# Chapter 5: Reducibility

## 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 C1, C2, ... Cf where C1 is the the start configuration of M on w, Cf is an accepting configuration of M, and each Ci+1 legally follows from Ci.
• Definition: A rejecting computation history for TM M on w is a sequence of configurations C1, C2, ... Cf where C1 is the the start configuration of M on w, Cf is a rejecting configuration of M, and each Ci+1 legally follows from Ci.
• 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 * gn distinct configurations of M for a tape of length n.
• ALBA = {<M, w> | M is an LBA that accepts string w} is decidable. What is the algorithm?
• ELBA = {<M> | M is an LBA where L(M) = ∅} is undecidable.

### Proof that ELBA is Undecidable

Assume that ELBA is decidable. We can then construct TM R to decide ELBA.

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

1. 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.
2. Run R on B.
3. If R rejects, then accept and if R accepts, then reject.

This provides an algorithm to decide ATM, a problem that we know to be undecidable. Contradiction! Thus, our assumption is wrong and consequently, ELBA 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 ATM to it. Understanding the proof is optional.