Chapter 5: Architecture - Computer and Information Science - CUNY

5.2. CLASSICAL GATES 15
A is a 2 m by 2 n matrix. B is a 2 m−p by 2 q matrix. Since nothing should be
done to the m − p bits, we might represent this as the 2 m−p by 2 m−p identity
matrix I m−p . We do not draw any gate for the identity matrix. The entire
circuit can be represented by the following matrix
Example 5.2.2 Consider the circuit
(B ⊗ I m−p ) ⋆ A. (5.47)
This is represented by
OR ⋆ (NOT ⊗ AND). (5.48)
Let us see how the operations look like as matrices. Calculating, we get:
⎡
⎤
0 0 0 0 1 1 1 0
⎡ ⎤ ⎡ ⎤
NOT ⊗ AND = ⎣ 0 1 ⎦ ⊗ ⎣ 1 1 1 0
0 0 0 0 0 0 0 1
⎦ =
.
1 0 0 0 0 1 ⎢
⎣
1 1 1 0 0 0 0 0⎥
⎦
0 0 0 1 0 0 0 0
(5.49)
And so we get
⎡
OR ⋆ (NOT ⊗ AND) = ⎣ 0 0 0 0 1 1 1 ⎤
0 ⎦ . (5.50)
1 1 1 1 0 0 0 1
Let us see if we can formulate DeMorgan’s laws in terms of matrices. One
of DeMorgan’s law states that ¬(¬P ∧ ¬Q) = P ∨ Q. In pictures this looks like
In terms of matrices this corresponds to
NOT ⋆ AND ⋆ (NOT ⊗ NOT) = OR. (5.51)