Primitive recursive set function

Last updated August 15, 2020

In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by Jensen & Karp (1971).

Definition

A primitive recursive set function is a function from sets to sets that can be obtained from the following basic functions by repeatedly applying the following rules of substitution and recursion:

The basic functions are:

Projection: P_n,m(x₁, ..., x_n) = x_m for 0 ≤ m ≤ n
Zero: F(x) = 0
Adjoining an element to a set: F(x, y) = x ∪ {y}
Testing membership: C(x, y, u, v) = x if u ∈ v, and C(x, y, u, v) = y otherwise.

The rules for generating new functions by substitution are

F(x, y) = G(x, H(x), y)
F(x, y) = G(H(x), y)

where x and y are finite sequences of variables.

The rule for generating new functions by recursion is

F(z, x) = G(∪_u ∈ z F(u, x), z, x)

A primitive recursive ordinal function is defined in the same way, except that the initial function F(x, y) = x ∪ {y} is replaced by F(x) = x ∪ {x} (the successor of x). The primitive recursive ordinal functions are the same as the primitive recursive set functions that map ordinals to ordinals.

One can also add more initial functions to obtain a larger class of functions. For example, the ordinal function ω^α is not primitive recursive, because the constant function with value ω (or any other infinite set) is not primitive recursive, so one might want to add this constant function to the initial functions.

Related Research Articles

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive.

Lambda calculus is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.

In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops. Primitive recursive functions form a strict subset of those general recursive functions that are also total functions.

Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances, it is often done in such a way that no infinite loop or infinite chain of references can occur.

In mathematical logic and computer science, a general recursive function or μ-recursive function, is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense. In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines. The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every μ-recursive function is a primitive recursive function—the most famous example is the Ackermann function.

In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations ; as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are a universal ordered field.

In mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function. In combinatory logic for computer science, a fixed-point combinator is a higher-order function $that returns some fixed point of its argument function, if one exists.$

In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem which constructs fixed points of a computable function is known as Rogers's theorem and is due to Hartley Rogers, Jr..

Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory.

In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if:

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel-Choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the five primitive recursive operators makes it possible to define all computable functions.

Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the μ-recursive functions.

In computability theory the S m
n theorem, is a basic result about programming languages. It was first proved by Stephen Cole Kleene (1943). The name S m
n comes from the occurrence of an S with subscript n and superscript m in the original formulation of the theorem.

In functional programming, fold refers to a family of higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing its constituent parts, building up a return value. Typically, a fold is presented with a combining function, a top node of a data structure, and possibly some default values to be used under certain conditions. The fold then proceeds to combine elements of the data structure's hierarchy, using the function in a systematic way.

Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Skolem as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε₀, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ω^ω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic.

In computer science and recursion theory the McCarthy Formalism (1963) of computer scientist John McCarthy clarifies the notion of recursive functions by use of the IF-THEN-ELSE construction common to computer science, together with four of the operators of primitive recursive functions: zero, successor, equality of numbers and composition. The conditional operator replaces both primitive recursion and the mu-operator.

In set theory, a mathematical discipline, the Jensen hierarchy or J-hierarchy is a modification of Gödel's constructible hierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchy figures prominently in fine structure theory, a field pioneered by Ronald Jensen, for whom the Jensen hierarchy is named.

The Grzegorczyk hierarchy, named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory. Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function appears in the hierarchy at some level. The hierarchy deals with the rate at which the values of the functions grow; intuitively, functions in lower level of the hierarchy grow slower than functions in the higher levels.

In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations that starts with a unary operation. The sequence continues with the binary operations of addition, multiplication, and exponentiation.

References

Jensen, Ronald B.; Karp, Carol (1971), "Primitive recursive set functions", Axiomatic Set Theory, Proc. Sympos. Pure Math., XIII, Part I, Providence, R.I.: Amer. Math. Soc., pp. 143–176, ISBN 9780821802458, MR 0281602

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.