Recursive function theory
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== Introduction == 
== Introduction == 

−  [http://planetmath.org/encyclopedia/RecursiveFunction.html PlanetMath article] 
+  * [http://planetmath.org/encyclopedia/RecursiveFunction.html PlanetMath article] 
+  * Dr Matt Fairtlough's [http://www.dcs.shef.ac.uk/~matt/teaching/04/com2030/lectures/tomlect13.pdf Introduction to recursive function theory] among his [http://www.dcs.shef.ac.uk/~matt/teaching/04/com2030/ lecture notes] 

−  == Plans towards a programming language == 
+  == Designed languages == 
+  * Dr Matt Fairtlough's [http://www.dcs.shef.ac.uk/~matt/teaching/04/com2030/lectures/tomlect16.pdf Minimal Programming Language (MIN)] is not exactly a recursive function theory language, but it is based on natural numbers, too and its equivalent power with partal recursive functions is shown in its description. 

−  Wellknown 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 zeroarity 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. 
+  == Implementations == 
+  [http://web.archive.org/web/20060713051445/http://www.dcs.shef.ac.uk/~matt/teaching/04/com2030/lectures/Recursion.hs In Haskell], among other implementations (e.g. written in Java) in Dr Matt Fairtlough's [http://web.archive.org/web/20060713073121/http://www.dcs.shef.ac.uk/~matt/teaching/04/com2030/ lecture notes] (see the bottom of the page). 

+  
+  == Motivations == 

+  
+  Wellknown concepts are taken from [Mon:MathLog], but several new notations (only notations, not concepts) are introduced to reflect all concepts described in [Mon:MathLog], and some simplifications are made (by allowing zeroarity generalizations). These are plans to achieve formalizations that can allow us in the future to incarnate the main concepts of recursive function theory in a ''toy'' programming language, to play with it so that some interesting concepts can be taught in a funny way. 

+  
+  The relatedness of this page to Haskell (and to functional programming) is very few. It seems to me that (programming in) recursive functional theory may be rather another world (e.g. currying is missing, too)  although the lack of variables (even lack of formal parameters) can yield a feeling resembling to [[pointfree]] style programming, or even to [[combinatory logic]]. 

+  
+  But despite of its weak (direct) relatedness to functional programming, maybe this page can be useful for someone in the future 

+  * e.g. when writing on quines (selfprinting programs or selfrepresenting formulas) etc. David Madore's [http://www.madore.org/~david/computers/quine.html Quines (selfreplicating programs)] page uses recursive function theory for explaining the theoretical roots of quines (e.g. [http://www.madore.org/~david/computers/quine.html#sec_fp fixed point theorem]) 

+  * or when writing on other general concepts of [[computer science]]. 

+  
+  Other few relatedness of this topic to Haskell may appear by the fact that the Haskell implementation of the mentioned toy programming language may use 

+  * tricks with types, [[type arithmetic]] 

+  * or metaprogramming concepts, at worst preprocessing steps 

+  because typesafe implementations of <math>\underline\mathbf\dot K^m_n</math> and <math>\underline\mathbf K^m_n</math> does not look straightforward for me. 

+  
+  == Notations == 

+  
+  ;<math>\mathbb N</math> 

+  :the set of natural numbers, incuding 0 

+  ;''Type system'' 

+  :I do not use this term in a precise way here: see the following items for explanation. Headlines ''Type system'' may suggest that the type system of the planned recursive function theory programming language implementation can prevent the user from applying a partial function to an outofdomain value  but I think that type safety in this sense cannot be achieved. 

+  ;<math>f:A\to B</math> 

+  :<math>f</math> is a total function from <math>A</math> to <math>B</math>. In an implemented recursion function theory language, it means a function that surely terminates. 

+  ;<math>f:A\supset\!\to B</math> 

+  :<math>f</math> is a partial function from <math>A</math> to <math>B</math> (it may be either a total function or a proper partial function). In an implemented recursion theory language, maybe this information (being partial) cannot be grasped by the type system. It may mean that proper partial functions simply fail to terminate, without reflecting this possibility in the type system in any way. 

+  ;<math>A\times B</math> 

+  :Descartesproduct of sets is used here for stressing the fact that recursive function theory does not know the concept of ''currying'' (much more other concepts will be needed to achieve something similar to currying, see [http://en.wikipedia.org/wiki/Smn_theorem Kleene's <math>s^m_n</math> theorem]). So using <math>f:A\times B\to C</math> (instead of <math>f:A\to B\to C</math>) is intended to stress the lack of currying (at this level). 

