires the development of schematic methods for resolution and unification which are defined in this book. The added value of proof schemata compared to other inductive approaches consists in the extension of Herbrands theorem to inductive proofs (in the form of Herbrand systems, which can be constructed effectively). An application to an analysis of mathematical proof is given. The work also contains and extends the newest results on schematic unification and corresponding algorithms.
Core topics covered:
first-order schemata cut-elimination by resolution point transition systems schematic resolution Herbrand systems inductive proof analysis
This volume is the first comprehensive work on first-order schemata and their applications. As such, it will be eminently suitable for researchers and PhD students in logic and computer science either working or with an interest in proof theory, inductive reasoning and automated deduction. Prerequisites are a firm knowledge of first-order logic, basic knowledge of automated deduction and a background in theoretical computer science.
Alexander Leitsch and Anela Lolic are affiliated with the Institute of Logic and Computation of the Technische Universität Wien, David M. Cerna with the Czech Academy of Sciences, Institute of Computer Science (Ústav informatiky AV R, v.v.i.).
1. Introduction.-
2. Schemata and Point Transition Systems.-
3. Term
schemata and formula schemata.-
4. Term Schemata and Unification.-
5. Proof
schemata.-
6. Proof schemata and arithmetic.-
7. Cut-Elimination and the
Method CERES.-
8. Schematic CERES (completely new - improves former
publications).-
9. An Application of Schematic CERES.-
10. Schematic
Reasoning in GAPT.-
11. Conclusion.
David Cerna is a computational logician who has held positions at several research institutions including Czech Academy of Sciences, Dynatrace Research, and the Research Institute for Symbolic Computation. In 2015, he earned his PhD from the Technical University of Vienna in the field of computational proof theory. He has expertise in the areas of Inductive Synthesis, Unification Theory, and automated reasoning.
Alexander Leitsch is retired professor of Mathematics and Theoretical Computer Science at the Technische Universität Wien. His main research areas are Computational Logic, Proof Theory and Automated Deduction. He is author of the book The Resolution calculus (Springer 1997) and of the book Methods of Cut-Elimination with coauthor Matthias Baaz (Springer 2011). He was the head of the Theory and Logic group in the Institute of Logic and Computation and the leader of seven research projects in various areas of Computational Logic supported by the Austrian Science Fund.
AnelaLoli is a logician at the Institute of Logic and Computation, TUWien, specializing in structural proof theory. She earned her PhD in Computer Science from TUWien in 2020, with a thesis focused on the CERES method for automated proof analysis. As principal investigator and researcher, she leads projects such as PANDAFOREST ("Proof analysis and automated deduction for recursive structures") and on Skolemization and Interpolation, supported by the Austrian Science Fund FWF and the Austrian Academy of Sciences.
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