By Heribert Vollmer
A complicated textbook giving a wide, smooth view of the computational complexity idea of boolean circuits, with broad references, for theoretical computing device scientists and mathematicians.
Read Online or Download Introduction to Circuit Complexity: A Uniform Approach PDF
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Extra info for Introduction to Circuit Complexity: A Uniform Approach
Sample text
Let m, n E N. :.. :.. b Problem: Input: Output: Show that SUB =def max{O, m - n } . E a and b FSIZE-DEPTH(n°< 1 l , 1) . 1 . 6 . Let A = (ai ,ih9,j� n be a Boolean matrix, and let A* = (ai,j h�i,j� n · Prove: For 1 � i, j � n, a;,3 = 1 iff i = j or there is a sequence k1 , k2 , . . , k1 E { 1 , . . , n} , 1 � l � n - 1 , such that ai,k 1 = ak 1 , k 2 = ak 2 , k 3 = · · · = a k, ,j = 1 . 1 . 7. 6 that A* = (A v I) n-l . 1 . 8 . Let B be a bounded fan-in basis, and let C = (V, E, a, ,B, w ) be a circuit over B of size O (n) with one output gate.
We write f =cd g if f ::=;cd g and g ::=;cd f . For f to be reducible to g there must be a polynomial size constant depth circuit family computing f where we allow, besides the standard unbounded fan-in basis, gates for g also. (These gates are sometimes called oracle gates for g, see also Sect. ) The requirement that a gate v computing the function g of fan-in k contributes with the number k to the size of a circuit reflects the "fact that for such a gate it may be reasonable (depending on g) to have more than one input wire from the same predecessor gate (for gates from the standard unbounded fan-in basis, this is not reasonable) .
Le t s(n) be the size of Cn · Then there is a polynomial p such that, for e ve ry n, the highest number of a gate in Cn is bounded by p(s(n)). ) The encoding of a gate v in Cn is now given by a tuple (g, b, 91 . . , 9 k) , whe re g is the number of v, b is the number of the type of v, and 9I , . . , 9 k are t he numbe rs of the prede c essor gates of v in the order of the edge numbe ring ofCn. F ix an arbitrary orde r of the gates of Cn. L et VI, . . , v8 be the gate s of in that order. Le t VI, .