lecture25

# lecture25 - Bounds on circuit computation so far CSci 5403...

This preview shows pages 1–3. Sign up to view the full content.

1 COMPLEXITY THEORY CSci 5403 LECTURE XXV: CIRCUIT LOWER BOUNDS “Bounds” on circuit computation so far: Size Hierarchy Theorem: For every ƒ(n) = o(2 n /n) there exists a function with complexity ω (ƒ(n)) Shannon bounds: “most” functions have complexity Θ (2 n /n) Relationships to uniform classes ( k, PH languages in SIZE( Θ (n k )), X P/poly Amazing(X), BPP P/poly, NC 1 L NL NC 2 …) This week: concrete bounds, specific languages. Theorem. Define (x 1 ,…,x n ) = i x i mod 2. Then any constant-depth circuit family for (with unbounded fan-in/out ( , ,¬ gates) has exponential size. Corollary. AC 0 . Proposition. NC 1 . Corollary. AC 0 NC 1 . Exponential lower bounds! For a simple function! Idea. A constant-depth ( , ,¬) poly(n)-size circuit can be set to a constant by fixing “few” input bits: But fixing even n-1 bits of the input to does not make the function constant!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Definition. A restriction ρ {0,1,*} n of function ƒ : {0,1} n {0,1} is the function given by ρ (x) i = ρ i , if ρ i {0,1} x i , if ρ i = * We let ƒ| ρ denote the map x ƒ( ρ (x)) Proposition. | ρ {0,1} iff ρ {0,1} n .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern