My original question was: Is Kappa calculus less powerful than Lambda calculus?

Does the lack of Higher-Order functions on a programming language excludes some programs that could only be written in functional programming (and therefore in a turing machine)?

By "powerful" I mean a Turing Complete programming language. For example, a language that can not contain infinite recursive expressions, ie infinite recursion or an infinite while loop, is not as powerful as Lambda-Calculus and Turiung Machines, since it would otherwise be a contradiction to the Halting Problem.

I can't find the reference right now, byt I remember something about an algorithm that on a non-lazy functional programming language has an $\Omega( n \log n )$ complexity, while the same algorithm in imperative programming is $\Omega( n )$

  • $\begingroup$ What is the Kappa calculus? Perhaps you could provide a link and perhaps some more details. $\endgroup$ Jun 5 '12 at 15:15
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    – Kaveh
    Jun 5 '12 at 20:13
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    $\begingroup$ Please consider posting general level CS questions on Computer Science. $\endgroup$
    – Kaveh
    Jun 5 '12 at 20:14
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    $\begingroup$ Higher-order functions are not necessary to obtain Turing-completeness. But recursion is. $\endgroup$
    – Uday Reddy
    Jun 5 '12 at 21:10
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    – Kaveh
    Jun 6 '12 at 21:30

Even the full simply typed lambda calculus with products is not Turing-complete (cf. http://en.wikipedia.org/wiki/Simply_typed_lambda_calculus#General_observations), and since the kappa calculus is a fragment of it (if I understand the Wikipedia page correctly), it isn't either.


As a concrete example, try representing the exponentiation function in this language (or even in the simply typed lambda calculus). If you try to type the exponentiation function for the untyped version, you'll see the difficulties that come up. On the other hand, if you have some sort of polymorphism then it is quite easy to represent exponentiation.

The functions on the natural numbers that the simply typed lambda calculus can represent is exactly that of the extended polynomial functions on the natural numbers, i.e. the class of functions generated by the constant functions outputting 0 and 1 and the operations of addition, multiplication and conditionals.

As Jan Johannsen says above, since the kappa calculus is (as I also am led to understand from the Wikipedia page) a fragment of the simply typed lambda calculus, its expressive power will consequently be less.

For the above result about the representable functions in the simply typed lambda calculus, see: H. Schwichtenberg. Definierbare Funktionen im $\lambda$-Kalkul mit Typen. Archiv fur mathematische Logik und Grundlagenforschung, 17 (1976), pp. 113{114

  • $\begingroup$ "since the kappa calculus is a fragment of the simply typed lambda calculus, its expressive power will consequently be lesser." -- surely you mean "no greater" rather than "lesser"? $\endgroup$ Jun 6 '12 at 9:32
  • $\begingroup$ :-) Yeah, I somehow figured though that if it was as expressive then people wouldn't study it by itself, but I think now that it might be a wrong assumption to make, since by expressive power I am talking about representing functions between the natural numbers, which might a priori be equally representable in both languages. $\endgroup$
    – tci
    Jun 6 '12 at 14:06
  • $\begingroup$ Note that the first word in that title should actually be "Definierbare" (fi ligature swallowed by copy/paste?) $\endgroup$ Jun 7 '12 at 11:57

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