Recursive Function Theory

Abhishek Kr Singh

TIFR Mumbai.

21 August 2014

Primitive Recursive Functions

IInitial Functions.

I s(x) = x + 1I n(x) = 0I uni (x1, ..., xn) = xi , where 1 i n .

IComposition:

I Let h(x1, ...., xn) = f (g1(x1, ....., xn), ....., gk(x1, ...., xn)).I Then h is said to be obtained from f and g1, ....., gk by

composition.

IPrimitive Recursion:

I Let h(x1, ....xn, 0) = f (x1, ....xn) , andh(x1, ....xn, t + 1) = g(t, h(x1, ....xn, t), x1, ....xn).

I Then h is said to be obtained from f and g by primitive

recursion, or simply recursion.

Definition: A function is called primitive recursive if it can be

obtained from the initial functions by a finite number

of applications of composition and recursion.

IBounded Quantifiers:

IIf the predicate P(t, x1, ....., xn) is primitive recursive then

so are the predicates (8t)yP(t, x1, ....., xn) and

(9t)yP(t, x1, ....., xn) .

IBounded Minimalization:

I mintyP(t, x1, ....., xn) is the least value of t y for whichP(t, x1, ....., xn) is true, if such exists; otherwise it assumes thedefault value 0.

I mintP(t, x1, ....., xn) is the unbounded version. However inthis case if there is no value of t for which P is true, thenmintP(t, x1, ....., xn) is undefined.

IIf the predicate P(t, x1, ....., xn) is primitive recursive then

so is the predicate mintyP(t, x1, ....., xn).

ISome Primitive Recursive Functions

1. x + y2. x .y3. x!4. xy

5. p(x) the predecessor function6. x . y7. |x � y |8. ↵(x) the IsZero predicate9. x = y

10. x y11. x < y12. y | x y divides x13. Prime(x)14. bx/yc15. R(x , y)16. pn the nth prime number17. < x , y > the pairing function18. [a1, ..., an] the Godel number19. Lt(x) where x = [a1, ..., an]20. ([a1, ..., an])i

Programs and Computable Functions

IProgramming language S .

I Our concept of computable function will be based onprogramming language S which has following instruction types.

1. V V2. V V + 13. V V � 14. IF V 6= 0 GOTO L

I A program in S is a sequence of labeled or unlabeledinstructions of above type.

ISyntax of the language S

I Conventions:I Input variables X1,X2,X3, ..... .I Output variable Y andI Local Variables Z1,Z2,Z3, .....

I State � and snapshot s = (i ,�) of program P .I A Computation of a program P is defined to be a sequence

s1, s2, s3, .....sk of snapshots of P such that si+1 is thesuccessor of si for each i and sk is the terminal snapshot.

Computable Functions

IFor any program P and any positive integer m, m

P (x1, ...., xm)represents the value of function computed by program P on

input x1, ...., xm.

I A given partial function g is said to be partially computable ifit is computed by some program.

I A function g is called computable if it is both total andpartially computable.

Primitive recursive Vs computable functions.

IEvery primitive recursive function is computable.

ICoding program by numbers

I #(I ) =< a, < b, c >>I #(P) = [#(I1),#(I2), ....,#(Ik)]� 1.

Theorem-1 Universality Theorem:

ILet �n(x1, ...., xn, y) = n

P(x1, ...., xn) , where

#(P) = y .

IThen for each n > 0 , the function

�n(x1, ...., xn, y) is partially computable.

Proof. Existence of a program Un, called the universal

progmam, such that

n+1Un

(x1, ...., xn, xn+1) = �n(x1, ...., xn, xn+1) = nP(x1, ...., xn) ,where xn+1 = #(P).

Theorem-2 Step-Counter Theorem:

ILet STPn(x1, ....., xn, y , t)() Program number

y halts after t or fewer steps on inputs

x1, ......, xn.Then for each n > 0, the predicate

STPn(x1, ....., xn, y , t) is primitive recursive.

Proof STPn(x1, ....., xn, y , t) ()Term(Snapn(x1, ....., xn, y , t), y)

where

Snapn(x1, ....., xn, y , 0) = Initn(x1, ....., xn) and

Snapn(x1, ....., xn, y , i + 1) =Succ(Snapn(x1, ....., xn, y , i), y).

