Computability complexity and Languages- Fundamentals of Theoretical Computer Science

3. Syntax 27
Suppose we have a program 9' and a state u of 9'. In order to say what
happens "next," we also need to know which instruction of 9' is about to
be executed. We therefore define a snapshot or instantaneous description
of a program 9' of length n to be a pair (i, u) where 1 ~ i ~ n + 1, and u
is a state of 9'. (Intuitively the number i indicates that it is the ith
instruction which is about to be executed; i = n + 1 corresponds to a
"stop" instruction.)
If s = (i, u) is a snapshot of 9' and V is a variable of 9', then the value
of Vats just means the value of V at u.
A snapshot (i, u) of a program 9' of length n is called terminal if
i = n + 1. If (i, u) is a nonterminal snapshot of 9', we define the successor
of (i, u) to be the snapshot (j, T) defined as follows:
Case 1. The ith instruction of 9' is V ~ V + 1 and u contains the
equation V = m. Then j = i + 1 and T is obtained from u by
replacing the equation V = m by V = m + 1 (i.e., the value of V
at T ism+ 1).
Case 2. The ith instruction of 9' is V ~ V- 1 and u contains the
equation V = m. Then j = i + 1 and T is obtained from u by
replacing the equation V = m by V = m - 1 if m -=!= 0; if m = 0,
T = U.
Case 3. The ith instruction of 9' is V ~ V. Then T = u and j = i + 1.
Case 4. The ith instruction of 9' is IF V-=!= 0 GOTO L. Then T = u, and
there are two subcases:
Case 4a. u contains the equation V = 0. Then j = i + 1.
Case 4b. u contains the equation V = m where m -=1= 0. Then, if there is
an instruction of 9' labeled L, j is the least number such that
the jth instruction of 9' is labeled L. Otherwise, j = n + 1.
For an example, we return to the program of (b), Section 2. Let u be
the state
X= 4, Y= 0, Z=O
and let us compute the successor of the snapshots (i, u) for various values
of i.
For i = 1, the successor is (4, u) where u is as above. For i = 2, the
successor is (3, T ), where T consists of the equations
X=4, Y= 0, Z=l.
For i = 7, the successor is (8, u ). This is a terminal snapshot.
A computation of a program 9' is defined to be a sequence (i.e., a list)
s 1 ,s 2 , ••• ,sk of snapshots of 9' such that s;+t is the successor of s; for
i = 1, 2, ... , k - 1 and sk is terminal.