Primitive recursive functional

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

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. 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. In recursion theory, the primitive recursive functionals are an example of higher-type computability, as primitive recursive functions are examples of Turing computability. 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. 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 x τ is assumed to have a certain type τ; the superscript may be omitted when the type is clear from context. The constant function f(n) = 0 is a primitive recursive functional The successor function g(n) = n + 1 is a primitive recursive functional For any type σ×τ, the functional K(x σ , y τ ) = x is a primitive recursive functional For any types ρ, σ, τ, the functional S(r ρ→σ→τ ,s ρ→σ , t ρ ) = (r(t))(s(t)) is a primitive recursive functional For any type τ, and f of type τ, and any g of type 0→τ→τ, the functional R(f,g) 0→τ defined recursively as R(f,g)(0) = f, R(f,g)(n+1) = g(n,R(f,g)(n)) is a primitive recursive functional

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