Design and Analysis of Algorithms - PowerPoint PPT Presentation

�������� Design and Analysis of Algorithms ����������������������� ������ �����ก��� ������ ������������ ���������������������� ��. ��. ����� ��������������ก�� ��������������������������������������������������ก�� �����������ก�������������� ����� ������ก�� ��������������� ������ก�������������� �������������กก���������ก���������� 2542 http://www.cp.eng.chula.ac.th/faculty/spj Outline Undecidable Problems � Noncomputability / Undecidability � Tiling Problem � Word Correspondence Problem Introduction � Halting Problem � Proving Undecidability http://www.cp.eng.chula.ac.th/faculty/spj

Algorithmic Problems Tiling Problem A finite set of T of tile descriptions � Tractable problems � admitting efficient algorithms Fixed orientation � Intractable problems Can any finite area of � admitting no efficient algorithms any size be �beatifully� � � Undecidable problems covered using only tiles � admitting no algorithms ! ! ! in T ? http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Tiling Problem Tiling Problem A finite set of T of tile descriptions A finite set of T of tile descriptions Fixed orientation Fixed orientation Can any finite area of Can any finite area of any size be �beatifully� any size be �beatifully� � � covered using only tiles covered using only tiles in T ? in T ? http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Tiling Problem Tiling Problem A finite set of T of tile descriptions � Tiling problem is undecidable � There is no algorithm (and never will be) for Fixed orientation solving the tiling problem ! Can any finite area of � If one claims A can solve the problem, any size be �beatifully� there will be input sets T upon which � covered using only tiles A runs forever or terminates with wrong answer in T ? http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Word Correspondence Word Correspondence abb bab abb bab a a baba baba aba aba Two groups of words over some finite alphabets aabbabbbabaabbaba bbab bbab ab ab aa a aa a http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Word Correspondence Unboundedness abb bab a � � ������� undecidable ���� ���������������� baba aba ก����������ก������������� � � � �������������������������ก������� � aabbabbbabaabbaba exponential �������� intractable problem� Hamiltonian path & Euler path bbab NP-Complete P ab aa a http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Domino Snake Domino Snake A finite set of T of tile descriptions A finite set of T of tile descriptions T Decidable Undecidable S http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Halting Problem Halting Problem for i=1 to n while x �� 1 do for j=1 to n if x is even for k=1 to j then x = x/2 sum += 1 else x = 3x+1 ��� n = 10000000000000000 ��� x = 7 ���ก���������� loop ������� ? ���ก���������� loop ������� ? 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, ... http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Proving Undecidablity Proving Undecidablity W program input P X S( W ) if Q(W,W) = ‘y’ W P W X loop forever else Q : Does P halt on X ? return S Q P ( X ) � P ( X ) � � � ��� W ( W ) ���� S ( W ) ��� loop Y N ��� W ( W ) ��� loop S ( W ) ���� �YES� �NO� http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Proving Undecidablity Proving Undecidablity W S W S ���ก������ S ( S ) ���� ���ก������ S ( S ) ���� ������� loop ? ������� loop ? W P S W X S P S W X W S S S ��� ��� Q Q S ( S ) ���� S ( S ) ��� loop � � ��� loop Y N Y N Q ���� Q ���� ���� http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Proving Undecidablity Diagonalization Method W 6 N Y N Y Y Y ... 5 Y N Y Y Y N ... all 4 Y Y Y Y N N ... Halting problem programs W P W X 3 N N N N N N ... is undecidable S 2 Y Y Y Y N N ... Q S ������������� 1 Y Y N N Y N ... 1 2 3 4 5 6 ... all inputs Y N program S N N Y N N N ... http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Computability � Undecidable � � Intractable � � Tractable � in principle in practice http://www.cp.eng.chula.ac.th/faculty/spj