Endterm
CS6520
Computational Complexity
Solve all 6 questions. The solutions are due by 5:00 pm on Monday, April
30. Some hints are given on page 3. All the best!
Problems
1. A DNF formula in (boolean) variables x1, x2, . . . , xnis of the form
φ=D1∨D2· · · ∨ Dm
where for 1 ≤i≤m, Di=yi1∧yi2. . . ∧yik
and each yjis a variable or its negation. Show that deciding if a DNF formula is
satisfiable is in Pbut counting the number of satisfying solutions is #P-complete.
2. Let Lbe the language accepted by a family of circuits {Cn}which consist of AND,
NOT and PARITY gates such that
•Circuit Cnhas ninputs, size 2nO(1) and depth O(1).
•AND gates have fan-in bounded by poly(n).
•PARITY gates have unbounded fanin.
•The circuits Cnare uniformly generated by a polynomial time DTM M.
Show that L∈ ⊕P. In other words show that there is a polynomial time NTM N
which has an odd number of accepting computations on input xiff x∈L.
3. Let Z3={0,1,−1}be the field of integers modulo 3. We say that a polynomial
P(X1,· · · , Xn) in nvariables is multilinear if the degree of each Xiin Pis at most
1. For instance P(X1, X2, X3) = X1X2+X2X3is multilinear but X2
1+X2
2is not.
•Show that every function f:{0,1}n→Z3is computed by a unique multilinear
polynomial in Z3[X1,· · · , Xn].
•Consider all Boolean functions f:{0,1}n→ {0,1}. Let the degree of function f
be the degree of the unique polynomial computing f. Show that AND and OR
functions have degree n.
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