Design and Analysis of Algorithms

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Design and Analysis of Algorithms

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2542

http://www.cp.eng.chula.ac.th/faculty/spj

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Undecidable Problems

Introduction

Outline

Noncomputability / Undecidability Tiling Problem

Word Correspondence Problem Halting Problem

Proving Undecidability

Algorithmic Problems

Tractable problems

admitting efficient algorithms

Intractable problems

admitting no efficient algorithms

Undecidable problems

admitting no algorithms ! ! !

Tiling Problem

Fixed orientation A finite set of Tof tile descriptions

Can any finite area of any size be beatifully covered using only tiles in T ?

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Tiling Problem

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Tiling Problem

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Tiling Problem

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Tiling Problem

Tiling problem is undecidable

There is no algorithm (and never will be) for solving the tiling problem !

If one claims

A

can solve the problem, there will be input sets

T

upon which

A

runs forever or terminates with wrong answer

Word Correspondence

abb a baba bab aba

bbab ab aa a

Two groups of words over some finite alphabets

Word Correspondence

abb a baba bab aba

bbab ab aa a

aabbabbbabaabbaba

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Word Correspondence

abb a bab

baba aba bbab ab aa a

aabbabbbabaabbaba -

Unboundedness

!"# undecidable $%& '()*+ !"# ),-."/

0)12 %*#&*0*3'45" 6 7

489:*;2< !"# ),-."/-=*,0)12'45 exponential !"#'45 intractable problem7

Hamiltonian path & Euler path

NP-Complete P

S

T

Domino Snake

A finite set of Tof tile descriptions

Domino Snake

A finite set of Tof tile descriptions

Undecidable Decidable

Halting Problem

for i=1 to n for j=1 to n

for k=1 to j sum += 1

A:!

n = 10000000000000000

"B0")C;D2E CF loop :)G"H% ?

Halting Problem

while x 1 do if x is even

then x = x/2 else x = 3x+1

A:!

_{x = 7}

"B0")C;D2E CF loop :)G"H% ?

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, ...

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Proving Undecidablity

Q: Does Phalt on X?

YES NO

P(X) P(X) _-

P

program X input

Proving Undecidablity

Q -

P X

Y N

W

W W

S

S( W )

if Q(W,W) = ‘y’

loop forever else

return

234 W(W) 5678 S( W) 9:8 loop 234 W(W) 9:8 loop S ( W) 5678

Q P X

Y N

W

W W

S

;64<=>3?@4 S( S) 5678 5=A;9:8 loop ? S

S S

TCF

- S( S) 5678

Q BCD?

loop

Q P X

Y N

W

W W

S

;64<=>3?@4 S( S) 5678 5=A;9:8 loop ? S

S S

TCF

S( S) 9:8 loop

Q BCD?

"

Q P X

Y N

W

W W

S

Proving Undecidablity

Halting problem is undecidable

Diagonalization Method

6 N Y N Y Y Y ...

5 Y N Y Y Y N ...

4 Y Y Y Y N N ...

3 N N N N N N ...

2 Y Y Y Y N N ...

1 Y Y N N Y N ...

1 2 3 4 5 6 ...

programsall

all inputs

N N Y N N N ...

program S

S

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Computability

Tractable Intractable Undecidable

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/ /

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- in principle in practice