Introduction: Inferences and Proofs

Unsound Inferences Make Proofs Shorter

We give examples of calculi that extend Gentzen’s sequent calculus LK by unsound quantifier inferences in such a way that (i) derivations lead only to true sequents, and (ii) proofs therein are non-elementarily shorter than LK-proofs.

1: Introduction to Proofs

By the end of this course, we would like to be able to understand and create proofs. Since logic is the foundation of a mathematical proof, we begin the course with basic logic. However, even before we discuss logic, we have to discuss precision of language. Precision of language is necessary in creating logical statements. To give an example of how logic and precision of language are important...

A Brief Introduction to Proofs

Proofs are perhaps the very heart of mathematics. Unlike the other sciences, mathematics adds a final step to the familiar scientific method. After experimenting, collecting data, creating a hypothesis, and checking that hypothesis through more experiments, mathematicians must prove their hypothesis is correct before it can be accepted as a true result. Similarly, mathematicians will only consi...

Introduction to Proofs: Homework Solutions

Introduction to Interactive Proofs & The Sumcheck Protocol

Traditional mathematical proofs are static objects: a prover P writes down a sequence of mathematical statements, and then at some later time a verifier V checks that these statements are consistent and correct. Over the years, computer science has changed the notion of a mathematical proof. The first such change was the observation that for all practical purposes, the verification procedure sh...