By Ingo Wegener

Finite capabilities (in specific, Boolean services) play a primary function in laptop technological know-how and discrete arithmetic. This ebook describes representations of Boolean services that experience small measurement for lots of very important features and which enable effective paintings with the represented services. The illustration measurement of vital and chosen capabilities is predicted, top and decrease sure ideas are studied, effective algorithms for operations on those representations are awarded, and the bounds of these strategies are thought of. This e-book is the 1st accomplished description of thought and purposes. examine parts like complexity idea, effective algorithms, facts constructions, and discrete arithmetic will enjoy the idea defined during this booklet. the implications defined inside have functions in verification, computer-aided layout, version checking, and discrete arithmetic. this can be the single e-book to enquire the illustration measurement of Boolean services and effective algorithms on those representations.

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We discuss our BP design with respect to the "variables" z^j. At the end, each "zy-node" is replaced as described in Fig. 1(b), which increases the size by a factor of 2. , ra — 1 (a bit in the /-column represents 2') is bounded generously by n(2°+2 1 H |-2m~1) < n2m, since each column contains at most n bits. , n — 1}. 3. Upper Bound Techniques for BPs 33 In nodes for the values 0 , . . , 2n — 1 are sufficient. We perform the counting in the same way as in the BP for HWBn. If we switch from one column to the next, the sums 2j and 2j + 1 lead to the carry j.

From the definition it follows that each path in an OBDD contains at most one Xj-node. Hence, OBDDs do not contain inconsistent paths. The BPs in 45 46 Chapter 3. 1 leading to an upper bound on the BP size of any / e Bn leads to an OBDD for the variable ordering zi,... , TT = id. The same construction works for each variable ordering. Thus we obtain the following results. 3. The iv-OBDD size of any Boolean function f G Bn is bounded for any variable ordering TT by (3 + o(l))2 n /n. Breitbart, Hunt III, and Rosenkrantz (1995) have improved this result by an upper bound of size (2 + o(l))2 n /n.

We may reformulate our result. We interpret NANDh as a function on N = [22 J variables where N—n variables are dummy variables. Then NAND^ has polynomial- 42 Chapter 2. 10. (i) FO(ZI) = xi and the bounds hold for h = 0. Since NAND(ar, y) = x+y, we get DNF(Ffc)j: 2-DNF(F/ l _i) - 2-CNF(F,,_ 1 ). Since NAND(x,y) =xy, we get DNF(Ffc) < DNF(F/ l _ 1 ) 2 . Now the bounds follow by induction. 8 for S = 0. 1,£44+2,141+3,0:41+4}. We calculate exactly the sum c^ of the absolute values of all Fourier coefficients for dense sets.

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