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

y the variable assignments). The notion of equivalent formulae formalizes this intuition.
Boolean Equivalences
Given a, b, c ∈ Ebool, ◦ ∈ {+, ∗}, let ˆ◦ :=
+ if ◦ = ∗
∗ else
We have the following equivalences in Boolean Algebra:
a ◦ b ≡ b ◦ a (commutativity)
(a ◦ b) ◦ c ≡ a ◦ (b ◦ c) (associativity)
a ◦ (bˆ◦c) ≡ (a ◦ b)ˆ◦(a ◦ c) (distributivity)
a ◦ (aˆ◦b) ≡ a (covering)
(a ◦ b)ˆ◦(a ◦ b) ≡ a (combining)
(a ◦ b)ˆ◦((a ◦ c)ˆ◦(b ◦ c)) ≡ (a ◦ b)ˆ◦(a ◦ c) (consensus)
a ◦ b ≡ aˆ◦b (De Morgan)
2.5.2 Boolean Functions
c○: Michael Kohlhase 139
We will now turn to “semantical” counterparts of Boolean expressions: Boolean functions. These
are just n-ary functions on the Boolean values.
Boolean functions are interesting, since can be used as computational devices; we will study
this extensively in the rest of the course. In particular, we can consider a computer CPU as
collection of Boolean functions (e.g. a modern CPU with 64 inputs and outputs can be viewed as
a sequence of 64 Boolean functions of arity 64: one function per output pin).
The theory we will develop now will help us understand how to “implement” Boolean functions
(as specifications of computer chips), viewing Boolean expressions very abstract representations of
configurations of logic gates and wiring. We will study the issues of representing such configurations
in more detail later 7 EdNote:7
Boolean Functions
Definition 246 A Boolean function is a function from B n to B.
Definition 247 Boolean functions f, g : B n → B are called equivalent, (write f ≡ g), iff
f(c) = g(c) for all c ∈ B n . (equal as functions)
Idea: We can turn any Boolean expression into a Boolean function by ordering the variables
(use the lexical ordering on {X} × {1, . . . , 9} + × {0, . . . , 9} ∗ )
Definition 248 Let e ∈ Ebool and {x1, . . . , xn} the set of variables in e, then call V L(e) :=
〈x1, . . . , xn〉 the variable list of e, iff (xi