Recursive function theory
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
This allows us to deal with a concept of zero in recursive function theory. In the literature (in [Mon:MathLog], too) this aim is achieved in another way: a
is defined instead. Is this approach superfluously overcomplicated? Can we avoid it and use the more simple and indirect looking
Are these approaches equivalent? Is the latter (more simple looking) one as powerful as the former one? Could we define a using the
approach? Let us try:
(see the definition of somewhat below). 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.
where 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 they be inferred? A practical approach to solve such questions: let us write a Haskell program which implements (at least partially) recursive function theory. Then we can see clearly which things have to be defined and things which are consequences. I think the construct is a rather straighforward thing.
Why all this can be important: it may be exactly that saves us from defining the concept of zero in recursive function theory as
-- it may be superfluous: if we need functions that throw away (some or all) of their arguments and return a constant, then we can combine them from , and , if we allow concepts like .
3.2.2 Successor function
3.2.3 Projection functions
For all :
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.
It resembles to the concept of inverse -- more exactly, to the existence part.
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]. As we can see, minimalization is a concept resembling somehow to the concept of inverse.
Existence of the required minimum value of the set -- a sufficient and satisfactory condition for this is that the set is never empty. And this is equivalent to the statement
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 constant? Is
Their definitions are straightforward extension of the corresponding total function based definitions.
Remark: these operations take partial functions as arguments, but they are total operations themselves in the sense that they always yield a result -- at worst an empty function (as an ultimate partial function).
- 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.