|
|
Line 21: |
Line 21: |
|
:<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 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? |
|
− |
Does it take a generalization to allow such cases, or can it be inferred? |
|
|
|
|
|
|
==== Successor function ==== |
|
==== Successor function ==== |
Line 37: |
Line 35: |
|
|
|
|
|
=== Operations === |
|
=== Operations === |
|
+ |
|
|
+ |
Are we allowed to generalize operations (especially composition) to deal with zero-arity cases in an appropriate way? |
|
⚫ |
E.g., The arity of <math>\underline\mathbf\dot K^ 0_n c\left\langle\right\rangle</math> and <math>\underline\mathbf K^ 0_n c</math> (where <math>c : \mathbb N</math> ) turns out to be undefined, so we may exclude such constructs. |
|
+ |
What about 0-arity cases, when can they be allowed or inferred? |
|
|
|
|
|
==== Composition ==== |
|
==== Composition ==== |
Revision as of 18:28, 23 April 2006
Introduction
PlanetMath article
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.
Primitive recursive functions
Type system
Base functions
Constant
Question: is the well-known approach superfluous? Can we avoid it and use the more simple and indirect approach?
Successor function
Projection functions
For all :
Operations
Are we allowed to generalize operations (especially composition) to deal with zero-arity cases in an appropriate way?
E.g., The arity of and (where ) turns out to be undefined, so we may exclude such constructs.
What about 0-arity cases, when can they be allowed or inferred?
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
Primitive recursion
-
The last equation resembles to the combinator of Combinatory logic (as described in [HasFeyCr:CombLog1, 169]):
General recursive functions
Everything seen above, and the new concepts:
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.
Operations
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].
Partial recursive functions
Everything seen above, but new constructs are provided, too.
Type system
Question: is there any sense to define
in another way than simply ? Partial constants?
Operations
Their definitions are straightforward.
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.