# Chapter 5: Reducibility

## For Your Enjoyment

## Chapter 5.1, Undecidable Problems from Language Theory

### Reductions Via Computation Histories

- Definition: An
**accepting computation history** for TM M on w
is a sequence of configurations C_{1}, C_{2}, ...
C_{f} where C_{1} is the the start configuration of
M on w, C_{f} is an accepting configuration of M, and
each C_{i+1} legally follows from C_{i}.
- Definition: A
**rejecting computation history** for TM M on w
is a sequence of configurations C_{1}, C_{2}, ...
C_{f} where C_{1} is the the start configuration of
M on w, C_{f} is a rejecting configuration of M, and
each C_{i+1} legally follows from C_{i}.
- Computation histories are finite sequence.

### Linear Bounded Automaton

- Definition: An
**LBA** is a restricted type of Turing Machine wherein the
tape head isn't permitted to move off the portion of the tape containing
the input. If the machine tries to move its head off either end of
the input, the head stays where it is.
- To give you more insight, here is a sample LBA 8-tuple.
- Lemma: Let M by an LBA with q states and g symbols in the tape
alphabet. There are exactly
**q * n * g**^{n} distinct
configurations of M for a tape of length n.
- A
_{LBA} = {<M, w> | M is an LBA that accepts string w}
is decidable. What is the algorithm?
- E
_{LBA} = {<M> | M is an LBA where L(M) = ∅} is
undecidable.

### Proof that E_{LBA} is Undecidable

Assume that E_{LBA} is decidable. We can then construct
TM R to decide E_{LBA}.

On input <M, w>, where M is a TM and w is a string:

- Construct LBA B from M and w. Let L(B) be the accepting
computation histories for M on w. Note: If L(B) is empty,
M does not accept w, otherwise it does.
- Run R on B.
- If R rejects, then accept and if R accepts, then reject.

This provides an algorithm to decide A_{TM}, a problem
that we know to be undecidable. Contradiction! Thus, our assumption
is wrong and consequently, E_{LBA} must be undecidable.

### The Post Correspondence Problem

- To understand the problem, look at examples 1 and 2 on
this Wikipedia page.
- PCP = {<P> | P is an instance of the Post Correspondence Problem
with a match.
- PCP is undecidable.
- Section 5.2 shows how to prove that PCP is undecidable by reducing
A
_{TM} to it. Understanding the proof is optional.