1. Lecture 1-4 Recursive Functions

2. Initial Functions Zero function ζ(n) = 0.
Successor σ(n) = n+1, for n ε N.
Projection πi (n1,…,nk) = ni. k

3. Composition Let m, k be two integers.
Given functions g: N → N and hi: N → N for I = 1, 2, …, m, define f: N → by
f(n1,…,nk) = g(h1(n1,…,nk),…,hm(n1,…,nk)
f is called the composition of g and h1, …, hm.
f = go(h1,…,hm) m k k

4. Primitive Recursion Let k > 0. Given g: N → N and h: N → N,
(when k=0, g is a constant),
define f : N → N by
f(n1,…nk,0) = g(n1,…,nk)
f(n1,…,nk,m+1) = h(n1,…,nk,m,f(n1,…nk,m)
k+1

5. Primitive Recursive Functions
All initial functions are primitive recursive.
If g and h1, …, hm are primitive recursive,
so is go(h1, …, hk).
If g and h are primitive recursive,
so is f obtained from g and h by primitive recursion.
Nothing else is primitive recursive.

6. Add(m, n) = m+n Add(m,0) = π1 (m)
Add(m, n+1) = σ( π3 (m,n, add(m, n))) 1 3

7. mult(m, n) = mn mult(m, 0) = ζ(m) Mult(m, n+1) =
add(π1(m,n,mult(m,n)), π3(m, n, mult(m,n)) 1 3

8. minus(m,n)= m-n if m > n; 0 if m<n
pred(m) = minus(m,1)
pred(0) = ζ(0)
pred(m+1) = m = π1 (m, pred(m))
minus(m,0) = π1 (m)
minus(m, n+1) = pred(π3(m,n,minus(m, n)) 2 1 3

9. Pairing Function
A function f: N → N is a pairing function if it satisfies the following conditions:
(1) f is 1-to-1 and onto;
(2) f is primitive recursive;
(3) f(i, j) < f(i+1, j), f(i, j) < f(i, j+1).
For example,

10. Two “inverse” functions i = g(f(i,j)) and 14 9 13 8 5 12 2 4 7 11 1 3 6 10
Two “inverse” functions i = g(f(i,j)) and
j = h(f(i, j)) are also primitive recursive.

11. Bounded minimization Given g: N → N, define f by
K+1
Given g: N → N, define f by
f(n1, …, nk, m) = min {i | g(n1, …, nk, i)=1}
if there exists i < m such
that g(n1, …, nk, i) =1;
= 0, otherwise.
If g is primitive recursive, so is f.

12. Unbounded minimization
K+1
Given g: N → N, define f by
f(n1, …, nk) = min {i | g(n1, …, nk, i)=1}
if there exists i such that
g(n1, …, nk, i) =1;
= ↑, otherwise.
f may not be primitive recursive, even if g is.

13. Partial Recursive Functions
All initial functions are partial recursive.
If g is a total recursive function, then f obtained from g by unbounded minimization is partial recursive.
If g and h1,…, hk are partial recursive, so is go(h1,…,hk).
If g and h are partial recursive, so is f obtained from g and h by primitive recursion.

14. Over Σ* Zero function ζ(x) = ε. Successor σa(x) = xa for a in Σ.
Projection (no change)
Composition (no change)
Primitive recursion (need a change, m+1 is replaced by ma)
Unbounded minimization (need to give a linear ordering of Σ*)

15. From Σ* to Γ* for Σ* c Γ* Zero function ζ(x) = ε.
Successor σa(x) = xa for a in Γ.
N can be considered as a subset of {0, 1}* if we represent n by 1…1 n

16. Theorem
A function is partial recursive iff it is Turing-computable.

17. Theorem The following are equivalent: A is r.e.
A is Turing-acceptable.
A is the range of a primitive recursive function.
A is the domain of a partial recursive function.

18. Pairing function on Σ* Let π(i, j) be a pairing function on N.
Let φ be a 1-to-1 onto mapping from N to Σ*.
Let μ be a 1-to-1 onto mapping from Σ* to N.
Then p(x, y) = φ(π(μ(x), μ(y)) is a pairing function on Σ*.