By Wegener I.

Offers a number of fresh examine effects formerly unavailable in e-book shape. in the beginning bargains with the wee-known computation versions, and is going directly to particular sorts of circuits, parallel pcs, and branching courses. comprises uncomplicated idea besides contemporary examine findings. every one bankruptcy contains workouts.

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Additional resources for The complexity of boolean functions

Example text

Let z0j ((2i − 1)L ≤ j ≤ 2iL ) and z1j be the output bits of P2i l −1 0 and P2i l −1 1 resp. 4) in depth 2 using 3 gates. Altogether step l can be realized in depth 2 using for each of the 2k−l · 2 problems 3(2l −1 + 1) gates. The circuit size of step l is 3(2k + 2k−l+1) . 2 : The Conditional Sum Adder may be realized (if n = 2k ) in depth 2 log n + 1 and size 3n log n + 10n − 6 . 42 According to our remark at the beginning of this chapter the depth of this circuit is only double the size of the lower bound.

We argue that with high probability no such algorithm exists. For this purpose we use the concept of NP-completeness (see Garey and Johnson (79)). 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.

Hence it happens that any sequence of Boolean functions fn ∈ Bn may be computed by (a sequence of) circuits while not all sequences fn ∈ Bn can be computed by a computer or a Turing machine. Furthermore, it is not astonishing that Turing machine programs may be simulated eﬃciently by circuits (see Ch. 9). Because of our way of thinking most of the sequences of circuits we design may be described uniformly and therefore can be simulated eﬃciently by Turing machines. EXERCISES 1. What is the cardinality of Bn m ?