# Chapter 5: Reducibility

## For Your Enjoyment

- A YouTube video on the
Banach-Tarski Paradox,
suggested by Joe Pagani. The first half covers concepts that we
have seen in CSCI 338.

## Chapter 5.3, Mapping Reducibility

- A function f: Σ
^{*} → Σ^{*} is a
**computable function** if some Turing Machine M, on every input w,
halts with just f(w) on its tape.
- Language A is
**mapping reducible** to language B, written
A ≤_{m} B, if there is a computable function
f: Σ^{*} → Σ^{*}, where for
every w,
w ∈ A ⇔ f(w) ∈ B

The function f is called a **reduction** from A to B.
- Theorem: If A ≤
_{m} B and B is decidable,
then A is decidable.
- Corollary: If A ≤
_{m} B and A is undecidable, then B is
undecidable.
- Theorem: If A ≤
_{m} B and B is Turing-recognizable,
then A is Turing-recognizable.
- Corollary: If A ≤
_{m} B and A is not Turing-recognizable,
then B is not Turing-recognizable.