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Monotone Network Complexity of the Boolean Convolution

A linear function is a polynomial of degree 1. An exponential function is a function of the form E(x) = ax for some real a – for example, 21.5 = 2.8284271 … . The logarithm to the base a of b, denoted loga b, is the value of x such that ax = b. For example, the logarithm to base 2 of 2.8284271 … is 1.5 and the logarithm to base 10 of 100 is 2. A function f(x) is polylogarithmic if for some polynomial p(x) we can write f(x) as p(log2 x); that is, it is a polynomial in the logarithm of x. Two other functions used often in this book are the floor and ceiling functions. Their domains are the reals, but their codomains are the integers. The ceiling function, denoted x : → , maps the real x to the smallest integer greater or equal to it. The floor function, denoted x : → , maps the real x to the largest integer less than or equal to it. Thus, 3.5 = 4 and 15.0001 = 16. Similarly, 3.5 = 3 and 15.0001 = 15. The following bounds apply to the floor and ceiling functions. f(x) − 1 ≤ f(x) ≤ f(x) f(x) ≤ f(x) ≤ f(x) + 1 As an example of the application of the ceiling function we note that log2 n is the number of bits necessary to represent the integer n. 1.2.8 Rate of Growth of Functions Throughout this book we derive mathematical expressions for quantities such as space, time, and circuit size. Generally these expressions describe functions f : → from the nonnegative integers to the reals, such as the functions f1(n) and f2(n) defined as f1(n) = 4.5n2 + 3n f2(n) = 3n + 4.5n2 When n is large we often wish to simplify expressions such as these to make explicit their dominant or most rapidly growing term. For example, for large values of n the dominant terms in f1(n) and f2(n) are 4.5n2 and 3n respectively, as we show. A term dominates when n is large if the value of the function is approximately the value of this term, that is, if the function is within some multiplicative factor of the term. To highlight dominant terms we introduce the big Oh, big Omega and big Theta notation. They are defined for functions whose domains and codomains are the integers or the reals