By Prof. Dr. Christoph Meinel, Dr. Thorsten Theobald (auth.)

One of the most difficulties in chip layout is the large variety of attainable mixtures of person chip parts, resulting in a combinatorial explosion as chips turn into extra advanced. New key ends up in theoretical computing device technological know-how and within the layout of knowledge buildings and effective algorithms may be utilized fruitfully right here. the appliance of ordered binary choice diagrams (OBDDs) has ended in dramatic functionality advancements in lots of computer-aided layout tasks. This textbook offers an advent to the principles of this interdisciplinary examine quarter with an emphasis on functions in computer-aided circuit layout and formal verification.

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**Example text**

The following characterization of monotone increasing and monotone decreasing functions could be equivalently used for defining monotony. 27. (1) A function f E JRn is monotone increasing if and only if for all a, bE JRn the property a S b implies f(a) S f(b). (2) A function f E JRn is monotone decreasing if and only if for all a, bE JRn the property a S b implies f(a) 2 f(b). Proof. Statements (1) and (2) can be proven by arguments that are completely analogous to each other. Hence, we only consider the first statement.

An n-variable function f : An -t A is called a Boolean function if it is induced by a Boolean formula F. We say that formula F represents the function f. The set of all Boolean functions over B is denoted by Pn(B). 10. Let us consider the set algebra B of the set {I, 2} and the formula F = Xl + Xl = {{I, 2}, U, n, -, 0, {I, 2}} . X2· The tabular representation of the induced Boolean function Fig. 2. f F is shown in 0 Let B = (A; +,', -,0,1) be a Boolean algebra. The set Pn(B) of all n-variable Boolean functions over the algebra B is a subset of the set Fn(A) of all functions from An to A.

C = c. Now we show that a (a + b) + (a· b) + b) . (a· b) + b) + 0:) . ((a + b) + b) = ((a + a) + b) . (a + (b + b)) (1 + b) . (a + 1) = 1·1 = 1. a· a· b + b· a' b = 0 + 0 = O. = ((a = (a + band 0:. b satisfy the two complement laws: = 30 3. , = a. a The last theorem of this section, which will be stated without proof, expresses that for each finite Boolean algebra there exists a set algebra of exactly the same structure. 6. ' ¢>(b) , --, ¢>(a) = ¢>(a) , ¢>(O) = 0', ¢>(1) = I'. 7. 2 Boolean Formulas and Boolean Functions Boolean functions are particular functions which can be described in terms of expressions over a Boolean algebra, so-called Boolean formulas.