Definitional schemes for primitive recursive and computable functions

Computability ◽  
2021 ◽  
pp. 1-8
Author(s):  
Pace P. Nielsen
Keyword(s):  
Initial Function ◽  
Unary Operation ◽  
Partial Functions ◽  
Recursive Functions ◽  
Binary Operations ◽  
Finite Set ◽  
Primitive Recursive ◽  
Computable Functions

The unary primitive recursive functions can be defined in terms of a finite set of initial functions together with a finite set of unary and binary operations that are primitive recursive in their inputs. We reduce arity considerations, by show that two fixed unary operations suffice, and a single initial function can be chosen arbitrarily. The method works for many other classes of functions, including the unary partial computable functions. For this class of partial functions we also show that a single unary operation (together with any finite set of initial functions) will never suffice.

Church's thesis without tears

1983 ◽  
Vol 48 (3) ◽  
pp. 797-803 ◽  
Author(s):  
Fred Richman
Keyword(s):  
Function Theory ◽  
Computable Function ◽  
Modern Theory ◽  
Central Feature ◽  
Partial Functions ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Church's Thesis ◽  
The Church ◽  
Computable Functions

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

scholarly journals Corrigendum to: V. A. Sokolov, “On the Existence Problem of Finite Bases of Identities in the Algebras of Recursive Functions”, Modeling and analysis of information systems, vol. 27, no. 3, pp. 304-315, 2020. DOI: https://doi.org/10.18255/1818-1015-2020-3-304-315

2020 ◽  
Vol 27 (4) ◽  
pp. 510-511
Author(s):  
Valery Anatolyevich Demidov
Keyword(s):  
New York ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Reference List ◽  
Equational Logic ◽  
Partial Functions ◽  
Article Title ◽  
Modeling And Analysis ◽  
Recursive Functions ◽  
Primitive Recursive

The author regrets that in the original list the references [3] and [4] are in the wrong places and they should be rearranged. In addition, [3] has the wrong article title. The corrected reference list is shown below.The author would like to apologize for an inconvenience caused.References[1] A. I. Mal'tsev, “Constructive algebras I”, Russian Mathematical Surveys, vol. 16, no. 3, pp. 77-129, 1961.[2] A. I. Mal'tsev, Algoritmy i rekursivnye funktsii. Moscow: Nauka, 1965, In Russian.[3] R. M. Robinson, “Primitive recursive functions”, Bulletin of the American Mathematical Society, vol. 53, no. 10,pp. 925-942, 1947.[4] J. Robinson, “General recursive functions”, Proceedings of the American Mathematical Society, vol. 1, no. 6,pp. 703-718, 1950.[5] V. A. Sokolov, “Ob odnom klasse tozhdestv v algebre Robinsona”, in 14-ya Vsesoyuznaya algebraicheskaya konferentsiya: tezisy dokladov, In Russian, vol. 2, Novosibirsk, 1977, pp. 123-124.[6] P. M. Cohn, Universal Algebra. New York, Evanston, and London: Harper & Row, 1965.[7] A. Robinson, “Equational logic for partial functions under Kleene equality: a complete and an incomplete set of rules”, The Journal of Symbolic Logic, vol. 54, no. 2, pp. 354-362, 1989.

N-ARY OPERATIONS WITH LONG PRE-PERIODS

2009 ◽  
Vol 02 (02) ◽  
pp. 201-212
Author(s):  
K. Denecke ◽  
Ch. Ratanaprasert
Keyword(s):  
Unary Operation ◽  
Equivalence Relations ◽  
Finite Set

Iterating a unary operation f defined on the finite set A with |A| = k one obtains the descending chain [Formula: see text] The least integer λ(f) with Imfλ(f) = Imfλ(f)+1 is called the pre-period of f. The pre-period of f is an integer between 0 and k - 1. If λ(f) = k - 1 and k ≥ 1, then f is called a long-tailed (LT)-operation and if λ(f) = k - 2 for k ≥ 2, f is said to be an (LT1)-operation. Unary (LT)- and (LT1)-operations and their invariant equivalence relations are characterized in [5]. In [6] these results are extended to partial operations. In this paper we consider the iteration of n-ary operations for n > 1, define and characterize (LT)- and (LT1)- operations and their invariant equivalence relations. The results can be applied in all fields where iteration and recursion plays a role.

scholarly journals Recursive density types and Nerode extensions of arithmetic

1975 ◽  
Vol 20 (2) ◽  
pp. 146-158 ◽  
Author(s):  
P. Aczel
Keyword(s):  
Equivalence Classes ◽  
Partial Functions ◽  
Recursive Functions ◽  
Cardinal Numbers ◽  
Partial Recursive Functions

The notion of a recursive density type (R.D.T.) was introduced by Medvedev and developed by Pavlova (1961). More recently the algebra of R.D.T.'s was initiated by Gonshor and Rice (1969). The R.D.T.'s are equivalence classes of sets of integers, similar in many respects to the R.E.T.'s. They may both be thought of as effective analogues of the cardinal numbers. While the equivalence relationfor R.E.T.'s is defined in terms of partial recursive functions, that for R.D.T.'s may be characterized in terms of recursively bounded partial functions (see 4.22a).

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).

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