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Re: Pure functions?

On 10/31/08 at 3:05 AM, siegman at (AES) wrote:

>In asking questions about the term pure function, I'm not trying to
>be contentious or argumentative.  I'm just trying to learn a bit
>about the concept.

>When I search on this term in Google, Wikipedia is the first hit
>that comes up, with the opening statement:

>In computer programming, a function may be described as pure if both
>these statements about the function hold:

>1. The function always evaluates the same result value given the
>same argument value(s). The function result value cannot depend on
>any hidden information or state that may change as program execution
>proceeds, nor can it depend on any external input from I/O devices.

>2. Evaluation of the result does not cause any semantically
>observable side effect or output, such as mutation of mutable
>objects or output to I/O devices.

>The result value need not depend on all (or any) of the argument
>values. However, it must depend on nothing other than the argument

>So let's consider the constructs:

>f1 = #1^3 + #2^4 &
>f2 = #1^x + #2^y &

>both of which function perfectly well (I think) as pure functions in
>the Mathematica sense.


>My interpretation is that f1 is also a pure function per the Wiki


>The function f2 is not pure in the Wiki sense,


>because f2[arg1,arg2] can give very different results when used in
>different parts of a program, since its result depends on "hidden"
>information (hidden in some sense, anyway) about the values that may
>have been assigned to the parameters x and y elsewhere in the program.

Once the values of x,y are fixed, it should be clear f2 is no
different than f1. In fact, f1 is literally a specific instance
of f2 with x = 3 and y = 4. Each selection for x,y defines a new
function and all of these are pure functions. The fact you can
write a function in terms of symbols instead of numbers does not
change it from pure to impure.

Contrast this with the functions Wikipedia gives as examples of
impure functions. For example, the Mathematica function
RandomReal[] gives a different result every time it is called
without changing any parameters. This is key. Here, the
information needed to determine what will result when
RandomReal[] is executed is not available to you at all. You
have no way to predict what RandomReal will return. In contrast,
you specify f2 and know exactly what it will return all of the time.

>Finally, Mathematica seems to put some focus on the _unnamed_ (or
>"anonymous") property of its pure functions, while the Wiki
>statement makes no mention at all of this.

Yes. In fact, many of the functions not considered pure
functions in Mathematica meet the Wikipedia definition for pure
function. For example, Sin[x] always computes the sine of x and
clearly has no I/O side effect. But a function need not be
unnamed to be a pure function in the Mathematica sense. That is both:

     NormalDistribution[#1, #2]&

have obvious names and are pure functions in the Mathematica
sense. And your f2 above has a name, specifically f2. Instead of
the function being unnamed, I see the focus as being on the
unnamed arguments to the function. That is for Sin[x] the
argument has a name, x and the function Sin[#]& is the function
without a named argument.

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