1. Mu-Recursive Functions
Chapter 13
Mu-Recursive Functions

2. Recursive Functions
Functions that can be computed by at least one computational system, regardless of the implementation & execution of the system.
Recursive function theory:
i) Start w/ a collection of initial functions, which are computable
ii) Combine the initial functions to form more functions which are also computable

3. Partial Functions & Total Functions
A partial function f : X  Y is a relation on X  Y such that y1 = y2 whenever (x, y1)  f & (x, y2)  f.
x  X is said to be defined in f if y Y such that (x, y)  f; o.w., x is undefined in f.
If the domain of f is the entire set X, then f is a total function on X.
A total function can be a one-to-one, onto, or bijection function.

4. Primitive Recursive Functions
Are more complex functions than the basic primitive recursive functions, which include
i) the successor function S: S(X) = X + 1
ii) the zero function Z: Z(X) = 0
iii) the projection functions Pi(n): Pi(n)(X1, ..., Xn) = Xi, 1  i  n.
These functions form the foundation of the hierarchy of computable functions.
Can be constructed from the basic functions by applications of
i) composition: the composition of two computable functions g & h, i.e., f = g  h, is a (computable) function (that applies the outputs of h as inputs of g )
ii) primitive recursion

5. Primitive Recursive Functions
Defn (Primitive Recursion). Let g & h be total functions w/ n & n+2 variables, respectively. The n+1-variable function f is defined by
i) f(X1, ..., Xn, 0) = g(X1, ..., Xn)
ii) f(X1, ..., Xn, y+1) = h(X1, ..., Xn, y, f (X1, ..., Xn, y))
is said to be obtained from g & h by primitive recursion, where Xi s (1  i  n) are called the parameters & y the recursive variable.
By definition, f (X1, ..., Xn, y+1) is obtained by the sequence of computations:
f (X1, ..., Xn, 0) = g(X1, ..., Xn)
f (X1, ..., Xn, 1) = h(X1, ..., Xn, 0, f (X1, ..., Xn, 0))
f (X1, ..., Xn, 2) = h(X1, ..., Xn, 1, f (X1, ..., Xn, 1)) :
f (X1, ..., Xn, y+1) = h(X1, ..., Xn, y, f (X1, ..., Xn, y))

6. Primitive Recursive Function
A function is primitive recursive (P.R.) if it can be obtained from the successor, zero & projection functions by a finite number of applications of composition & primitive recursion.
e.g., let add be the function defined by P.R. from g(x) = x & h(x, y, z) = z + 1. Then
add(x, 0) = g(x) = x
add(x, y+1) = h(x, y, add(x, y)) = add(x, y) + 1
Hence, add is P.R. since g = P1(1) & h = S  P3(3)

7. Primitive Recursive Function
e.g., add(2, 4) = add(2, 3) + 1
= (add(2, 2) + 1) + 1
= ((add(2, 1) + 1) + 1) + 1
= (((add(2, 0) + 1) + 1) + 1) + 1 =
= 6
Example Let g & h be the total functions g = z & h = add  (P3(3), P1(3)), mult, i.e., , can be defined by primitive recursion from g & h as
mult(x, 0) = g(x) = 0
mult(x, y+1) = h(x, y, mult(x, y)) = mult(x, y) + x

8. Primitive Recursive Function
Example Let fact be the factorial function defined by
if y = 0
fact(y) = o.w.
fact is p.r. Let h(x, y) = mult  (P2(2), SP1(2)) = y  (x+1).
Then, fact can be defined using p.r. from h by
fact(0) = 1 & fact(y+1) = h(y, fact(y)) = fact(y)  (y+1)
hence, fact(0) = 1,
fact(1) = fact(0)  1 = 1
fact(2) = fact(1)  2 = 2,
fact(3) = fact(2)  3 = 6
fact(4) = fact(3)  4 = 24, …
h, the function used in a p.r. function, can be any function that has previously be shown to be p.r. Õ = y i 1

9. Table 13.2.1 Primitive Recursive Arithmetic Functions
Description Function Definition
Addition add(x, y) add(x, 0) = x
x + y add(x, y+1) = add(x, y) + 1
Multiplication mult(x, y) mult(x,0) = 0
x  y mult(x, y+1) = mult(x, y) + x
Predecessor pred(y) pred(0) = 0
pred(y+1) = y
Proper sub(x, y) sub(x, 0) = x
Subtraction x - y sub(x, y+1) = pred(sub(x, y))
Exponentation exp(x, y) exp(x, 0) = 1
xy exp(x, y+1) = exp(x, y)  x .

10. TABLE 13.2.2 Primitive Recursive Predicates
Description Predicate Definition
Sign sg(x) sg(0) = 0
sg(y + 1) = 1
Sign Complement cosg(x) cosg(0) = 1
cosg(y + 1) = 0
Less than lt(x, y) sg(y - x)
Greater than gt(x, y) sg(x - y)
Equal to eq(x, y) cosg(lt(x, y) + gt(x, y))
Not equal to ne(x, y) cosg(eq(x, y)) . .

