# The Complexity of Boolean Functions (Wiley Teubner on by Ingo Wegener

By Ingo Wegener

Offers various contemporary learn effects formerly unavailable in publication shape. in the beginning bargains with the wee-known computation types, and is going directly to particular sorts of circuits, parallel desktops, and branching courses. comprises easy concept to boot contemporary study findings. each one bankruptcy contains workouts.

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**Example text**

For the other rectangles let j′ = d(l j) − (r − 1) 2h(l ) − 1 . By definition of d(l j) we can find some k such that d(l j) = k 2h(l ) , thus j′ = (k − r) 2h(l ) + 2h(l ) − 1 . Furthermore d(l j) − r 2h(l ) = (k − r) 2h(l ) = d(l j′) by definition of d . So also the other rectangles are of the form Vj′′ d(l j′ ) and are computed before step l . The triangles are computed in t − 1 steps (2 ≤ l ≤ t). 25) is correct by our standard arguments. The rectangles are computed before step l as has been shown above.

So log3 2 n + 1 steps are sufficient to reduce the number of summands to 2 . 1 : The school method for multiplication implemented with CSA gates and a Krapchenko adder leads to a circuit of size O(n2) and depth O(log n) . This multiplication circuit is asymptotically optimal with respect to depth. It is hard to imagine that o(n2) gates are sufficient for multiplication. Let us try a divide-and-conquer algorithm. Let n = 2k and let x = (x′ x′′ ) and y = (y′ y′′ ) be divided into two parts of length n 2 .

For all those who are not familiar with the NP-theory we give the following short explanation. Many (more than 1000) problems are known to be NP-complete. It can be proved that one of the following statements is correct. Either all NP-complete problems have polynomial algorithms or no NP-complete problem may be solved by a polynomial algorithm. The conjecture of most of the experts is that the second statement holds. One of the well-known NP-complete problems is the set cover problem which we prove to be equivalent to our minimization problem.