Chapter 10: Mathematical proofs Introduction 10.1 Preliminaries 10.2 Mathematical deduction 10.3 Specific fields 10.4 Conclusion

Introduction One of few ways to demonstrate absolute truth Research that proves assertions mathematically is highly regarded

10.1 Preliminaries Need for mathematical notation Sets, functions, and relations Propositional logic (p59) First-order predicate logic Automata Petri nets Formal grammars Specification languages

10.2 Mathematical deduction Definitions describe what is meant by a specific term Axioms express truths that are either obvious or assumed Theorems represent interesting or useful derived facts Lemmas are similar to theorems but are only used as stepping stones to derive more general or more powerful theorems Corollaries are rephrased theorems or lemmas

10.3 Specific fields Computability and algorithmic complexity Can a given problem be solved by a computer? If so, prove it by means of an algorithm. Algorithmic complexity refers to time and/or memory requirements of an algorithm for a computable problem Correctness: Proving a program correct Combinatorics and graph theory Formal languages and automata theory

10.4 Conclusion A little mathematics increase other people’s regard of your work