22C:131 Final Exam
Close books and notes, except three sheets of notes
Total points = 100
1. (30) Decide if the following language L1is decidable. If yes, provide an algorithm to solve
the problem; if not, prove it formally (you may assume any undecidable language discussed
in the class).
L1={hM, si | Mis a TM, sis a state to be used by M}
2. (30) Let SET-SPLITTING ={hS, Ci | Sis a finite set and C={C1, ..., Ck}is a collection
of subsets of S, for some k > 0, such that elements of Scan be colored red or blue so that
no Cihas all its elements colored with the same color.}Show that (a) SET-SPLITTING
is in NP; (b) 3SAT ≤PSET-SPLITTING.
3. (20) Given a set X={x1, x2, ..., xn}of nreal numbers, we have two computation jobs to
perform on X:
(a) sort X;
(b) compute P1≤i,j≤n√xixj/n2.
Show that there exists a function f(n) such that one job is in TIME(f(n)) but the other
job is not in TIME(f(n)).
4. (20) Let Mbe a probabilistic polynomial time turing machine and let Cbe a language
where, for some fixed 0 < 1< 2<1,
(a) w6∈ Cimplies Pr[Maccepts w]≤1, and
(b) w∈Cimplies Pr[Maccepts w]≥2.
Show that C∈BPP.
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