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.
