Primitive recursive functional explained

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.

Background

Every primitive recursive functional has a type, which says 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.

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.

Definition

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

• 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

See also

• Dialectica interpretation
• Higher-order function
• Primitive recursive function
• Simply typed lambda calculus

References

• Book: Gödel's functional ("Dialectica") interpretation. Jeremy Avigad and Solomon Feferman. in S. Buss ed., The Handbook of Proof Theory, North-Holland. 1999. 337 - 405.