Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 236 — #244
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Chapter 7
Recursive Data Types
(g) Verify that if R and S are boring, then so is R [ S.
(h) Verify that if R and S are boring, then so is R S.
Hint: By cases: whether R and S are both 0-finite, whether R or S contains no
all-0 words at all (including the empty word ), and whether neither of these cases
hold.
(i) Conclude by structural induction that all C-D languages are boring.
So we have proved that the set .00/ of even length all-0 words is not a C-D
language.
Problem 7.23.
We can explain in a simple and precise way how digital circuits work, and gain the
powerful proof method of structural induction to verify their properties, by defining
digital circuits as a recursive data type DigCirc. The definition is a little easier to
state if all the gates in the circuit take two inputs, so we will use the two-input NOR
gate rather than a one-input NOT, and let the set of gates be
Gates WWD fNOR; AND; OR; XORg:
A digital circuit will be a recursively defined list of gate connections of the form
.x; y; G; I / where G is a gate, x and y are the input wires, and I is the set of wires
that the gate output feeds into as illustrated in Figure 7.5.
Formally, we let W be a set w 0 ; w 1 ; : : : whose elements are called wires, and
O … W be an object called the output.
Definition. The set of digital circuit DigCirc, and their inputs and internal wires,
are defined recursively as follows:
Base case: If x; y 2 W , then C 2 DigCirc, where
Constructor cases: If
C D list..x; y; G; fOg// for some G 2 Gates;
inputs.C/ WWD fx; yg;
internal.C/ WWD ;:
C 2 DigCirc;
I inputs.C/; I ¤ ;;
x; y 2 W .I [ internal.C//