1. Theory of Computation Lecture 5: Programs and Computable Functions III
Sample Questions
Question 1:
Write a program P that computes P(1)(r) = r-2 using no macros.
Notice that we only consider natural numbers! This means that, for example, the expression (1-2) is undefined. Any program computing such an expression must never halt.
Sample solution:
X  X-1
IF X0 GOTO A2
[A1] Z  Z+1
IF Z0 GOTO A1
[A2] X  X-1
Y  Y+1
IF X  0 GOTO A2
Y  Y-1
February 14, 2019
Theory of Computation Lecture 5: Programs and Computable Functions III

2. Theory of Computation Lecture 5: Programs and Computable Functions III
Sample Questions
Question 2:
a) Write a program S that computes S(1)(r) = r Use a macro based on program P of the form W  p(V) (which has the effect W  V-2).
Sample solution:
Y  p(X)
Y  p(Y)
February 14, 2019
Theory of Computation Lecture 5: Programs and Computable Functions III

3. Theory of Computation Lecture 5: Programs and Computable Functions III
Sample Questions
b) Now comes the best part: Expand all macros in S using the method we discussed in the lecture.
Z2  0 // expansion of Y  p(X) with m = 2
Z3  X
Z4  0
Z3  Z3-1
IF Z30 GOTO A4
[A3] Z4  Z4-1
IF Z40 GOTO A3
[A4] Z3  Z3-1
Z2  Z2+1
Z2  Z2-1
Y  Z2
Z5  0 // exp. of Y  p(Y) with m = 5
Z6  Y
Z7  0
Z6  Z6-1
IF Z60 GOTO A7
[A6] Z7  Z7+1
IF Z70 GOTO A6
[A7] Z6  Z6-1
Z5  Z5+1
Z5  Z5-1
Y  Z5
February 14, 2019
Theory of Computation Lecture 5: Programs and Computable Functions III

4. Theory of Computation Lecture 5: Programs and Computable Functions III
Macros
Now let us introduce macros of the form
IF P(V1, …,Vn) GOTO L ,
where P(x1, …,xn) is a computable predicate.
This will be based on the convention that TRUE = 1, FALSE = 0.
According to this convention, predicates are just total functions whose values are always either 0 or 1.
February 14, 2019
Theory of Computation Lecture 5: Programs and Computable Functions III

5. Theory of Computation Lecture 5: Programs and Computable Functions III
Macros
Let P(x1, …,xn) be a computable predicate.
Then we expand the macro
IF P(V1, …,Vn) GOTO L to
Z  P(V1, …,Vn)
IF Z0 GOTO L
As usual, the variable Z has to be chosen to create no conflicts with the main program.
February 14, 2019
Theory of Computation Lecture 5: Programs and Computable Functions III

6. Theory of Computation Lecture 5: Programs and Computable Functions III
Macros
Example:
How can we expand the macro
IF V=0 GOTO L ?
V = 0 corresponds to the following predicate P(x):
P(x) = TRUE , if x = 0
= FALSE, otherwise
This can be computed by the following program:
IF X0 GOTO E
Y  Y+1
February 14, 2019
Theory of Computation Lecture 5: Programs and Computable Functions III

7. (Partially) Computable Functions
By introducing macros, we have seen that it is possible to compute complex functions with our very simple programming language L .
Notice that macros do not change the specification of the language, but they just simplify writing down programs.
As you know, we can always replace macros with actual code.
So what are the limitations of the language L ?
In order to find out, we need to do some mathematics.
February 14, 2019
Theory of Computation Lecture 5: Programs and Computable Functions III