Recursive Function Theory

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1 Recursive Function Theory Abhishek Kr Singh TFR Mumbai. 21 August 2014

2 Primitive Recursive Functions nitial Functions. s(x) =x + 1 n(x) =0 u n i (x 1,...,x n )=x i,where1apple i apple n. Composition: Let h(x 1,...,x n )=f(g 1 (x 1,...,x n ),...,g k (x 1,...,x n )). Then h is said to be obtained from f and g 1,...,g k by composition. Primitive Recursion: Let h(x1,...x n, 0) =f (x 1,...x n ),and h(x 1,...x n, t + 1) =g(t, h(x 1,...x n, t), x 1,...x n ). 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.

3 Bounded Quantifiers: f the predicate P(t, x1,...,x n ) is primitive recursive then so are the predicates (8t) appley P(t, x 1,...,x n ) and (9t) appley P(t, x 1,...,x n ). Bounded Minimalization: min tappley P(t, x 1,...,x n ) is the least value of t apple y for which P(t, x 1,...,x n ) is true, if such exists; otherwise it assumes the default value 0. mint P(t, x 1,...,x n ) is the unbounded version. However in this case if there is no value of t for which P is true, then min t P(t, x 1,...,x n ) is undefined. f the predicate P(t, x 1,...,x n ) is primitive recursive then so is the predicate min tappley P(t, x 1,...,x n ).

4 Some Primitive Recursive Functions 1. x + y 2. x.y 3. x! 4. x y 5. p(x) the predecessor function 6. x. y 7. x y 8. (x) the szero predicate 9. x = y 10. x apple y 11. x < y 12. y x y divides x 13. Prime(x) 14. bx/y c 15. R(x, y) 16. p n the nth prime number 17. < x, y > the pairing function 18. [a 1,...,a n ] the Godel number 19. Lt(x) where x =[a 1,...,a n ] 20. ([a 1,...,a n ]) i

5 Programs and Computable Functions Programming language S. Our concept of computable function will be based on programming language S which has following instruction types. 1. V V 2. V V V V 1 4. F V 6= 0 GOTO L AprograminS is a sequence of labeled or unlabeled instructions of above type. Syntax of the language S Conventions: nput variables X 1, X 2, X 3,.... Output variable Y and Local Variables Z 1, Z 2, Z 3,... State and snapshot s =(i, ) of program P. A Computation of a program P is defined to be a sequence s 1, s 2, s 3,...s k of snapshots of P such that s i+1 is the successor of s i for each i and s k is the terminal snapshot.

6 Computable Functions For any program P and any positive integer m, m P (x 1,...,x m ) represents the value of function computed by program P on input x 1,...,x m. A given partial function g is said to be partially computable if it is computed by some program. A function g is called computable if it is both total and partially computable. Primitive recursive Vs computable functions. Every primitive recursive function is computable. Coding program by numbers #( )=< a,<b, c >> #(P) =[#(1 ), #( 2 ),...,#( k )] 1.

7 Theorem-1 Universality Theorem: Let n (x 1,...,x n, y) = P n (x 1,...,x n ),where #(P) =y. Then for each n > 0,thefunction n (x 1,...,x n, y) is partially computable. Proof. Existence of a program U n,calledtheuniversal progmam, such that n+1 U n (x 1,...,x n, x n+1 )= n (x 1,...,x n, x n+1 )= n P (x 1,...,x n ),where x n+1 =#(P).

8 Theorem-2 Step-Counter Theorem: Let STP n (x 1,...,x n, y, t) () Program number y halts after t or fewer steps on inputs x 1,...,x n. Then for each n > 0, the predicate STP n (x 1,...,x n, y, t) is primitive recursive. Proof STP n (x 1,...,x n, y, t) () Term(Snap n (x 1,...,x n, y, t), y) where Snap n (x 1,...,x n, y, 0) =nit n (x 1,...,x n ) and Snap n (x 1,...,x n, y, i + 1) = Succ(Snap n (x 1,...,x n, y, i), y).

9 Theorem-3 Normal Form Theorem: Letf (x 1,...,x n ) be a partially computable function. Then there is a primitive recursive predicate R(x 1,...,x n, y) such that f (x 1,...,x n )=l(min z R(x 1,...,x n, z)). Proof. Let y 0 =#(P) where P is the program that computes f. Consider the following predicate, call it R(x 1,...,x n, z), STP n (x 1,...,x n, y 0, r(z)) &(r(snap n (x 1,...,x n, y 0, r(z)))) 1 = l(z) Then we have f (x 1,...,x n )=l(min z R(x 1,...,x n, z)) Note that if there is no value of z for which min z R(x 1,...,x n, z) is true, then according to the definition of minimalization min z R(x 1,...,x n, z) is undefined.

10 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. When min z R(x 1,...,x n, z) is a total function, we say that we are applying proper minimalization to R.

11 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 W n = {x 2 N (x, n) #}. We define K = {n 2 N n 2 W n }. Therefore, n 2 W n () (n, n) #() HALT (n, n). Since we have Godel numbering functions [x 1,...,x m ] 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 N m, m 1. Then B belongs to C if and only if B 0 = {[x 1,...,x m ] 2 N (x 1,...,x m ) 2 B} belongs to C. Proof. P B 0 (x) () (Lt(x) =m)&p B ((x) 1,...,(x) m ) P B (x 1,...,x m ) () P B 0([x 1,...,x m ])

12 Theorem-5 Enumeration Theorem: AsetB is r.e if and only if there is an n for which B = W n. Theorem-6 The set B is recursive if and only if B and B are both r.e. Proof. Let P and Q be the programs coresponding to B and B. Considerthefollowingprogramwhere p =#(P) and q =#(Q) [A] f STP(X, p, T ) Goto C f 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 = W n.thenb = {x 2 N (9t) STP(x, n, t)}. Theorem-8 f B and C are r.e sets so are B [ C and B \ C.

13 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),...}. Thatis,S is the range of f. Proof. By Theorem-7 we can write, S = {x (9t)R(x, t)}. Let x 0 2 S and u =< x, t >. Consider ( the following function x if R(x, t) f (u) = otherwise x 0 That is, f (< x, t >) =x.r(x, t)+x 0. R(x, t) Which is same as f (u) =l(u).r(l(u), r(u)) + x 0. (R(l(u), r(u))).

14 Theorem-10 Let f (x) be a partially computable function and let S = {f (x) f (x) #}. (Thatis,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] F STP(l(Z), p, r(z)) Goto B F f (l(z)) = X Goto E [B] Z Z + 1 Goto A

15 Theorem-11 Suppose that S 6= are all equivalent: 1. S is r.e..thenthefollowingstatements

Recursive Functions. Recursive functions are built up from basic functions by some operations.

Recursive Functions. Recursive functions are built up from basic functions by some operations. Recursive Functions Recursive functions are built up from basic functions by some operations. The Successor Function Let s get very primitive. Suppose we have 0 defined, and want to build the nonnegative