CS748 - ARITHMETIC CIRCUIT COMPLEXITY

NITIN SAXENA

ASSIGNMENT 2

POINTS: 40

DATE GIVEN: 05-FEB-2016 DUE: 15-FEB-2016

Rules:• You are strongly encouraged to work independently.• You are allowed to discuss; or, refer to articles. But write the so-lutions on your own and honorably acknowledge the sources if any.http://cse.iitk.ac.in/pages/AntiCheatingPolicy.html

Question 1: [6 points] Recall the proof of O(log d)-depth reduction, ofsize-s degree-d n-variate circuits, done in the class. Fill in the detailsshowing that the final size bound is indeed poly(sn log d).

Question 2: [5+3 points] Given a circuit C of size s and depth d, showthat the circuit D computing all the first-order partial derivatives of Chas depth O(d), and size O(s). [The latter was covered in the class.]

What is the best that you can say about D in terms of degree δ ofC (ignoring d)?

Question 3: [9 points] Recall the proof of “det ∈ VP” that we sawusing Newton’s identities. It was claimed that a linear system of n equa-tions that looks triangular can be solved by O(log n)-depth poly(n)-sizearithmetic circuits. Prove this claim.

Question 4: [7 points] Show that, over an integral domain R, thematrix (αj

i )i,j∈[n] is invertible iff αi’s are distinct in R.

Question 5: [10 points] Over a field F, show that the monomialx1 . . . xn has a Σ ∧ Σ expression iff n! 6= 0 in F.

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