== Primitive recursive functions == 
== Primitive recursive functions == 

Line 14:  Line 14:  
:<math>\begin{matrix}\left\lfloor n + 1\right\rfloor = \underbrace{\mathbb N\times\dots\times\mathbb N}\to\mathbb N\\\;\;\;\;\;\;\;\;n+1\end{matrix}</math> 
:<math>\begin{matrix}\left\lfloor n + 1\right\rfloor = \underbrace{\mathbb N\times\dots\times\mathbb N}\to\mathbb N\\\;\;\;\;\;\;\;\;n+1\end{matrix}</math> 

−  === Base functions === 
+  === Initial functions === 
==== Constant ==== 
==== Constant ==== 

Line 21:  Line 21:  
:<math>\mathbf 0 = 0</math> 
:<math>\mathbf 0 = 0</math> 

−  Question: is the wellknown <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? 
+  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 

+  :<math>\mathbf{zero} : \left\lfloor1\right\rfloor</math> 

+  :<math>\mathbf{zero}\;x = 0</math> 

+  is defined instead. Is this approach superfluously overcomplicated? Can we avoid it and use the more simple and indirect looking 

+  :<math>\mathbf 0 : \left\lfloor0\right\rfloor</math> 

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

+  approach? 

−  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? 
+  Are these approaches equivalent? Is the latter (more simple looking) one as powerful as the former one? Could we define a <math>\mathbf{zero}</math> using the 
+  :<math>\mathbf 0 : \left\lfloor0\right\rfloor</math> 

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

+  approach? 

Let us try: 
Let us try: 

:<math>\mathbf{zero} = \underline\mathbf K^0_1 \mathbf0</math> 
:<math>\mathbf{zero} = \underline\mathbf K^0_1 \mathbf0</math> 

−  +  (see the definition of <math>\underline\mathbf K^m_n</math> somewhat below). 

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

:<math>\underline\mathbf\dot K^0_n c\left\langle\right\rangle</math> 
:<math>\underline\mathbf\dot K^0_n c\left\langle\right\rangle</math> 

Line 34:  Line 34:  
can be regarded as <math>n</math>ary functions throwing all their <math>n</math> arguments away and returning <math>c</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? 
+  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 <math>\underline\mathbf K^0_n c</math> construct is a rather straighforward thing. 

+  
+  Why all this can be important: it may be exactly <math>\underline\mathbf K^0_n c</math> that saves us from defining the concept of zero in recursive function theory as 

+  :<math>\mathbf{zero} : \left\lfloor1\right\rfloor</math> 

+  :<math>\mathbf{zero}\;x = 0</math> 

+   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 <math>\underline\mathbf K^m_n</math>, <math>\mathbf s</math> and <math>\mathbf 0</math>, if we allow concepts like <math>\underline\mathbf K^0_m</math>. 

==== Successor function ==== 
==== Successor function ==== 

Line 113:  Line 113:  
:<math>\overline\mathbf\dot K^m_n : \left\lceil m\right\rceil \times \left\lceil n\right\rceil^m \to \left\lceil n\right\rceil</math> 
:<math>\overline\mathbf\dot K^m_n : \left\lceil m\right\rceil \times \left\lceil n\right\rceil^m \to \left\lceil n\right\rceil</math> 

+  :<math>\begin{matrix}\overline\mathbf K^m_n : \left\lceil m\right\rceil \times \underbrace{\left\lceil n\right\rceil\times\dots\times\left\lceil n\right\rceil} \to \left\lceil n\right\rceil\\\;\;\;\;\;\;\;\;\;\;m\end{matrix}</math> 

+  
:<math>\overline\mathbf R^m : \left\lceil m\right\rceil \times \left\lceil m+2\right\rceil \to \left\lceil m+1\right\rceil</math> 
:<math>\overline\mathbf R^m : \left\lceil m\right\rceil \times \left\lceil m+2\right\rceil \to \left\lceil m+1\right\rceil</math> 

:<math>\overline\mu^m : \left\lceil m+1\right\rceil \to \left\lceil m\right\rceil</math> 
:<math>\overline\mu^m : \left\lceil m+1\right\rceil \to \left\lceil m\right\rceil</math> 

Their definitions are straightforward extension of the corresponding total function based definitions. 
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). 