Theorem-3 Normal Form Theorem: Let f (x1, ......, xn) be a

partially computable function. Then there is a

primitive recursive predicate R(x1, ...., xn, y) such that

f (x1, ......, xn) = l(minzR(x1, ...., xn, z)).

Proof. Let y0 = #(P) where P is the program that

computes f .Consider the following predicate, call it

R(x1, ...., xn, z) ,

STPn(x1, ....., xn, y0, r(z))

& (r(Snapn(x1, ....., xn, y0, r(z))))1 = l(z)

Then we have

f (x1, ......, xn) = l(minzR(x1, ...., xn, z))

Note that if there is no value of z for which

minzR(x1, ...., xn, z) is true, then according to the

definition of minimalization minzR(x1, ...., xn, z) is

undefined.

Corollary-3.1 A function is partially computable if and only if it can

be obtained from the initial functions by a finite

number of applications of composition, recursion, and

minimalization.

Corollary-3.2 A function is computable if and only if it can be

obtained from the initial functions by a finite number

of applications of composition, recursion, and proper

minimalization.

IWhen minzR(x1, ...., xn, z) is a total function, we

say that we are applying proper minimalization

to R .

Recursively Enumerable Sets

Definition The set B ⇢ N is called recursively enumerable if

there is a partially computable function g(x) such

that B = {x 2 N|g(x) #}.Definition We write Wn = {x 2 N|�(x , n) #}.

We define K = {n 2 N|n 2Wn}.Therefore, n 2Wn () �(n, n) # () HALT (n, n).

Since we have Godel numbering functions [x1, ...., xm] and (x)i , we

can restrict to the subsets of N in our discussion of Computabilty

theory. Therefore, we have,

Theorem-4 Let C be a PRC class, and let B be a subset of Nm,

m � 1. Then B belongs to C if and only if

B0= {[x1, ...., xm] 2 N|(x1, ...., xm) 2 B} belongs to

C .

Proof. PB0 (x) () (Lt(x) = m)&PB((x)1, ....., (x)m)PB(x1, ...., xm) () PB0([x1, ...., xm])

Theorem-5 Enumeration Theorem: A set B is r.e if and only if

there is an n for which B = Wn.

Theorem-6 The set B is recursive if and only if B and B̄ are both

r.e.

Proof. Let P and Q be the programs coresponding to B and

B̄ . Consider the following program where

p = #(P) and q = #(Q)

[A] If STP(X , p,T ) Goto C

If STP(X , q,T ) Goto E

T T + 1

Goto A

[C ] Y 1

Theorem-7 Let B be an r.e. set. Then there is a primitive

recursive predicate R(x , t) such that

B = {x 2 N|(9t)R(x , t)}.Proof. Let B = Wn. Then B = {x 2 N|(9t) STP(x , n, t)}.

Theorem-8 If B and C are r.e sets so are B [ C and B \ C .

Theorem-9 Let S be a nonempty r.e. set. Then there is a

primitive recursive function f (u) such that

S = {f (n)|n 2 N} = {f (0), f (1), .....}. That is, S is

the range of f .

Proof. By Theorem-7 we can write, S = {x |(9t)R(x , t)}.

Let x0 2 S and u =< x , t > .Consider the following function

f (u) =

(x if R(x , t)

x0 otherwise

That is,

f (< x , t >) = x .R(x , t) + x0. ⇠ R(x , t)Which is same as

f (u) = l(u).R(l(u), r(u)) + x0.↵(R(l(u), r(u))).

Theorem-10 Let f (x) be a partially computable function and let

S = {f (x)|f (x) #}. (That is, S is the range of f .)Then S is r.e.

Proof Let program P computes f and p = #(P).Then we need to demonstrate a program, say Q,

which behaves as follow,

Q stops at x () 9u9t, (STP(u, p, t)& f (u) = x)

() 9 < u, t >, (STP(u, p, t)& f (u) = x)() 9n, (STP(l(n), p, r(n))& f (l(n)) = x)

Thus Q can be the following program,

[A] IF ⇠ STP(l(Z ), p, r(Z )) Goto B

IF f (l(Z )) = X Goto E

[B] Z Z + 1

Goto A

Theorem-11 Suppose that S 6= �. Then the following statements

are all equivalent:

1. S is r.e.

2. S is the range of a primitive recursive function.

3. S is the range of a recursive function.

4. S is the range of a partial recursive function.

Proof. By Theorem-9, (1) =) (2).(2) =) (3) and (3) =) (4) are obvious. By