Honors Theory of Computation, Spring'99
Homework 5
-- DATE DUE: May 3, 1999
-- PROBLEMS:
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This simple exercise (suggested by Emily Chapman)
tests your basic understanding of probabilistic computation.
To determine if a binary number n is odd,
the obvious algorithm is, of course, to determine the predicate
LSB(n)=``the least significant bit of n is 1''.
Consider the following (unusual) algorithm:
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LOOP:
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(1) Toss a coin. If head, then output LSB(n).
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(2) If tail, let n\ass n+2 and go back to LOOP.
- (i) Let T be the complete computation tree on input n.
Determine the least fixed-point valuation ValT.
- (ii) Does this algorithm have bounded-error? zero-error?
- (iii) Determine the running time t(m) of this algorithm
on inputs of length m,
assuming LSB(n) takes lg(n) steps to compute.
- (iv) Determine the average running time t'(m)
of this algorithm.
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Show that BPP is closed under union.
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Consider the following ``counting'' version of 3SAT:
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#SAT := { <F,k> : F is a 3CNF formula
with exactly k satisfying assignments.}
Prove that #SAT is complete for the class PP (under Karp
reducibility).