Reducibility

Sipser 5.1 (pages 187-198)

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Reducibility

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Driving directions

Western MassCambridge

Boston

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If you can’t drive to London…

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If something’s impossible…

• Theorem 4.11: ATM = {<M,w> | M is a TM and M accepts w}

is undecidable.

• Define:HALTTM =

{<M,w> | M is a TM and M halts on input w}• Is HALTTM decidable?

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The Halting Problem (again!)

• Theorem 5.1: HALTTM is undecidable.

• Proof Idea: –We know ATM is undecidable.

–We need to reduce one of HALTTM or ATMto the other. • Which way to go?

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HALTTM is undecidable.

• Proof: Suppose R decides HALTTM. Define

S = "On input <M,w>, where M is a TM and w a string: 1. Run TM R on input <M,w>. 2. If R rejects, then reject. 3. If R accepts, simulate M on input w until it

halts.4. If M enters its accept state, accept;

if M enters its reject state, reject."

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What about emptiness?

• ETM = {<M> | M is a TM and L(M) = Ø}

• Theorem 5.2: ETM is undecidable.

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A step along the way

• Given an input <M,w>, define a machine Mw as follows.

• Mw = "On input x: 1. If x ≠ w, reject.2. If x = w, run M on input w and

accept if M does.”

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ETM = {<M> | M is a TM and L(M) = Ø}

• Proof:Suppose TM R decides ETM. Define a TM to decide ATM

S = "On input <M,w>: 1. Use the description of M and w to

construct Mw.

2. Run R on input <Mw>.

3. If R accepts, reject; if R rejects, accept."

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With power comes uncertaintyM accepts w L(M) = Ø L(M1) = L(M2)

Turing machines ✗ ✗ ✗

PDA ✔ ✔ ✗

Finite automata ✔ ✔ ✔

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Is there anything that can be done?

• Rice’s Theorem: Testing any nontrivial property of the languages recognized by Turing machines is undecidable!

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We can’t even tell when something’s regular!

• REGULARTM =

{<M> | M is a TM and L(M) is regular} • Theorem 5.3: REGULARTM is

undecidable.

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REGULARTM is undecidable

• Proof: Assume R is a TM that decides REGULARTM.

Define S = "On input <M,w>: 1. Construct TM

M2 = "On input x: 1. If x has the form 0n1n, accept. 2. Otherwise, run M on input w and accept if M

accepts w."

2. Run R on input <M2>.

3. If R accepts, accept; if R rejects, reject."