By Ingo Wegener

Provides a good number of contemporary study effects formerly unavailable in e-book shape. in the beginning bargains with the wee-known computation types, and is going directly to precise different types of circuits, parallel desktops, and branching courses. contains easy idea to boot contemporary study findings. each one bankruptcy contains routines.

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Output : The nonempty sets Qk and Pk of all implicants and prime implicants resp. of f with length k . In particular PI(f) is the union of all Pk . Qn is the set of all minterms ma such that f(a) = 1 , i = n . While Qi = ◦ i := i − 1 ; Qi := {m | ∃ j : xj , xj are not in m but m xj m xj ∈ Qi+1 } ; Pi+1 := {m ∈ Qi+1 | ∀ m′ ∈ Qi : m′ is not a proper submonom of m } . 1 the sets Qk are computed correctly. Also the sets of prime implicants Pk are computed correctly. If an implicant of length k has no proper shortening of length k − 1 which is an implicant, then it has no proper shortening which is an implicant and therefore it is a prime implicant.

If we still have 3k + i summands where i ∈{0,1,2} we may reduce the number of summands to 2k + i by k parallel CSA gates. Let n0 = n and let nj be the number of summands after the j -th step. 2) For j = log3 2 n , we conclude nj ≤ 3 . 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.

2) for j + 1 times we can compute cj from the u- and v-parameters. 8) ui vi+1 · · · vj This can be interpreted in the following way. Carry cj = 1 iff for some i ≤ j at position i we have two ones (ui = 1) and at the positions 43 i+1 j exactly a zero and a one (vi+1 = · · · = vj = 1). 10) In particular cj = Gj 0 . In Fig. 1 we give a triangle representation for Gd a where one has to combine the rows by disjunctions. Let a b d . Since Gd a : ua va+1 vb−1 vb ua+1 vb−1 vb ········· ··· ub vb+1 vb+1 ··· vb+1 vb+2 vd−1 vb+2 vd−1 ········· vb+2 vd−1 vd vd Vd b+1 ··· vd ub+1 vb+2 · · · vd−1 vd ········· ··· ud−1 vd ud Gb a Gd b+1 Fig.

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