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

c○: Michael Kohlhase 140
The definition above shows us that in theory every Boolean Expression induces a Boolean function.
The simplest way to compute this is to compute the truth table for the expression and then read
off the function from the table.
Boolean Functions and Truth Tables
The truth table of a Boolean function is defined in the obvious way:
x1 x2 x3 f x1∗(x2+x3)
T T T T
T T F F
T F T T
T F F T
F T T F
F T F F
F F T F
F F F F
compute this by assigning values and evaluating
Question: can we also go the other way? (from function to expression?)
Idea: read expression of a special form from truth tables (Boolean Polynomials)
c○: Michael Kohlhase 141
Computing a Boolean expression from a given Boolean function is more interesting — there are
many possible candidates to choose from; after all any two equivalent expressions induce the same
function. To simplify the problem, we will restrict the space of Boolean expressions that realize a
given Boolean function by looking only for expressions of a given form.
Boolean Polynomials
special form Boolean Expressions
a literal is a variable or the negation of a variable
a monomial or product term is a literal or the product of literals
a clause or sum term is a literal or the sum of literals
a Boolean polynomial or sum of products is a product term or the sum of product terms
a clause set or product of sums is a sum term or the product of sum terms
For literals xi, write x1 i , for xi write x0 i . ( not exponentials, but intended truth values)
Notation 251 Write xixj instead of xi ∗ xj. (like in math)
c○: Michael Kohlhase 142
Armed with this normal form, we can now define an way of realizing 8 Boolean functions. EdNote:8
Normal Forms of Boolean Functions
Definition 252 Let f : Bn → B be a Boolean function and c ∈ Bn , then Mc := n j=1 xcj j
and Sc := n j=1 x1−cj j
8 EdNote: define that formally above
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