On additive complexity of a sequence of matrices
نویسنده
چکیده
The present paper deals with the complexity of computation of a sequence of Boolean matrices via universal commutative additive circuits, i.e. circuits of binary additions over the group (Z, +) (an additive circuit implementing a matrix over (Z, +), implements the same matrix over any commutative semigroup (S, +).) Basic notions of circuit and complexity see in [3, 5]. Denote the complexity of a matrix A over (Z, +) as L(A). Consider a sequence of n × n-matrices An with zeros on the leading diagonal and ones in other positions. It is known that L(An) = 3n− 6, see e.g. [2]. In [4] it was proposed a sequence of matrices Bp,q,n more general than An and the question of complexity of the sequence was investigated. Matrix Bp,q,n has C q n rows and C p n columns. Rows are indexed by q-element subsets of [1..n]; columns are indexed by p-element subsets of [1..n] (here [k..l] stands for {k, k+1, . . . , l}). A matrix entry at the intersection of Q-th row and P -th column is 1 if Q ∩ P = ∅ and 0 otherwise. Consider some simple examples of Bp,q,n. If n < p + q then Bp,q,n is zero matrix. Evidently, B1,1,n = An. By the symmetry of definition Bp,q,n = B q,p,n. Matrices Bp,0,n and B0,q,n are all-ones row and column respectively. So, L(Bp,0,n) = C p n − 1, L(B0,q,n) = 0. Note that by the transposition principle (see e.g. [3]) complexity of matrices Bp,q,n and Bq,p,n satisfies the identity
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ورودعنوان ژورنال:
- CoRR
دوره abs/1209.1645 شماره
صفحات -
تاریخ انتشار 2012