Statistical approach to proof theory
Ivan Kramosil · Czech digital mathematics library · 1979
The aim of this work is to survey various approaches, methods and results connected with application of probability theory and mathematical statistics in proof theory.A special attention is devoted to various methods of statistical deducibility testing and their applications.The historical background of this branch of applied mathematics as well as a short description of the particular chapters of this work and an intuitive motivation can be found in the introductory chapter.CONTENTS 1. Introduction 2. Formalized Theories 3. Resolution-Based Theorem-Proving 4. A General Model of Statistical Theorem-Proving 5. Statistical Deducibility Testing in Random Extensions 6.The Role of Experience in Statistical Deducibility Testing 7. Other Statistical Approaches to Deducibility Testing 8. Application of Statistical Deducibility Testing 9. Other Conceptions of Statistical Approximations in Proof Theory 10.Conclusive Remarks be found in [6.2]..." refers to the second item of the reference list at the end of Chapter 6. Theorems, definitions, examples and relations are numbered by the usual double enumeration, the first numeral referring to the chapter in question.simple and genial: to choose a small number of basic and immediately observable sentences from which all the other valid sentences could be derived by some rules of reasoning.Euclides was successful in applying this idea to geometry, he chose five elementary geometric assertions, usually called axioms or postulates and derived from them all valid geometric sentences known in his age.A mathematical, or, in general, scientific theory described in this way is called axiomatic theory, it can be formalized by a pair