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Recursive function theory

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1 Introduction

PlanetMath article

2 Plans towards a programming language

Well-known concepts are taken from [Mon:MatLog], but several new notations (only notations, not concepts) are introduced to reflect all concepts described in [Mon:MatLog], and some simplification are made (by allowing zero-arity generalizations). These are plans to achive formalizations that can allow us in the future to incarnate the main concepts of recursive function theory in a programming language.

2.1 Primitive recursive functions

2.1.1 Type system

\left\lfloor0\right\rfloor = \mathbb N
\begin{matrix}\left\lfloor n + 1\right\rfloor = \underbrace{\mathbb N\times\dots\times\mathbb N}\to\mathbb N\\\;\;\;\;\;\;\;\;n+1\end{matrix}

2.1.2 Base functions Constant
\mathbf 0 : \left\lfloor0\right\rfloor
\mathbf 0 = 0

Question: is the well-known \mathbf{zero}(x)=0 approach superfluous? Can we avoid it and use the more simple and indirect \mathbf{0} = 0 approach, if we generalize operations (especially composition) to deal with zero-arity cases in an approprate way? E.g., \underline\mathbf\dot K^0_1\mathbf0\left\langle\right\rangle = n and \underline\mathbf K^0_1\mathbf0 = n, too? Does it take a generalization to allow, or can it be inferred? Succesor function
\mathbf s : \left\lfloor1\right\rfloor
\mathbf s = \lambda x . x + 1 Projection functions

For all 0\leq i<m:

\mathbf U^m_i : \left\lfloor m\right\rfloor
\mathbf U^m_i x_0\dots x_i \dots x_{m-1} = x_i

2.1.3 Operations Composition
\underline\mathbf\dot K^m_n : \left\lfloor m\right\rfloor \times \left\lfloor n\right\rfloor^m \to \left\lfloor n\right\rfloor
\underline\mathbf\dot K^m_n f \left\langle g_0,\dots, g_{m-1}\right\rangle x_0 \dots x_{n-1} = f \left(g_0 x_0 \dots x_{n-1}\right) \dots \left(g_{m-1} x_0 \dots x_{n-1}\right)

This resembles to the \mathbf\Phi^n_m combinator of Combinatory logic (as described in [HasFeyCr:CombLog1, 171]). If we prefer avoiding the notion of tuple, and use a style more resembling to currying:

\begin{matrix}\underline\mathbf K^m_n : \left\lfloor m\right\rfloor \times \underbrace{\left\lfloor n\right\rfloor\times\dots\times\left\lfloor n\right\rfloor} \to \left\lfloor n\right\rfloor\\\;\;\;\;\;\;\;\;\;\;m\end{matrix}

Let underbrace not mislead us -- it does not mean any bracing.

\underline\mathbf K^m_n f g_0\dots g_{m-1} x_0 \dots x_{n-1} = f \left(g_0 x_0 \dots x_{n-1}\right) \dots \left(g_{m-1} x_0 \dots x_{n-1}\right)

remembering us to

\underline\mathbf K^m_n f g_0\dots g_{m-1} x_0 \dots x_{n-1} = \mathbf \Phi^n_m f g_0 \dots g_{m-1} x_0 \dots x_{n-1} Primitive recursion
\underline\mathbf R^m : \left\lfloor m\right\rfloor \times \left\lfloor m+2\right\rfloor \to \left\lfloor m+1\right\rfloor
\underline\mathbf R^m f h = g\;\mathrm{where}
g x_0 \dots x_{m-1} 0 = f x_0 \dots x_{m-1}
g x_0 \dots x_{m-1} \left(\mathbf s y\right) = h x_0 \dots x_{m-1} y \left(g x_0\dots x_{m-1} y\right)

The last equation resembles to the \mathbf S_n combinator of Combinatory logic (as described in [HasFeyCr:CombLog1, 169]):

g x_0 \dots x_{m-1} \left(\mathbf s y\right) = \mathbf S_{m+1} h g x_0 \dots x_{m-1} y

2.2 General recursive functions

Everything seen above, and the new concepts:

2.2.1 Type system

 \widehat{\,m\,} = \left\{ f : \left\lfloor m+1\right\rfloor\;\vert\;f \mathrm{\ is\ special}\right\}

See the definition of being special [Mon:MathLog, 45]. This property ensures, that minimalization does not lead us out of the world of total functions. Its definition is the rather straightforward formalization of this expectation.

2.2.2 Operations Minimalization
\underline\mu^m : \widehat m \to \left\lfloor m\right\rfloor
\underline\mu^m f = \min \left\{y\in\mathbb N\;\vert\;f x_0 \dots x_{m-1} y = 0\right\}

Minimalization does not lead us out of the word of total functions, if we use it only for special functions -- the property of being special is defined exactly for this purpose [Mon:MatLog, 45].

2.3 Partial recursive functions

Everything seen above, but new constructs are provided, too.

2.3.1 Type system

\begin{matrix}\left\lceil n + 1\right\rceil = \underbrace{\mathbb N\times\dots\times\mathbb N}\supset\!\to\mathbb N\\\;\;\;\;\;\;n+1\end{matrix}

Question: is there any sense to define \left\lceil0\right\rceil in another way than simply \left\lceil0\right\rceil = \left\lfloor0\right\rfloor = \mathbb N? Partial constants?

2.3.2 Operations

\overline\mathbf\dot K^m_n : \left\lceil m\right\rceil \times \left\lceil n\right\rceil^m \to \left\lceil n\right\rceil
\overline\mathbf R^m : \left\lceil m\right\rceil \times \left\lceil m+2\right\rceil \to \left\lceil m+1\right\rceil
\overline\mu^m : \left\lceil m+1\right\rceil \to \left\lceil m\right\rceil

Their definitions are straightforward.

3 Bibliography

Curry, Haskell B; Feys, Robert; Craig, William: Combinatory Logic. Volume I. North-Holland Publishing Company, Amsterdam, 1958.
Monk, J. Donald: Mathematical Logic. Springer-Verlag, New York * Heidelberg * Berlin, 1976.