Primitive recursive functional

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In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist of a collection of functions in all pure finite types.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type.

Contents

The primitive recursive functionals are important in proof theory and constructive mathematics They are a central part of the Dialectica interpretation of intuitionistic arithmetic developed by Kurt Gödel.

Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic arithmetic into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic. The name of the interpretation comes from the journal Dialectica, where Gödel's paper was published in a 1958 special issue dedicated to Paul Bernays on his 70th birthday.

Kurt Gödel logician and mathematician

Kurt Friedrich Gödel was an Austro-Hungarian-born Austrian, and later American, logician, mathematician, and philosopher. Considered along with Aristotle, Alfred Tarski and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor.

In recursion theory, the primitive recursive functionals are an example of higher-type computability, as primitive recursive functions are examples of Turing computability.

Background

Every primitive recursive functional has a type, which tells what kind of inputs it takes and what kind of output it produces. An object of type 0 is simply a natural number; it can also be viewed as a constant function that takes no input and returns an output in the set N of natural numbers.

For any two types σ and τ, the type σ→τ represents a function that takes an input of type σ and returns an output of type τ. Thus the function f(n) = n+1 is of type 0→0. The types (0→0)→0 and 0→(0→0) are different; by convention, the notation 0→0→0 refers to 0→(0→0). In the jargon of type theory, objects of type 0→0 are called functions and objects that take inputs of type other than 0 are called functionals.

For any two types σ and τ, the type σ×τ represents an ordered pair, the first element of which has type σ and the second element of which has type τ. For example, consider the functional A takes as inputs a function f from N to N, and a natural number n, and returns f(n). Then A has type (0 × (0→0))→0. This type can also be written as 0→(0→0)→0, by Currying.

In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, a function that takes two arguments, one from X and one from Y, and produces outputs in Z, by currying is translated into a function that takes a single argument from X and produces as outputs functions from Y to Z. Currying is related to, but not the same as, partial application.

The set of (pure) finite types is the smallest collection of types that includes 0 and is closed under the operations of × and →. A superscript is used to indicate that a variable xτ is assumed to have a certain type τ; the superscript may be omitted when the type is clear from context.

Definition

The primitive recursive functionals are the smallest collection of objects of finite type such that:

See also

In mathematics and computer science, a higher-order function is a function that does at least one of the following:

In computability theory, primitive recursive functions are a class of functions that are defined using composition and primitive recursion – described below – as central operations. They are a strict subset of those µ-recursive functions which are also total functions. Primitive recursive functions form an important building block on the way to a full formalization of computability. These functions are also important in proof theory.

The simply typed lambda calculus, a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor: that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties.

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References

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