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hwsol2.1

# hwsol2.1 - CSci 5403 Spring 2010 Stefan Nelson-Lindall...

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CSci 5403, Spring 2010 Stefan Nelson-Lindall Homework 2 Example Solution 1. Games. Prove that the following problems are PSPACE -complete: (a) A Stochastic Satisfiability instance is a quantified boolean formula, only the universal quantifier is replaced by the random R quantifier (meaning, “for a random x”). An instance x 1 R x 2 x 3 . . . Q n x n φ is in ssat if x 1 R x 2 x 3 . . . Q n x n Pr[ φ ( x 1 , . . . , x n )] 1 2 . Show that ssat is PSPACE -complete. In some sense, ssat is a “game against nature,” e.g. we can think of the player in a formula game instance playing against an opponent who is not explicitly adversarial, but simply moves randomly at each stage. It might seem like this should be an easier game to win, but your reduction shows this is not always true. Pf: ssat is PSPACE -hard Reduction from fg . Map the formula game instance f = Q 1 x 1 . . . Q i x i φ ( x 1 , . . . x i ) to the ssat problem f = R x 0 Q 1 x 1 ... Q i x i ( x 0 φ ( x 1 , . . . x i )) where Q k = if Q k is R if Q k is . x 0 will be assigned 1 with probability 1 2 , and Pr[ x 0 φ ] = Pr[ x 0 ] · Pr[ φ ]. If f fg , then for any possible assignment of random variables the resulting formula is satisfiable, so Pr[ φ ] = 1, Pr[ x 0 φ ] = 1 2 , and hence f ssat . If f fg , then there exists some assignment of random variables such that the resulting formula is not satisfiable. This makes Pr[ φ ] < 1, so Pr[ x 0 φ ] < 1 2 , and hence f ssat . Thus, fg P ssat . ssat PSPACE Iterate through each possible assignment of variables with the R quantifier, in each checking each possible assignment of variables with the quantifier to determine whether or not φ can be satisfied.

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hwsol2.1 - CSci 5403 Spring 2010 Stefan Nelson-Lindall...

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