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
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