A word circuit [1] is a directed acyclic graph in which each edge holds a w-bit word (i.e. some x {0,1} w ) and each node is a gate computing some binary function g:{0,1} w ×{0,1} w →{0,1} w . The following problem was studied in [1]: How many binary gates are needed to compute a ternary function f:({0,1} w ) 3→{0,1} w . They proved that (2+o(1))2 w binary gates are enough for any ternary function, and there exists a ternary function which requires word circuits of size (1-o(1))2 w . One of the open problems in [1] is to get these bounds tight within a low order term. In this paper we solved this problem by constructing new word circuits for ternary functions of size (1+o(1))2 w . We investigate the problem in a general setting: How many k-input word gates are needed for computing an n-input word function f:({0,1} w ) n →{0,1} w (here n≥ k). We show that for any fixed n, (1-o(1))2 (n-k)w basic gates are necessary and (1+o(1))2(n-k)w gates are sufficient (assume w is sufficiently large). Since word circuit is a natural generalization of boolean circuit, we also consider the case when w is a constant and the number of inputs n is sufficiently large. We show that basic gates are necessary and sufficient in this case. © 2010 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Chen, X., Hu, G., & Sun, X. (2010). The complexity of word circuits. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6196 LNCS, pp. 308–317). https://doi.org/10.1007/978-3-642-14031-0_34
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