On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions
On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions
复制标题
DOI：
10.1016/s0166-218x(99)00099-2
复制
发表时间：
1999-10-15
影响因子：
1.1
通讯作者：
Khachiyan, L
中科院分区：
文献类型：
--
作者：
Gurvich, V;Khachiyan, L
令f：{0，1}（n） - > {0，1}为单调布尔函数，其值在任何点x是{0，1}（n）的元素，可以在时间t中确定。用c = boolean表示，（i是c的元素）boolean ori是i x的元素（i）f的不redlectimenlent cnf，其中c是f的主要含义的集合。同样，令d = boolean orj是d boolean的元素，（j是j）x（j）的元素是同一函数的不reredluctiond dnf，其中d是f的主要隐含物的集合。我们表明，给定的子集C c'子集或等于c和d'子集或等于d的子集，以至于（c'，d'）不等于（c，d），是（c \ c'）布尔值的新术语或（d \ d'）可以在时间O（n（t + n）） + m（o（log m））中找到，其中m = \ c'\ + + \ d'\。特别是，如果每个X可以评估f（x）是多项式时间{0，1}（n）的元素，则可以在逐步的准级别时间时间中共同生成表格C和D。另一方面，即使对于布尔值和布尔式or-formulae f of Depth 2（即，对于CNFS或DNF），也不太可能携带F素中含义和本性的均匀采样的均匀采样。在f的输入尺寸中，以准多项式2（Polylog（。））的限制。我们还表明，对于某些类别的多项式计算单调布尔函数，测试d'= d或c'= C的条件中的NP hard是np-hard。这提供了证据，表明这些类别既不结合也不是不合时宜的。正常形式可以在总（或增量）准多物理时间中产生。这样的单调布尔值在游戏理论，网络和继电器接触电路，凸面编程中自然出现，并包括布尔值的子集和深度的布尔值或形式。
Let f : {0, 1}(n) --> {0, 1} be a monotone Boolean function whose value at any point x is an element of {0, 1}(n) can be determined in time t. Denote by c = boolean AND(i is an element of C) boolean ORi is an element of I x(i) the irredundant CNF of f, where C is the set of the prime implicates of f. Similarly, let d = boolean ORJ is an element of D boolean AND(j is an element of J) x(j) be the irredundant DNF of the same function, where D is the set of the prime implicants of f. We show that given subsets C' subset of or equal to C and D' subset of or equal to D such that (C', D') not equal (C, D), a new term in (C\C')boolean OR(D\D') can be found in time O(n(t + n))+ m(o(log m)), where m = \C'\ + \D'\. In particular, if f(x) can be evaluated for every x is an element of {0, 1}(n) in polynomial time, then the forms c and d can be jointly generated in incremental quasi-polynomial time. On the other hand, even for the class of boolean AND, boolean OR-formulae f of depth 2, i.e., for CNFs or DNFs, it is unlikely that uniform sampling from within the set of the prime implicates and implicants of f can be carried out in time bounded by a quasi-polynomial 2(polylog(.)) in the input size of f. We also show that for some classes of polynomial-time computable monotone Boolean functions it is NP-hard to test either of the conditions D' = D or C' = C. This provides evidence that for each of these classes neither conjunctive nor disjunctive irredundant normal forms can be generated in total (or incremental) quasi-polynomial time. Such classes of monotone Boolean functions naturally arise in game theory, networks and relay contact circuits, convex programming, and include a subset of boolean AND, boolean OR-formulae of depth 3. (C) 1999 Elsevier Science B.V. All rights reserved.