General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Definition 253 The disjunctive normal form (DNF) of f is
c∈f −1 (1) Mc
(also called the canonical sum (written as DNF(f)))
Definition 254 The conjunctive normal form (CNF) of f is
c∈f −1 (0) Sc
(also called the canonical product (written as CNF(f)))
x1 x2 x3 f monomials clauses
0 0 0 1 x 0 1 x0 2 x0 3
0 0 1 1 x 0 1 x0 2 x1 3
0 1 0 0 x 1 1 + x0 2 + x1 3
0 1 1 0 x 1 1 + x0 2 + x0 3
1 0 0 1 x 1 1 x0 2 x0 3
1 0 1 1 x 1 1 x0 2 x1 3
1 1 0 0 x 0 1 + x0 2 + x1 3
1 1 1 1 x 1 1 x1 2 x1 3
DNF of f: x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3 + x1 x2 x3
CNF of f: (x1 + x2 + x3) (x1 + x2 + x3) (x1 + x2 + x3)
c○: Michael Kohlhase 143
In the light of the argument of understanding Boolean expressions as implementations of Boolean
functions, the process becomes interesting while realizing specifications of chips. In particular it
also becomes interesting, which of the possible Boolean expressions we choose for realizing a given
Boolean function. We will analyze the choice in terms of the “cost” of a Boolean expression.
Costs of Boolean Expressions
Idea: Complexity Analysis is about the estimation of resource needs
if we have two expressions for a Boolean function, which one to choose?
Idea: Let us just measure the size of the expression (after all it needs to be written down)
Better Idea: count the number of operators (computation elements)
Definition 255 The cost C(e) of e ∈ Ebool is the number of operators in e.
Example 256 C(x1 + x3) = 2, C(x1 ∗ x2 + x3 ∗ x4) = 4,
C((x1 + x2) + (x1 ∗ x2 + x3 ∗ x4)) = 7
Definition 257 Let f : B n → B be a Boolean function, then C(f) :=
min({C(e) | f = fe}) is the cost of f.
Note: We can find expressions of arbitrarily high cost for a given Boolean function.(e ≡ e ∗ 1)
but how to find such an e with minimal cost for f?
c○: Michael Kohlhase 144
2.5.3 Complexity Analysis for Boolean Expressions
The Landau Notations (aka. “big-O” Notation)
Definition 258 Let f, g : N → N, we say that f is asymptotically bounded by g, written as
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