Limits of Algorithmic Computation Wen-Guey Tzeng Computer Science - - PowerPoint PPT Presentation

Limits of Algorithmic Computation

Wen-Guey Tzeng Computer Science Department National Chiao Tung University

The halting problem

• Problem: given an encoding string <M, w>,

determine whether M(w) halts or not.

– Solvable? – We want an “yes/no” answer for each input <M, w>.

• <M, w> = 10110101011 00 11011010111011 … 000

11011101110111011

• Methods:

– Run M(w) in TMU – Analyze M – …

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The halting problem: really hard?

• Can we just run M(w) for a long time?

– If it hasn’t halted yet, we just claim that M(w) does not halt.

• The busy beaver problem will show that it is

indeed M(w) can run a very very very long time and halt.

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Theorem: The halting problem cannot be solved by any algorithm (Turing machine). Proof. Step 1: Assume that it can be solved by some TM H.

– The initial state q0. – qY and qN are both final states. – q0 <M,w> |-* x1qY x2 (halt) if M(w) halts – q0 <M,w> |-* y1qN y2 (halt) if M(w) does not halt

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4

<M, w>

H

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Step 2: We construct H’, from H, as follows:

– q0 <M,w> |-* x1qYx2 |-*  if M(w) halts – q0 <M,w> |-* y1qN y2 (halt) if M(w) does not halt

H’

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<M, w>

Step 3: We construct , from H’, as follows:

– Its input is the encoding <M> of a TM – It copies the input <M> and runs H’ on <M><M>

• q0 <M> |-* q0 <M><M> |-* 

if M(<M>) halts

• q0 <M> |-* q0 <M><M> |-* y1qN y

if M(<M>) does not halt <M><M>

<M>

Ｈ’

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Step 4: Now, feed < > into :

– q0 < > |-* q0 < > < > |-*  if (< >) halts – q0 < > |-* q0 < > < > |-* y1qNy2 (halt) if (< >) does not halt

Conclusion:

• It is a contradiction !!!
• Thus, H does not exist. 

< >< >

<𝐼 >

Ｈ’

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