by an O(logk n)-depth, O(nO(1) )-size Boolean circuit Cn with fan-in 2. The class ACk is de ned similarly substituting unbounded fan-in AND- and OR-gates for arbitrary gates with fan-in at most 2. (So AC0 is the class of Boolean functions computed by constant-depth polynomial-size circuits with NOT-gates and with unbounded fan-in AND- and OR-gates.) In Section 1.6 the topic is Boolean circuits and formal languages. It is shown that (the characteristic function of) any regular (context-free) language is in nonuniform NC1 (AC1 ). (Chapter 1 considers nonuniform circuit families only.) Section 1.7 reviews circuits for arithmetic operations. Addition of just two n-bit integers is shown to be in AC0 , and their multiplication in NC1 . As for integer division, an algorithm based on Newton iteration and due to Brent puts it in NC2 whereas a more complex technique involving Chinese remaindering and due to Beame, Cook, and Hoover improves this to NC1 . (The very recent result of Allender, Barrington, and Hesse putting integer division in uniform TC0 is not covered.) Section 1.8 takes up the design of circuits for arbitrary Boolean functions. First, there is the standard technique based upon disjunctive normal form. The existence
Review of Boolean Functions and Computation Models by Peter Clote and Evangelos Kranakis, Springer-Verlag, 2002
by an O(logk n)-depth, O(nO(1) )-size Boolean circuit Cn with fan-in 2. The class ACk is de ned similarly substituting unbounded fan-in AND- and OR-gates for arbitrary gates with fan-in at most 2. (So AC0 is the class of Boolean functions computed by constant-depth polynomial-size circuits with NOT-gates and with unbounded fan-in AND- and OR-gates.) In Section 1.6 the topic is Boolean circuits and formal languages. It is shown that (the characteristic function of) any regular (context-free) language is in nonuniform NC1 (AC1 ). (Chapter 1 considers nonuniform circuit families only.) Section 1.7 reviews circuits for arithmetic operations. Addition of just two n-bit integers is shown to be in AC0 , and their multiplication in NC1 . As for integer division, an algorithm based on Newton iteration and due to Brent puts it in NC2 whereas a more complex technique involving Chinese remaindering and due to Beame, Cook, and Hoover improves this to NC1 . (The very recent result of Allender, Barrington, and Hesse putting integer division in uniform TC0 is not covered.) Section 1.8 takes up the design of circuits for arbitrary Boolean functions. First, there is the standard technique based upon disjunctive normal form. The existence of universal circuits is n demonstrated, speci cally, a size-O(22 ) circuit with multiple outputs such that any n-ary Boolean circuit is computed by one of them. It is then shown that, with such a universal circuit in hand, any n-ary Boolean function f is computable by a circuit with fan-in 2 and size O(2n /n) (Shannon s Method). Another technique due to Lupanov improves this to 2n /n + o(2n /n). If f happens to be symmetric, then a size O(n) circuit is possible still fan-in 2 and it is mentioned in passing that depth O(log n) may be assumed. Section 1.9 shows that, given Boolean circuit C of fan-in 2 but arbitrary fan-out having n inputand m output-gates, it is possible to construct an equivalent fan-in-2 fan-out-2 circuit C whose...
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Review of Boolean Functions and Computation Models by Peter Clote and Evangelos Kranakis, Springer-Verlag, 2002
Taylor, R. Gregory
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, Volume 35 (4)
Association for Computing Machinery – Dec 1, 2004
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