Homework Questions

1. Homework 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: • IF X0 GOTO B • [A] Z  Z+1 • IF Z0 GOTO A • [B] X  X-1 • IF X0 GOTO D [C] Z  Z+1 IF Z0 GOTO C [D] X  X-1 Y  Y+1 IF X0 GOTO D Y  Y-1 Theory of Computation Lecture 6: Primitive Recursive Functions II

2. Homework Questions • Question 2: • a) Write a program S that computes S(1)(r) = r-4. 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) Theory of Computation Lecture 6: Primitive Recursive Functions II

3. Homework Questions Z7  0 // exp. of Y  p(Y) with m = 7 Z8  Y Z9  0 IF Z80 GOTO A9 [A8] Z9  Z9+1 IF Z90 GOTO A8 [A9] Z8  Z8-1 IF Z80 GOTO A11 [A10] Z9  Z9+1 IF Z90 GOTO A10 [A11] Z8  Z8-1 Z7  Z7+1 IF Z80 GOTO A11 Z7  Z7-1 Y  Z7 • 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 • IF Z30 GOTO A4 • [A3] Z4  Z4+1 • IF Z40 GOTO A3 • [A4] Z3  Z3-1 • IF Z30 GOTO A6 • [A5] Z4  Z4+1 • IF Z40 GOTO A5 • [A6] Z3  Z3-1 • Z2  Z2+1 • IF Z30 GOTO A6 • Z2  Z2-1 • Y  Z2 Theory of Computation Lecture 6: Primitive Recursive Functions II

4. Some Primitive Recursive Functions • So we have learned that every primitive recursive function is computable. • We will now look at some examples of primitive recursive functions. • In order to prove that a function is primitive recursive, we have to show how that function can be derived from the initial functions by using composition and recursion. Theory of Computation Lecture 6: Primitive Recursive Functions II

5. Some Primitive Recursive Functions • Remember? The recursive definition of a function looks like this: • h(x1, …, xn, 0) = f(x1, …, xn) • h(x1, …, xn, t + 1) = g(t, h(x1, …, xn, t), x1, …, xn) . • If f is a function of k variables, g1, …, gk are functions of n variables, and • h(x1, …, xn) = f(g1(x1, …, xn), …, gk(x1, …, xn)). • Then h is said to be obtained from f and g1, …, gkby composition. Theory of Computation Lecture 6: Primitive Recursive Functions II

6. Some Primitive Recursive Functions • We defined the following initial functions: • s(x) = x + 1 • n(x) = 0 • uin(x1, …, xn) = xi , 1 ≤ i ≤ n. Theory of Computation Lecture 6: Primitive Recursive Functions II

7. Some Primitive Recursive Functions • Example 1: f(x, y) = x + y • We can transform this into a recursive definition: • f(x, 0) = x • f(x, y + 1) = f(x, y) + 1 • This can be rewritten as: • f(x, 0) = u11(x) • f(x, y + 1) = g(y, f(x, y), x) • where g(x1, x2, x3) = s(u23(x1, x2, x3)). • Obviously, u11(x), u23(x1, x2, x3), and s(x) are primitive recursive functions – they are initial functions. • g(x1, x2, x3) is obtained by composition of primitive recursive functions, so it is primitive recursive itself. • Therefore, f(x, y) = x + y is primitive recursive. Theory of Computation Lecture 6: Primitive Recursive Functions II

8. Some Primitive Recursive Functions • Example 2: h(x, y) = x  y • We can obtain the following recursive definition: • h(x, 0) = 0 • h(x, y + 1) = h(x, y) + x • This can be rewritten as: • h(x, 0) = n(x) • h(x, y + 1) = g(y, h(x, y), x) • where • f(x1, x2) = x1 + x2 and • g(x1, x2, x3) = f(u23(x1, x2, x3), u33(x1, x2, x3)). • Obviously, these are all primitive recursive functions. • Therefore, h(x, y) = x  y is primitive recursive. Theory of Computation Lecture 6: Primitive Recursive Functions II