Chapter 5: Reducibility

For Your Enjoyment

• A Wikipedia article on Reduction.

Chapter 5.1, Undecidable Problems from Language Theory

What is a Reduction?

• A reduction is a way of converting problem A to problem B in such a way that problem B can be used to solve problem A.
• Could problem A be easier than problem B?
• Could problem A have the same difficulty as problem B?
• Could problem A be more difficult than problem B?
• The previous chapter proved that A_TM = { <M, w> | M is a TM and M accepts w} is undecidable.
• A problem can be shown to be undecidable by reducing a known undecidable problem to it.
• (Also, a problem can be shown to be decidable by reducing it to a known decidable problem.)

The Halting Problem is Undecidable

• HALT_TM = { <M, w> | M is a TM and M halts on input w}.
• Proof by Contradiction: Assume HALT_TM is decidable. If this assumption tells us that A_TM is decidable, we have a contradiction and we would then know that HALT_TM is undecidable.
• If HALT_TM is decidable, we can build TM R that decides HALT_TM.
• How does this help us know if TM S that takes input <M, w>, an encoding of TM M and string w, will accept or reject?

E_TM is Undecidable

• E_TM = {<M> | M is a TM and L(M) = empty}.
• Assume TM R can decide E_TM.
• The string that encodes the problem, S = <M, w>
• Construct M₁ such that it rejects all inputs except for w.
• Where does the argument go from here?

EQ_TM is Undecidable

• EQ_TM = {<M₁, M₂> | M₁ and M₂ are TMs and L(M₁) = L(M₂)}.
• Assume TM R can decide EQ_TM.
• The string that encodes the problem is S = <M>
• Construct M₁ to reject all inputs.
• Run R on <M, M₁>. If R accepts, M is empty. If R rejects, M is not empty.

REGULAR_TM is Undecidable

• REGULAR_TM = {<M> | M is a TM and L(M) is a regular language}.
• The proof appears on page 219.

In-Class Notes