# The Power of Self-Reducibility: Selectivity, Information, and Approximation

## TL;DR

This paper offers a tutorial on self-reducibility, guiding readers through proofs of key theorems related to selectivity, information, and approximation, aiming to enhance understanding and teaching of this proof technique.

## Contribution

It provides a hands-on tutorial and lesson plan to help readers learn and apply self-reducibility in complexity theory proofs, especially for educational purposes.

## Key findings

- Guides readers to discover proofs using self-reducibility
- Includes challenge problems related to selectivity and approximation
- Provides educational resources for teaching complexity theory

## Abstract

This chapter provides a hands-on tutorial on the important technique known as self-reducibility. Through a series of "Challenge Problems" that are theorems that the reader will---after being given definitions and tools---try to prove, the tutorial will ask the reader not to read proofs that use self-reducibility, but rather to discover proofs that use self-reducibility. In particular, the chapter will seek to guide the reader to the discovery of proofs of four interesting theorems---whose focus areas range from selectivity to information to approximation---from the literature, whose proofs draw on self-reducibility. The chapter's goal is to allow interested readers to add self-reducibility to their collection of proof tools. The chapter simultaneously has a related but different goal, namely, to provide a "lesson plan" (and a coordinated set of slides is available online to support this use [Hem19]) for a lecture to a two-lecture series that can be given to undergraduate students---even those with no background other than basic discrete mathematics and an understanding of what polynomial-time computation is---to immerse them in hands-on proving, and by doing that, to serve as an invitation to them to take courses on Models of Computation or Complexity Theory.