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LECTURE 12-13 Complete Problems in PH. THEOREM. PROOF. BOUNDED HALTING PROBLEM (BHP). THEOREM. PROOF. A. BHP. LEMMA. PROOF. LEMMA. PROOF. DEFINITION. THEOREM. PROOF by induction on k. Natural Problem. (Machine unrelated). Machine Related Problem. SAT k. THEOREM.

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## LECTURE 12-13 Complete Problems in PH

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**THEOREM**PROOF**BOUNDED HALTING PROBLEM (BHP)**THEOREM PROOF**A**BHP LEMMA PROOF**LEMMA**PROOF**DEFINITION**THEOREM PROOF by induction on k**Natural Problem**(Machine unrelated) Machine Related Problem**SAT k**THEOREM**3-CNF SAT k**THEOREM**THEOREM**PROOF Construction Such that**Variables**Clauses**Variables**Clauses**Variables**Clauses**Variables**Clauses**Variables**Clauses**Variables**Clauses**Variables**Clauses**Case 1**Answer:**Case 2**Answer:**Case 2**Answer:**Consider**To prove Define a coloring c’: No red clique of size K exists.**Consider**To prove Define a coloring c’: No red clique of size K exists. There must exist a blue clique Q of size K.**PPROPERTY OF BLUE CLIQUE Q**Assignment for y Then F(X,Y) is satisfied

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