1 / 5

Circuit complexity

Giorgi Japaridze Theory of Computability. Circuit complexity. Section 9.3. 9.3.a. Giorgi Japaridze Theory of Computability. Boolean circuits. A Boolean circuit is a collection of gates and inputs connected by wires . Cycles

maina
Download Presentation

Circuit complexity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Giorgi Japaridze Theory of Computability Circuit complexity Section 9.3

  2. 9.3.a Giorgi JaparidzeTheory of Computability Boolean circuits A Boolean circuit is a collection of gates and inputs connected by wires. Cycles aren’t permitted. Gates take thee forms: AND, OR, and NOT. One of the gates (sometimes more) is designated the output gate. Every circuit C with n inputs computes some n-ary Boolean function, i.e. a function of the type {0,1}n{0,1}.The following animation illustrates a Boolean circuit C at work. We see that C(010)=1. 0 1 0 inputs x1 x2 x3 ¬ ∧ ∧ 1 0 0 ∨ ¬ 1 1 output ∧ 1

  3. 9.3.b Giorgi JaparidzeTheory of Computability Examples What functions are computed by the following circuits? x1 x2 x3 ∧ ∧ ∧ ∨ x1 x2 x3 ∨ ∨ ∨ ∧

  4. 9.3.c Giorgi JaparidzeTheory of Computability Main definitions A circuit family C is an infinite list (C0,C1,C2,…) of circuits where Cn has n input variables. C is said to decide a language A over {0,1} if, for every bitstring w of length n, wA iff Cn(w)=1. The size of a circuit is the number of its gates. The depth of a circuit is the length (number of wires) of the longest path from an input variable to the output gate. A circuit is size minimal if no smaller circuit is equivalent to it (i.e. computes the same function). “Depth minimal” is defined similarly. The size complexity of a circuit family (C0,C1,C2,…) is the function f: NN, where f(n) is the size of Cn. Depth complexity is defined similarly. The circuit size complexity of a language is the size complexity of a minimal circuit family for that language. Circuit depth complexity is defined similarly.

  5. 9.3.d Giorgi JaparidzeTheory of Computability Main theorems Theorem 9.30 Let t: NN be a function, where t(n) ≥ n. If ATIME(t(n)), then A has circuit size complexity O(t2(n)). Corollary Every problem in P has a polynomial circuit size complexity. Lots of efforts have gone into attempts to show that certain NP-problems have no polynomial size circuits (and hence, in view of the above corollary, P≠NP). However, the progress has been very slow. A circuit is said to be satisfiable if there is an assignment for its variables such that the circuit outputs 1. Theorem 9.33 The circuit satisfiability problem is NP-complete.

More Related