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 September 4th, 2018, 02:36 PM #1 Newbie   Joined: Sep 2018 From: Guadalajara Posts: 8 Thanks: 0 Function decomposition Greetings all, I read in a book that “a logical, mathematical fact about functions is that functions of two arguments, which include all of the basic arithmetic functions, cannot be decomposed into functions of one-argument”. The authors give no demonstration of this, nor do they cite any reference where a demonstration is given. Are the authors right? Is there a demonstration of this? Where? Many thanks in advance, Jose
September 4th, 2018, 04:11 PM   #2
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Quote:
 Originally Posted by JoseEnrique Greetings all, I read in a book that “a logical, mathematical fact about functions is that functions of two arguments, which include all of the basic arithmetic functions, cannot be decomposed into functions of one-argument”. The authors give no demonstration of this, nor do they cite any reference where a demonstration is given. Are the authors right? Is there a demonstration of this? Where? Many thanks in advance, Jose
It would be helpful to know the context. I don't see it as being true in general but they may mean something else.

Given three sets $X$, $Y$, and $Z$ and a function $f : X \times Y \to Z$ given by $z = f(x,y)$, we can consider a collection of functions $\{f_x : x \in X\}$ where $f_x : Y \to Z$ and $f_x(y) = f(x,y)$.

Then given an expression $f(x,y)$ we can just input $y$ into the appropriate $f_x$. In fact we have two functions: one that inputs an element $x \in X$ and outputs some function $f_x$; and some particular $f_x$, into which we input $y$. In this way we transform a two-variable function into a composition of one-variable functions. In computer science this is called Currying.

That's one way to interpret your statement. But your book might be intending an entirely different interpretation. That's why we need more context. What's the subject, what's the book, what is the chapter about, what point are they trying to make.

Last edited by Maschke; September 4th, 2018 at 04:17 PM.

September 4th, 2018, 07:29 PM   #3
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Quote:
 Originally Posted by Maschke In computer science this is called Currying.
I never knew this had a name. Cool!

 September 4th, 2018, 09:40 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,285 Thanks: 1968 The book is presumably Memory and the Computational Brain (by C. R. Gallistel, Adam Philip King). It can be found online. Thanks from JoseEnrique
 September 6th, 2018, 05:57 AM #5 Newbie   Joined: Sep 2018 From: Guadalajara Posts: 8 Thanks: 0 Yes, that's it.
 September 6th, 2018, 06:20 AM #7 Newbie   Joined: Sep 2018 From: Guadalajara Posts: 8 Thanks: 0 The other possibility I thought to refute the authors' claim for the basic arithmetic operations was this (e.g., for the case of multiplication): Let m(x,y)=x*y One way to decompose this two-argument function into two one-argument functions is by introducing the identity function f(x)=x. We could thus write m(x,y)=f(x)*f(y) Alas, I do not know if this is a correct refutation. I am not a professional mathematician. Cheers, Jose
 September 6th, 2018, 06:30 AM #8 Newbie   Joined: Sep 2018 From: Guadalajara Posts: 8 Thanks: 0 Perhaps this is the same that Maschke showed before? Last edited by JoseEnrique; September 6th, 2018 at 06:41 AM.
 September 6th, 2018, 08:09 AM #9 Newbie   Joined: Sep 2018 From: Guadalajara Posts: 8 Thanks: 0 Thank you for posting the context. The key part is found in the section The Limits to Functional Decomposition, where they assert this (I have added the special symbols that didn't make the cut and paste): "Functions of one argument cannot combine to do the work of functions with two. Take the integer multiplication function, f*: Z × Z → Z, which maps pairs of integers to a single integer. It is inherent in this function that the two arguments are combined into one value and cannot remain separate within two separate functions. If f*(x, y) = z, there is no way that we can create two functions f*_ part1: Z → Z and f*_ part2: Z → Z that enable us to determine z without eventually using some function that can be applied to multiple arguments. Thus, we cannot realize this and many other two-argument functions by composing one-argument functions. This elementary mathematical truth is a critical part of our argument as to why the architecture of a powerful computing device such as the brain must make provision for bringing the values of variables to the machinery that implements some primitive two-argument functions. The ability to realize functions of at least two arguments is a necessary condition for realizing functions of a non-trivial nature" (p. 49, emphasis mine). But then again, from Maschke's reply above, one can curry multiplication. Last edited by skipjack; September 6th, 2018 at 09:22 AM.
September 6th, 2018, 10:02 AM   #10
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Quote:
 Originally Posted by JoseEnrique But then again, from Maschke's reply above, one can curry multiplication.
In currying one of the functions inputs an element of a set and outputs a function. That's disallowed by the text.

Still, they seem to be going a long way to make a trivial point and then extrapolating that to something about minds or brains.

After all, every function can be regarded as taking one input, namely an element of the Cartesian product. The multiplication takes the ordered pair (3,5) and outputs 15. But the ordered pair is a single element in the Cartesian product of the reals with itself.

To me this seems like a long way to stretch a point. Indeed, to say "... a powerful computing device such as the brain ..." already makes the assumption that the brain is a computing device. There's no actual proof of this. Brains work very differently than Turing machines. [Arguments about machine learning algorithms won't help, since ML always reduces to a TM. ML programs run on conventional computing hardware].

So the author has already baked his conclusion into his argument. I'm not buying this.

Last edited by Maschke; September 6th, 2018 at 10:08 AM.

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