Chapter 5: Reducibility

For Your Enjoyment

Chapter 5.1, Undecidable Problems from Language Theory

Reductions Via Computation Histories

Linear Bounded Automaton

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

In-Class Notes

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