Computability complexity and Languages- Fundamentals of Theoretical Computer Science

4. Computable Functions 29
together with the equations V = 0 for each variable V in go other than
X 1 , ••• , Xm, Y. We will call this the initial state, and the snapshot (1, u ),
the initial snapshot.
Case 1. There is a computation s 1 , s 2 , ••• , s k of go beginning with the initial
snapshot. Then we write r/J~m>(r 1 ,r 2 , ••• ,rm) for the value of the
variable Y at the (terminal) snapshot sk.
Case 2. There is no such computation; i.e., there is an infinite sequence
s 1 ,s 2 ,s 3 , ••• beginning with the initial snapshot where each si+l
is the successor of s;. In this case r/J~m>(r 1 , ••• , r m) is undefined.
Let us reexamine the examples in Section 2 from the point of view of
this definition. We begin with the program of (b). For this program go, we
have
1/J~l)(x)
= x
for all x. For this one example, we give a detailed treatment. The following
list of snapshots is a computation of go:
(1,{X = r, Y= O,Z = 0}),
(4, {X= r, Y = 0, Z = 0}),
(5, {X= r- 1, Y = 0, Z = 0}),
(6, {X= r- 1, Y = 1, Z = 0}),
(7, {X= r- 1, Y = 1, Z = 1}),
(1, {X= r- 1, Y = 1, Z = 1}),
(1, {X= 0, Y = r, Z = r}),
(2, {X= 0, Y = r, Z = r}),
(3, {X= 0, Y = r, Z = r + 1}),
(8, {X= 0, Y = r, Z = r + 1}).
We have included a copy of go showing line numbers:
[A]
[B]
IF X =1= 0 GOTO B
z ~z + 1
IF Z =I= 0 GOTO E
x~x-1
Y~ Y+ 1
z ~z + 1
IF Z =/= 0 GOTO A
(1)
(2)
(3)
(4)
(5)
(6)
{7)