1. Primitive Recursive ≤ Turing Machine • Every instance of Primitive Recursive can be replaced by an equivalent instance of Turing Machine • Primitive Recursive • Base functions • Composition • Iteration • Bounded Minimization

2. Turing Computable • a Turing computation of some n-ary function F is to assume that the machine starts with a tape • containing the n inputs, x1, ... , xn in the form • …01x101x20…01xn0… • and ends with • …01x101x20…01xn01y0… • where y = F(x1, ... , xn).

3. TM - Base Machine • R -- move right over any scanned symbol • L -- move left over any scanned symbol • 0 -- write a 0 in current scanned square • 1 -- write a 1 in current scanned square

4. TM - SubMachine • R -- move right to next 0 • L -- move left to next 0 • Ck -- copies k-th preceding value.

5. PR - SubMachine • Translate -- moves a value left one tape square • Shift -- shift a rightmost value left, destroying value to its left • Rotk -- Rotate a k value sequence one slot to the left

6. PR - Base Function • Ca(x1,...,xn) = a : constant functions • (R1)aR • Iin(x1,...,xn) = xi : identity functions • Cn-i+1 • S(x) = x+1 : an increment function • C11R

7. PR - Composition • If G, H1, … , Hk are already known to be Turing computable, then so is F, where • F(x1,…,xn) = G(H1(x1,…,xn), … , Hk(x1,…,xn))‏ • <1> if E(x1,…,xn) is Turing computable then so is E<m>(x1,…,xn, y1,…,ym) = E(x1,…,xn)‏ • Ln+m (Rotn+m)nRn+m E Ln+m+1 (Rotn+m)mRn+m+1 • <2> F can be defined by • H1 H2<1> H3<2> … Hk<k-1> G Shiftk

8. PR - Minimization • If G is already known to be Turing computable, then so is F, where • F(x1,…,xn) = y (G(x1,…,xn, y) == 1)‏

9. PR - Iteration • If G, H are already known to be primitive recursive, then so is F, where • F(0, x1,...,xn) = G(x1,...,xn)‏ • F(y+1, x1,...,xn) = H(y, x1,...,xn, F(y, x1,...,xn))‏ • Ln+2 T L T Rn+2 G Ln+3 L--->Rn+4 H Shift1Ln+3 1 L0 Rn+2 • 0 Rn+4