== Bibliography == 
== Bibliography == 
Latest revision as of 13:13, 29 January 2012
Contents 
[edit] 1 Introduction
 PlanetMath article
 Dr Matt Fairtlough's Introduction to recursive function theory among his lecture notes
[edit] 2 Designed languages
 Dr Matt Fairtlough's Minimal Programming Language (MIN) is not exactly a recursive function theory language, but it is based on natural numbers, too and its equivalent power with partal recursive functions is shown in its description.
[edit] 3 Implementations
In Haskell, among other implementations (e.g. written in Java) in Dr Matt Fairtlough's lecture notes (see the bottom of the page).
[edit] 4 Motivations
Wellknown concepts are taken from [Mon:MathLog], but several new notations (only notations, not concepts) are introduced to reflect all concepts described in [Mon:MathLog], and some simplifications are made (by allowing zeroarity generalizations). These are plans to achieve formalizations that can allow us in the future to incarnate the main concepts of recursive function theory in a toy programming language, to play with it so that some interesting concepts can be taught in a funny way.
The relatedness of this page to Haskell (and to functional programming) is very few. It seems to me that (programming in) recursive functional theory may be rather another world (e.g. currying is missing, too)  although the lack of variables (even lack of formal parameters) can yield a feeling resembling to pointfree style programming, or even to combinatory logic.
But despite of its weak (direct) relatedness to functional programming, maybe this page can be useful for someone in the future
 e.g. when writing on quines (selfprinting programs or selfrepresenting formulas) etc. David Madore's Quines (selfreplicating programs) page uses recursive function theory for explaining the theoretical roots of quines (e.g. fixed point theorem)
 or when writing on other general concepts of computer science.
Other few relatedness of this topic to Haskell may appear by the fact that the Haskell implementation of the mentioned toy programming language may use
 tricks with types, type arithmetic
 or metaprogramming concepts, at worst preprocessing steps
because typesafe implementations of and does not look straightforward for me.
[edit] 5 Notations
 the set of natural numbers, incuding 0
 Type system
 I do not use this term in a precise way here: see the following items for explanation. Headlines Type system may suggest that the type system of the planned recursive function theory programming language implementation can prevent the user from applying a partial function to an outofdomain value  but I think that type safety in this sense cannot be achieved.
 f is a total function from A to B. In an implemented recursion function theory language, it means a function that surely terminates.
 f is a partial function from A to B (it may be either a total function or a proper partial function). In an implemented recursion theory language, maybe this information (being partial) cannot be grasped by the type system. It may mean that proper partial functions simply fail to terminate, without reflecting this possibility in the type system in any way.
 Descartesproduct of sets is used here for stressing the fact that recursive function theory does not know the concept of currying (much more other concepts will be needed to achieve something similar to currying, see Kleene's theorem). So using (instead of ) is intended to stress the lack of currying (at this level).
[edit] 6 Primitive recursive functions
[edit] 6.1 Type system
[edit] 6.2 Initial functions
[edit] 6.2.1 Constant
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
approach?
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 zeroarity cases in an appropriate way? E.g.
where can be regarded as nary 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 .
[edit] 6.2.2 Successor function
[edit] 6.2.3 Projection functions
For all :
[edit] 6.3 Operations
[edit] 6.3.1 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
[edit] 6.3.2 Primitive recursion
The last equation resembles to the combinator of Combinatory logic (as described in [HasFeyCr:CombLog1, 169]):
[edit] 7 General recursive functions
Everything seen above, and the new concepts:
[edit] 7.1 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.
[edit] 7.2 Operations
[edit] 7.2.1 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]. 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
[edit] 8 Partial recursive functions
Everything seen above, but new constructs are provided, too.
[edit] 8.1 Type system
Question: is there any sense to define in another way than simply ? Partial constant? Is
or
 ?
[edit] 8.2 Operations
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).
[edit] 9 Bibliography
 [HasFeyCr:CombLog1]
 Curry, Haskell B; Feys, Robert; Craig, William: Combinatory Logic. Volume I. NorthHolland Publishing Company, Amsterdam, 1958.
 [Mon:MathLog]
 Monk, J. Donald: Mathematical Logic. SpringerVerlag, New York * Heidelberg * Berlin, 1976.