# Recursive function theory

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:<math>\mathbf 0 = 0</math> | :<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 | + | 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? |

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? | 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? | ||

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

− | ==== | + | ==== Successor function ==== |

:<math>\mathbf s : \left\lfloor1\right\rfloor</math> | :<math>\mathbf s : \left\lfloor1\right\rfloor</math> | ||

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

− | See the definition of being special <nowiki>[Mon:MathLog, 45]</nowiki>. 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. | + | See the definition of being ''special'' <nowiki>[Mon:MathLog, 45]</nowiki>. 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. |

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:<math>\underline\mu^m : \widehat m \to \left\lfloor m\right\rfloor</math> | :<math>\underline\mu^m : \widehat m \to \left\lfloor m\right\rfloor</math> | ||

:<math>\underline\mu^m f = \min \left\{y\in\mathbb N\;\vert\;f x_0 \dots x_{m-1} y = 0\right\}</math> | :<math>\underline\mu^m f = \min \left\{y\in\mathbb N\;\vert\;f x_0 \dots x_{m-1} y = 0\right\}</math> | ||

− | 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]. | + | 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]. |

== Partial recursive functions == | == Partial recursive functions == |

## Revision as of 17:40, 23 April 2006

## Contents |

## 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

### 3.2 Base functions

#### 3.2.1 Constant

Question: is the well-known approach superfluous? Can we avoid it and use the more simple and indirect approach, if we generalize operations (especially composition) to deal with zero-arity cases in an appropriate way? E.g., and , too? Does it take a generalization to allow such cases, or can it be inferred?

#### 3.2.2 Successor function

#### 3.2.3 Projection functions

For all :

### 3.3 Operations

#### 3.3.1 Composition

This resembles to the 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):

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

remembering us to

#### 3.3.2 Primitive recursion

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

## 4 General recursive functions

Everything seen above, and the new concepts:

### 4.1 Type system

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

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

Question: is there any sense to define in another way than simply ? Partial constants?

### 5.2 Operations

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.