On primitive recursive permutations and their inverses

1970 ◽  
Vol 34 (4) ◽  
pp. 634-638 ◽  
Author(s):  
Frank B. Cannonito ◽  
Mark Finkelstein
Keyword(s):  
Recursive Function ◽  
Primitive Recursive ◽  
First Time

It has been known for some time that there is a primitive recursive permutation of the nonnegative integers whose inverse is recursive but not primitive recursive. For example one has this result apparently for the first time in Kuznecov [1] and implicitly in Kent [2] or J. Robinson [3], who shows that every singularly recursive function ƒ is representable aswhere A, B, C are primitive recursive and B is a permutation.

On the Gödel class with identity

1981 ◽  
Vol 46 (2) ◽  
pp. 354-364 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Recursive Function ◽  
Decision Problem ◽  
Decision Procedure ◽  
Finite Model ◽  
Quantification Theory ◽  
Free Variables ◽  
Primitive Recursive

The Gödel Class is the class of prenex formulas of pure quantification theory whose prefixes have the form ∀y1∀y2∃x1 … ∃xn. The Gödel Class with Identity, or GCI, is the corresponding class of formulas of quantification theory extended by inclusion of the identity-sign “ = ”. Although the Gödel Class has long been kndwn to be solvable, the decision problem for the Gödel Class with Identity is open. In this paper we prove that there is no primitive recursive decision procedure for the GCI, or, indeed, for the subclass of the GCI containing just those formulas with prefixes ∀y1∀y2∃x.Throughout this paper we take quantification theory to include, aside from logical signs, infinitely many k-place predicate letters for each k > 0, but no function signs or constants. Moreover, by “prenex formula” we include only those without free variables. A decision procedure for a class of formulas is a recursive function that carries a formula in the class to 0 if the formula is satisfiable and to 1 if not. A class is solvable iff there exists a decision procedure for it. A class is finitely controllable iff every satisfiable formula in the class has a finite model. Since we speak only of effectively specified classes, finite controllability implies solvability (but not conversely).The GCI has a curious history. Gödel showed the Gödel Class (without identity) solvable in 1932 [4] and finitely controllable in 1933 [5].

On definition trees of ordinal recursive functionals: Reduction of the recursion orders by means of type level raising

1982 ◽  
Vol 47 (2) ◽  
pp. 395-402 ◽  
Author(s):  
Jan Terlouw
Keyword(s):  
Lower Order ◽  
Recursive Function ◽  
Functional Interpretation ◽  
Crucial Step ◽  
Recursive Functions ◽  
Order Types ◽  
Primitive Recursive ◽  
Primitive Recursive Functionals

It is known that every < ε0-recursive function is also a primitive recursive functional. Kreisel has proved this by means of Gödel's functional-interpretation, using that every < ε0-recursive function is provably recursive in Heyting's arithmetic [2, §3.4]. Parsons obtained a refinement of Kreisel's result by a further examination of Gödel's interpretation with regard to type levels [3, Theorem 5], [4, §4]. A quite different proof is provided by the research into extensions of the Grzegorczyk hierarchy as done by Schwichtenberg and Wainer: this yields another characterization of the < ε0-recursive functions from which easily appears that these are primitive recursive functionals (see [5] in combination with [6, Chapter II]).However, these proofs are indirect and do not show how, in general, given a definition tree of an ordinal recursive functional, transfinite recursions can be replaced (in a straightforward way) by recursions over wellorderings of lower order types. The argument given by Tait in [9, pp. 189–191] seems to be an improvement in this respect, but the crucial step in it is (at least in my opinion) not very clear.

The recursive irrationality of π

1954 ◽  
Vol 19 (4) ◽  
pp. 267-274 ◽  
Author(s):  
R. L. Goodstein
Keyword(s):  
Real Number ◽  
Recursive Function ◽  
Present Note ◽  
Rational Numbers ◽  
Negative Integer ◽  
Exponential Series ◽  
General Recursive Function ◽  
Recursive Sequence ◽  
Image Position ◽  
Primitive Recursive

A primitive-recursive sequence of rational numbers sn is said to be primitive-recursively irrational, if there are primitive recursive functions n(k), i(p, q) > 0 and N(p, q) such that:1. (k)(n ≥ n(k) → ∣sn – sn(k)∣ < 2−k).2. (p)(q)(q > 0 & n ≥ N(p, q) → ∣sn ± p/q∣ > 1/i(p, q)).The object of the present note is to establish the primitive-recursive irrationality of a sequence which converges to π. In a previous paper we proved the primitive-recursive irrationality of the exponential series Σxn/n!, for all rational values of x, and showed that a primitive-(general-) recursively irrational sequence sn is strongly primitive-(general-)recursive convergent in any scale, where a recursive sequence sn is said to be strongly primitive-(general-)recursive convergent in the scale r (r ≥ 2), if there is a non-decreasing primitive-(general-) recursive function r(k) such that,where [x] is the greatest integer contained in x, i.e. [x] = i if i ≤ x < i + 1, [x] = —i if i ≤ —x < i+1, where i is a non-negative integer.A rational recursive sequence sn is said to be recursive convergent, if there is a recursive function n(k) such that.If a sequence sn is strongly recursive convergent in a scale r, then it is recursive convergent and its limit is the recursive real number where, for any k ≥ 0,.

Incompleteness along paths in progressions of theories

1962 ◽  
Vol 27 (4) ◽  
pp. 383-390 ◽  
Author(s):  
S. Feferman ◽  
C. Spector
Keyword(s):  
Number Theory ◽  
Recursive Function ◽  
Order Number ◽  
Basic Logic ◽  
Natural Numbers ◽  
Logical Deduction ◽  
First Order ◽  
Primitive Recursive Function ◽  
Primitive Recursive ◽  
Recursive Definitions

We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf. [1] §2 (C)).

scholarly journals The first example of a recursive function which is not primitive recursive

1979 ◽  
Vol 6 (4) ◽  
pp. 380-384 ◽  
Author(s):  
Cristian Calude ◽  
Solomon Marcus ◽  
Ionel Tevy
Keyword(s):  
Recursive Function ◽  
Primitive Recursive

On axiomatizability within a system

1953 ◽  
Vol 18 (1) ◽  
pp. 30-32 ◽  
Author(s):  
William Craig
Keyword(s):  
Recursive Function ◽  
Recursive Relation ◽  
Recursively Enumerable Set ◽  
Formal Systems ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Primitive Recursive ◽  
Recursive Set

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.

Arithmetical representation of recursively enumerable sets

1956 ◽  
Vol 21 (2) ◽  
pp. 162-186 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

Pseudo-complements and ordinal logics based on consistency statements

1966 ◽  
Vol 31 (3) ◽  
pp. 359-364 ◽  
Author(s):  
Robert A. Di Paola
Keyword(s):  
Recursive Function ◽  
Peano Arithmetic ◽  
Additional Property ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Primitive Recursive

Following [1] we write {n} for the nth recursively enumerable (re) set; that is, {n} = {x|VyT(n, x, y)}. By a “pair (T, α)” we mean a consistent re extension T of Peano arithmetic P and an RE-formula α which numerates the non-logical axioms of T in P [4]. Given a pair (T, α) and a particular formula which binumerates the Kleene T predicate in P, there can be defined a primitive recursive function Nα such that and which has the additional property that {Nα(Nα(n))} = ø for all n.

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