Chapter 5: Reducibility

For Your Enjoyment

• A Wikipedia article on Gödel's Incompleteness Theorem.

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, ... C_f where C₁ 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 C₁, C₂, ... C_f where C₁ 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 * gⁿ 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:

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 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.

In-Class Notes

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