Summary of previous class

1
Summary of previous class

• Completeness for coNP co-SAT and Taut
• Proof systems for SAT and super proof systems for
  SAT
• Resolution soundness and completeness
• Hard formulas for resolution
• PTM definition use of random numbers
• Probability of one trace and runtime
• Class of Las Vegas algorithms - ZPP

2
Monte Carlo algorithms main direction

• Bounded runtime but one sided-error algorithm
  never output 1 when x?L but may output 0 when
  x?L.
• Define PTM M? RTIME(T(n)) if the following
  conditions hold
• x?L ? Pr(M accepts x)?1/2 ? for some ?gt0
• x?L ? Pr(M accepts x)0
• TimeM(n) O(T(n))
• One-sided Monte Carlo class (main direction)
  RPUcgt0RTIME(nc).

3
Independent Runs of Monte Carlo algorithm

• Let a1, , ak be answers of independent runs of a
  Monte-Carlo algorithm A on x
• If aiyes for some 1?i?k then we know with
  certainty that x?L
• If a1 akno then the probability x?L is at
  most (1/2)k. So we decide that x?L. This is
  correct with probability at least
• 1-(1/2)k.

4
Example of Monte Carlo Algorithms

• Polynomial Identity Problem Given two
  polynomials P(x) and Q(x), does the identity P(x)
  ? Q(x) hold?
• Example of instance
• (x1)(x-2)(x3)(x-4)(x5)(x-6) ? x6-7x325

5
Algorithm for Polynomial Identity (PI)

• Input PI polynomials P(x),Q(x)
• Output PI accept if P(x)?Q(x) or reject
  otherwise
• Pick a prime n s.t. ngtgt3d and n is bigger than
  all coefficients in G(x)P(x)-Q(x)
• Choose a value a from Zn0,1,,n
• Evaluate G(x/a) doing all computations mod n.
• If G(x/a)?0 then output P(x)?Q(x) (accept) else
  output P(x)Q(x) (reject)

6
Analysis -Polynomial Identity

• Accept polynomials are over the field Zn that
  are non-identical i.e. the language is
• L(P(x), Q(x)) P(x), Q(x)?Znx and P(x)
  ?Q(x))
• P(x), Q(x) are both of degree d. For
  G(x)P(x)-Q(x) either G(x) ? 0 or (G(x) ? 0).
• If (G(x) ? 0) then G(X) has at most d roots.
• Base if d0 then G(x) has 0 roots.
• Induction step if a is the root of G(x) then
  ?R(x) of degree d-1 s.t. G(x)(x-a)R(x).
• If P(x) ?Q(x) then (G(x) ? 0)
  Prob(G(x/a)0)?d/n so Prob(PI accpt P(x)?Q(x))1-
  Prob(G(x/a)0)?1-1/3
• If P(x) ?Q(x) the algorithm never errs

7
Monte Carlo algorithms reverse direction

• Bounded runtime but one sided-error algorithm
  may output 1 when x?L but never output 0 when
  x?L.
• Define PTM M? coRTIME(T(n)) if the following
  conditions hold
• x?L ? Pr(M accepts x)1
• x?L ? Pr(M rejects x)? 1/2 ? for some ?gt0
• TimeM(n) O(T(n))
• One-sided Monte Carlo class (main direction)
  coRPUcgt0 coRTIME(nc) and coRPL L?RP

8
Monte Carlo and Las Vegas

• Suppose we have
• A one-sided error Monte Carlo algorithm for
  PRIMES main direction
• no is always correct
• yes is correct with small probability of error.
• B one-sided error Monte Carlo algorithm for
  PRIMES reverse direction
• yes is always correct
• no is correct with small probability of error.
• Can we use A and B as subroutines to get a Las
  Vegas algorithm for PRIMES?
• Theorem ZPPRP?coRP (prove!)

9
Two-sided error algorithms

• Bounded runtime but two sided-error algorithm
  may output 1 when x?L but may output 0 when x?L.
• Define PTM M? BPTIME(T(n)) if the following
  conditions hold
• Pr(M(x)L(x))?1/2 ? for some ?gt0, where we
  denote by L(x) value accept if x?L and reject
  if x?L
• TimeM(n) O(T(n))
• Class of two-sided error algorithms is defined as
  BPPUcgt0BPTIME(nc).

10
Example primality testing

• Theorem 1 Let n?i1k qi?I be the factorization
  of a number that is not a prime or a prime power.
  Then the number of solutions of an equation x2a
  mod n is either 0 or 2k.
• In particular, given a prime (or prime power) p
  then the an equation x2a mod p has either 0 or
  2 solutions.

11
Example primality-testing (continued)

• Input a number p and a safety parameter s
• Output prime or composite
• If P either even or prime power return prime
• Choose x?Zp uniformly at random
• If gcd(x,p) ? 1 return composite
• Compute a x2 mod p
• V MSQRT(a,p,s)
• If (Vfail) or (V2 ? x2 mod p) or (V?x,-x)
  then return composite else return prime

12
Example primality-testing (continued)

• Here we use algorithm MSQRT(a,p,s) that solves
  the equation x2a mod p in polytime with
  probability of failure 2-s.
• If p is prime then primality-test returns prime
  with probability 1-2-s (error when MSQRT fails)
• If p is composite primality-test returns
  composite with probability gt1/2 MSQRT could
  output 2k values (theorem) so probability that it
  returned either x or x is at most (1-2s)2/2k.
  Since k?2 we have P(prime)lt1/2-?

13
Independent Runs of two-sided error algorithm

• A randomized two-sided error algorithm deciding
  L with probability ½?
• For an input x run t times A on x saving outputs
  a1,,at (take t odd)
• If among a1,,at at least ?t/2? answers are
  accept then output accept otherwise output
  reject

14
Analysis of independent runs

• Probability that A outputs incorrect answer
  (t-i)-out-of-t times (i.e i time correct) is

Therefore, due to the way algorithm chooses
answer