By Heribert Vollmer

ISBN-10: 3540643109

ISBN-13: 9783540643104

A complicated textbook giving a extensive, glossy view of the computational complexity idea of boolean circuits, with huge 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**

UCOUNT :5 cd SORT . :5 i :5 n . Sort these n- 1 numbers. The sequence of most significant bits in the ordered sequence is unn ( 2:�=1 ai ) reversed. Cl Proof. Given a1 , . . , a n , define A i 0 · · 0 for 1 =de£ a i .... 4 Constant-Depth Reductions 25 We now prove the converse. 44. SORT Proof. Given are ai �cd UCOUNT . = ai , n - 1 · · · ai , o , 1 � i � n. For 1 � i , j � n define For fixed j , 1 � j � n, determine ci =ctef UCOUNT(c 1 ,j , . . , Cn,j ) · Then Cj is the n-bit unary representation of the position of ai in the ordered output sequence.

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. Show that lEI = 8(1VI) . Hence to evaluate the size of C it is sufficient to determine lEI .

17. SIZE(n° < 1l ) = P/Poly. We now come to a simulation of dept h-bounded circuits by non-uniform space-bounde d Turing machines. I t will be conve nient if we assume that the graph underlying our circu it is actually a tree . An arbitrary circuit can be unwound into a t re e as follows: Le t C = (V, E, a, {3, w ) be a Boole an circuit with n inputs and one ou tput. Le t d be the depth of C. 3 Non-uniform Turing machines 45 After the completion of stage m we will have ensured that all nodes v at depth m (i.

### Introduction to Circuit Complexity: A Uniform Approach by Heribert Vollmer

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