A slide from lecture 12: Fundamentele Informatica 3 Definition 10.1. Initial Functions The initial functions are the following: voorjaar 2014 1. Constant functions: For each k ≥ 0 and each a ≥ 0, the a : N k → N is defined by the formula constant function C k http://www.liacs.nl/home/rvvliet/fi3/ C k for every X ∈ N k a ( X ) = a Rudy van Vliet kamer 124 Snellius, tel. 071-527 5777 2. The successor function s : N → N is defined by the formula rvvliet(at)liacs(dot)nl s ( x ) = x + 1 college 14, 12 mei 2014 3. Projection functions: For each k ≥ 1 and each i with 1 ≤ i : N k → N is defined by the 10. Computable Functions i ≤ k , the projection function p k 10.2. Quantification, Minimalization, and µ -Recursive formula Functions p k i ( x 1 , x 2 , . . . , x k ) = x i 10.3. G¨ odel Numbering 1 2 A slide from lecture 12: A slide from lecture 12: Definition 10.2. The Operations of Composition and Primitive Recursion (continued) Definition 10.2. The Operations of Composition and Primitive Recursion 2. Suppose n ≥ 0 and g and h are functions of n and n + 2 variables, respectively. (By “a function of 0 variables,” we 1. Suppose f is a partial function from N k to N , and for each i mean simply a constant.) with 1 ≤ i ≤ k , g i is a partial function from N m to N . The function obtained from g and h by the operation of primitive recursion is the function f : N n +1 → N defined by The partial function obtained from f and g 1 , g 2 , . . . , g k by composition is the partial function h from N m to N defined the formulas by the formula f ( X, 0) = g ( X ) h ( X ) = f ( g 1 ( X ) , g 2 ( X ) , . . . , g k ( X )) for every X ∈ N m f ( X, k + 1) = h ( X, k, f ( X, k )) for every X ∈ N n and every k ≥ 0. 3 4 A slide from lecture 12: Theorem 10.4. A slide from lecture 12: Every primitive recursive function is total and computable. n -place predicate P is function from N n to { true , false } characteristic function χ P defined by � 1 if P ( X ) is true χ P ( X ) = 0 if P ( X ) is false We say P is primitive recursive. . . PR : Turing-computable functions: total and computable not necessarily total 5 6 A slide from lecture 13: Definition 10.11. Bounded Minimalization For an ( n + 1)-place predicate P , the bounded minimalization of P is the function m P : N n +1 → N defined by � min { y | 0 ≤ y ≤ k and P ( X, y ) } if this set is not empty 10.2. Quantification, Minimalization, and m P ( X, k ) = k + 1 otherwise µ -Recursive Functions The symbol µ is often used for the minimalization operator, and we sometimes write k µ y [ P ( X, y )] m P ( X, k ) = An important special case is that in which P ( X, y ) is ( f ( X, y ) = 0), for some f : N n +1 → N . In this case m P is written m f and referred to as the bounded minimalization of f . 7 8
A slide from lecture 13: A slide from lecture 13: Example 10.13. The n th Prime Number Theorem 10.12. PrNo (0) = 2 If P is a primitive recursive ( n + 1)-place predicate, PrNo (1) = 3 PrNo (2) = 5 its bounded minimalization m P is a primitive recursive function. Prime ( n ) = ( n ≥ 2) ∧ ¬ (there exists y such that Proof. . . y ≥ 2 ∧ y ≤ n − 1 ∧ Mod ( n, y ) = 0) 9 10 A slide from lecture 12: A slide from lecture 13: Theorem 10.4. Example 10.13. The n th Prime Number Every primitive recursive function is total and computable. Let P ( x, y ) = ( y > x ∧ Prime ( y )) Then PrNo (0) = 2 PrNo ( k + 1) = m P ( PrNo ( k ) , ( PrNo ( k ))! + 1) is primitive recursive, with h ( x 1 , x 2 ) = . . . PR : Turing-computable functions: total and computable not necessarily total 11 12 Unbounded minimalization Total? Unbounded minimalization A possible definition: Total? � (min { y | P ( X, y ) is true } ) + 1 if this set is not empty M ( X ) = 0 otherwise Computable? 13 14 Definition 10.14. Unbounded Minimalization A slide from lecture 13: If P is an ( n + 1)-place predicate, the unbounded minimalization Unbounded quantification of P is the partial function M P : N n → N defined by ( y 2 = x ) Sq ( x, y ) = M P ( X ) = min { y | P ( X, y ) is true } M P ( X ) is undefined at any X ∈ N n for which there is no y satis- H ( x, y ) = T u stopt na precies y stappen voor invoer s x fying P ( X, y ). 15 16
Definition 10.15. µ -Recursive Functions Definition 10.14. Unbounded Minimalization The set M of µ -recursive, or simply recursive , partial functions is defined as follows. If P is an ( n + 1)-place predicate, the unbounded minimalization of P is the partial function M P : N n → N defined by 1. Every initial function is an element of M . M P ( X ) = min { y | P ( X, y ) is true } M P ( X ) is undefined at any X ∈ N n for which there is no y satis- 2. Every function obtained from elements of M by composition fying P ( X, y ). or primitive recursion is an element of M . 3. For every n ≥ 0 and every total function f : N n +1 → N in M , The notation µ y [ P ( X, y )] is also used for M P ( X ). the function M f : N n → N defined by In the special case in which P ( X, y ) = ( f ( X, y ) = 0), we write M P = M f and refer to this function as the unbounded minimal- M f ( X ) = µ y [ f ( X, y ) = 0] ization of f . is an element of M . 17 18 Exercise. Example. a. Give an example of a non-total function f and another func- Let 1 ( x, k ) . tion g , such that the composition of f and g is total. f ( x, k ) = p 2 − C 2 1 ( x, k ) b. Can you also find an example of a non-total function f and M f ( x ) . . . another function g , such that the composition of g and f is total? 19 20 Theorem 10.16. 10.3. G¨ odel Numbering All µ -recursive partial functions are computable. Proof. . . 21 22 Definition 10.17. Example 10.18. The G¨ odel Number of a Sequence of Natural Numbers The Power to Which a Prime is Raised in the Factorization of x For every n ≥ 1 and every finite sequence x 0 , x 1 , . . . , x n − 1 of n natural numbers, the G¨ odel number of the sequence is the Function Exponent : N 2 → N defined as follows: number � the exp. of PrNo ( i ) in x ’s prime fact. if x > 0 gn ( x 0 , x 1 , . . . , x n − 1 ) = 2 x 0 3 x 1 5 x 2 . . . ( PrNo ( n − 1)) x n − 1 Exponent ( i, x ) = 0 if x = 0 where PrNo ( i ) is the i th prime (Example 10.13). 23 24
Theorem 10.19. Suppose that g : N n → N and h : N n +2 → N are primitive recursive Example. functions, and f : N n +1 → N is obtained from g and h by course- of-values recursion; that is Fibonacci f ( X, 0) = g ( X ) 0 if n = 0 f ( n ) = 1 if n = 1 f ( X, k + 1) = h ( X, k, gn ( f ( X, 0) , . . . , f ( X, k ))) f ( n − 1) + f ( n − 2) if n ≥ 2 Then f is primitive recursive. Proof. . . 25 26
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