Primitive Recursive and μ-Recursive Functions

Computability ◽  
1987 ◽  
pp. 42-52
Author(s):  
Klaus Weihrauch
Keyword(s):  
Recursive Functions ◽  
Primitive Recursive

scholarly journals Primitive recursive functions. II

1955 ◽  
Vol 6 (4) ◽  
pp. 663-663 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Functions ◽  
Primitive Recursive

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

A Macro Program for the Primitive Recursive Functions

1991 ◽  
Vol 37 (8) ◽  
pp. 121-124
Author(s):  
Hilbert Levitz ◽  
Warren Nichols ◽  
Robert F. Smith
Keyword(s):  
Recursive Functions ◽  
Primitive Recursive

The relation of A to Prov ˹A˺ in the Lindenbaum sentence algebra

1973 ◽  
Vol 38 (2) ◽  
pp. 295-298 ◽  
Author(s):  
C. F. Kent
Keyword(s):  
Boolean Algebra ◽  
Normal Forms ◽  
Order Relation ◽  
Mathematical Induction ◽  
Axiomatic Theory ◽  
Recursive Functions ◽  
Primitive Recursive

Let U be a consistent axiomatic theory containing Robinson's Q [TMRUT, p. 51]. In order for the results below to be of interest, U must be powerful enough to carry out certain arguments involving versions of the “derivability conditions,” DC(i) to DC(iii) below, of [HBGM, p. 285], [F60, Theorem 4.7], or [L55]. Thus it must contain, at least, mathematical induction for formulas whose prenex normal forms contain at most existential quantifiers. For convenience, U is assumed also to contain symbols for primitive recursive functions and relations, and their defining equations. One of these is used to form the standard provability predicate, Prov ˹A˺, “there exists a number which is the Gödel number of a proof of A.” Upper corners denote numerals for Gödel numbers for the enclosed sentences, and parentheses are often omitted in their presence.This paper contains some results concerning the relation between the sentence A, and the sentence Prov ˹A˺ in the Lindenbaum Sentence Algebra (LSA) for U, the Boolean algebra induced by the pre-order relation A ≤ B ⇔ ⊦A → B. Half of the answer is provided by a theorem of Löb [L55], which states that ⊦Prov ˹A˺ → A ⇔ ⊦A. Hence, in the presence of DC(iii), below, it is never true that Prov ˹A˺ < A in the LSA. However, there is a large and interesting set of sentences, denoted here by Γ, for which A < Prov ⌜A⌝.

Recursive functions in basic logic

1956 ◽  
Vol 21 (4) ◽  
pp. 337-346 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Basic Logic ◽  
Recursive Functions ◽  
Numerical Equality ◽  
Partial Recursive Functions ◽  
Image Position ◽  
Primitive Recursive ◽  
Double Arrow

1.1 The system K* of basic logic, as presented in a previous paper, will be shown to be formalizable in an alternative way according to which the rule [E],is replaced by the rule [F],1.2. General recursive functions will be shown to be definable in K* in a way that retains functional notation, so that the equation,will be formalized in K* by the formula,where ‘f’, ‘p1’, … ‘pn’ respectively denote φ, x1, …, xn, and where ‘≐’ plays the role of numerical equality. Partial recursive functions may be handled in a similar way. The rule [E] is not required for dealing with primitive recursive functions by this method.1.3. An operator ‘G’ will be defined such that ‘[Ga ≐ p]’ is a theorem of K* if and only if ‘p’ denotes the Gödel number of ‘a’.1.4. In reformulating K* we assume ‘o0’, ‘o1,’ ‘o2’, …, have been so chosen that we can determine effectively whether or not a given U-expression is the mth member of the above series. The revised rules for K* are then as follows. (Double-arrow forms of these rules are derivable, except in the case of rule [V].)

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