11. Primitive Recursive Function
Table Primitive Recursive Arithmetic Functions
Theorem Every p.r. function is Turing computable
Example: Let f be the function defined by
x + 1 if x is even
x - 1 o.w.
a) Give the state transitive diagram of a TM M that computes f
b) Show that f is p.r.
f(x) =
b) oe(x) = add(mult(ev(x), s(x)), mult(cosq(ev(x)), pred(x))) where
ev(0) = 1 = s ° z
ev(x+1) = h(x, ev(x))
= eq(P2(2)(x, ev(x)), 0)
(eq is defined in Table ) q0 q1 q2 q4 q3
B/B R
B/1 R
B/B L
1/B R
1/1 R a)
ev(x) =
if x is even
o.w.

12. Primitive Recursive Function
Theorem Let g be a p.r. function & f a total function that is identical to g for all but a finite # of input values. Then f is p.r.
Proof: Let f be defined by
y1 if x = n1
y2 if x = n2
f(x) =
yk if x = nk
g(x) o.w.
f is p.r. since f can be written as
f(x) = eq(x, n1) · y1 + eq(x, n2) · y eq(x, nk) · yk ne(x, n1) · ne(x, n2) · ... · ne(x, nk) · g(x) 

13. Tabular Functions
Output values are listed along w/ their corresponding input values, and a single common value is associated w/ all other inputs, i.e.,
g1(x1, ..., xn) if P1 (x1, ..., xn)
f(x1, ..., xn) =
gm(x1, ..., xn) if Pm (x1, ..., xn)
gm+1(x1, ..., xn) o.w.
f is p.r. because f can be expressed in the form
f(x1, ..., xn) = g1(x1, ..., xn) · C1 (x1, ..., xn)
gm(x1, ..., xn) · Cm (x1, ..., xn) +
gm+1(x1,...,xn) · cosq(C1 (x1,..., xn) Cm(x1, ..., xn))
where
if Pp(x1, ..., xn) holds
o.w.
& Cp is called the characteristic function 
Cp(x1, ..., xn) =

14. Computable Partial Functions
All primitive recursive functions are total, but not vice versa.
Example: The Ackerman’s function is computable & total, but not p.r.
A(0, y) = y (*basis*)
A(x+1, 0) = A(x, 1)
A(x+1, y+1) = A(x, A(x+1, y))
To illustrate, consider
A(2,1) = A(1, A(2,0))
= A(1, A(1,1)) & A(1,1) = A(0, A(1, 0))
= A(1,3) = A(0, A(0, 1))
= A(0, A(1,2)) = A(0, 2)
= A(0, A(0, A(1,1))) = 3
= A(0, A(0, 3))
= A(0, 4)
= 5

15. Computable Partial Functions
Theorem: There is a computable total function from N to N that is not primitive recursive.
Proof. Each p.r. function from N to N can be defined by a finite string of symbols. Thus, we can assign an order to the p.r. functions by arranging their definitions according to their lengths (short strings first) w/ the strings of the same length by alphabetical order, i.e., f1, f2, ..., fn, ...
Let f(n) = fn(n) + 1 be a function from N to N. f is total & computable. However, f is not p.r. (If it were, f would have to be fm, for some m  N+. But then f(m) would be equal to fm(m), which cannot be true since f(m) = fm(m) + 1.)

16. Partial Recursive Functions
The class of partial recursive functions is the class of partial functions that can be constructed from the initial (i.e., basic) functions by applying a finite number of compositions, primitive recursions, & minimalizations.
minimalization allows us to construct a function f: Nn  N from function g: Nn+1  N .. f( ) is the smallest y  N, g( , y) = 0 & g( , z) is defined for all z ( N) < y, written f( ) = µy[g( , y) = 0].
e.g., g(0,0) = g(1,0) = g(2,0) = g(3,0) = 2
g(0,1) = g(1,1) = g(2,1) = g(3,1) = 6
g(0,2) = g(1,2) = g(2,2) = undefined g(3,2) = 7
g(0,3) = g(1,3) = g(2,3) = g(3,3) = 2
g(0,4) = g(1,4) = g(2,4) = g(3,4) = 8
g(0,5) = g(1,5) = g(2,5) = g(3,5) = 4
 f(0) =  f(1) =  f(2) is undefined x  

17. Partial Recursive Functions
Example. Let f: N  N be defined as f(x) = µy[plus(x, y) = 0]. Thus f(0) = 0, but f(x) is undefined for x > 0
Example. Let div: N2  N be defined as
integer portion of x/y, if y  0
undefined, if y = 0
using minimalization, div can be constructed as
div(x, y) = µt[((x + 1) - (mult(t, y) + y)) = 0]
div(x, y) = .

18. -Recursive Functions
-recursive functions yield a family of computable partial recursive functions.
Defn The family of -recursive functions is defined as:
i) The successor, zero, & projection functions are -recursive
ii) If h is an n-variable -recursive function & g1, ..., gn are k variable -recursive functions, then f = h ° (g1, ..., gn) is -recursive.
iii) If g & h are n & n+2 variable -recursive functions, then f defined from g & h by primitive recursion is -recursive.
iv) If p(x1, ..., xn, y) is a total -recursive predicate, then f = z [ p(x1, ..., xn, z)] is -recursive
v) Each -recursive function can be obtained from i) by a finite number of applications of the rules in ii), iii) & iv)
Theorem Every -recursive function is Turing computable
Theorem Every Turing computable function is -recursive