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4 in , the total number of integers is at most So the total number of changes we made on is at most for some constant This concludes the proof. 3 Achieving Logarithmic Amortized Query Time In this section we show how to modify the algorithm to achieve better amortized query time. The worst case query time for a single query remains the same. We need a technical lemma first. Lemma 5. For any integers such that interval let be a integer. Let be two Then there is a witness to badness in the 24 N.
For a monotone function let be the set of all prime implicants of whose length is The covering number of denoted by cov(h), is defined as the minimal such that there are pairs of sets of variables that satisfy (i) (ii) and (iii) In such a case, we say that a set of pairs covers Obviously, for every G. K. Amano and A. Maruoka 34 We define the operation as follows: Suppose that Then is the disjunction of all prime implicants of whose length is at most 2. Let C* be a circuit obtained from C by replacing each gate in C with gate.
Amano and A. Maruoka functions are also proved (Theorem 4). 3, we describe the way of lower bounding the single level circuit complexity (Theorem 8), and give a set of quadratic functions whose monotone circuit complexity is strictly smaller than its single level complexity (Theorem 10). Finally, in Section 4, we discuss the relationship between the complexity of monotone circuits and of non-monotone circuits for quadratic functions based on the notion of pseudo complements (Theorem 11). 2 Boolean Circuits A Boolean circuit is a directed acyclic graph.