Recursive function theory

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Revision as of 19:46, 23 April 2006

1 Introduction
2 Plans towards a programming language
3 Primitive recursive functions
4 General recursive functions
- 4.1 Type system
- 4.2 Operations
  - 4.2.1 Minimalization
5 Partial recursive functions
- 5.1 Type system
- 5.2 Operations
6 Bibliography

1 Introduction

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.

3 Primitive recursive functions

3.1 Type system

$\left\lfloor0\right\rfloor = \mathbb N$

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

3.2 Base functions

3.2.1 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?

Are these approaches equivalent? Is the latter (more simple) one as powerful as the former one? Could we define a $\mathbf{zero}\;x = 0$ using the $\mathbf{0} = 0$ approach? Let us try:

$\mathbf{zero} = \underline\mathbf K^0_1 \mathbf0$

This looks like working, but raises new questions: what about generalizing operations (here composition) to deal with zero-arity cases in an appropriate way? E.g.

$\underline\mathbf\dot K^0_n c\left\langle\right\rangle$

$\underline\mathbf K^0_n c$

where $c : \left\lfloor0\right\rfloor, n \in \mathbb N$ can be regarded as $n$ -ary functions throwing all their $n$ arguments away and returning $c$ .

Does it take a generalization to allow such cases, or can it be inferred?

3.2.2 Successor function

$\mathbf s : \left\lfloor1\right\rfloor$

$\mathbf s = \lambda x . x + 1$

3.2.3 Projection functions

For all $0\leq i<m$ :

$\mathbf U^m_i : \left\lfloor m\right\rfloor$

$\mathbf U^m_i x_0\dots x_i \dots x_{m-1} = x_i$

3.3 Operations

3.3.1 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 the nested tuple, and use a more homogenous style (somewhat 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

$\underline\mathbf K^m_n f g_0\dots g_{m-1} x_0 \dots x_{n-1} = \mathbf \Phi^n_m f g_0 \dots g_{m-1} x_0 \dots x_{n-1}$

3.3.2 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]):

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

4 General recursive functions

Everything seen above, and the new concepts:

4.1 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.

4.2 Operations

4.2.1 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].

5 Partial recursive functions

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

5.1 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?

5.2 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.

6 Bibliography

[HasFeyCr:CombLog1]: Curry, Haskell B; Feys, Robert; Craig, William: Combinatory Logic. Volume I. North-Holland Publishing Company, Amsterdam, 1958.
[Mon:MathLog]: Monk, J. Donald: Mathematical Logic. Springer-Verlag, New York * Heidelberg * Berlin, 1976.

Retrieved from "http://www.haskell.org/haskellwiki/Recursive_function_theory"

Category: Theoretical foundations

@@ Line 21: / Line 21: @@
 :<math>\mathbf 0 = 0</math>
-Question: is the well-known <math>\mathbf{zero}(x)=0</math> approach superfluous? Can we avoid it and use the more simple and indirect <math>\mathbf{0} = 0</math> approach, if we generalize operations (especially composition) to deal with zero-arity cases in an appropriate way?
+Question: is the well-known <math>\mathbf{zero}\;x = 0</math> approach superfluous? Can we avoid it and use the more simple and indirect <math>\mathbf{0} = 0</math> approach?
-E.g., <math>\underline\mathbf\dot K^0_1\mathbf0\left\langle\right\rangle = n</math> and <math>\underline\mathbf K^0_1\mathbf0 = n</math>, too?
+Are these approaches equivalent? Is the latter (more simple) one as powerful as the former one? Could we define a <math>\mathbf{zero}\;x = 0</math> using the <math>\mathbf{0} = 0</math> approach?
+Let us try:
+:<math>\mathbf{zero} = \underline\mathbf K^0_1 \mathbf0</math>
+This looks like working, but raises new questions: what about generalizing operations (here composition) to deal with zero-arity cases in an appropriate way?
+E.g.
+:<math>\underline\mathbf\dot K^0_n c\left\langle\right\rangle</math>
+:<math>\underline\mathbf K^0_n c</math>
+where <math>c : \left\lfloor0\right\rfloor, n \in \mathbb N</math>
+can be regarded as <math>n</math>-ary functions throwing all their <math>n</math> arguments away and returning <math>c</math>.
 Does it take a generalization to allow such cases, or can it be inferred?

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