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(unknown charset) Re: Content markups, piecewise functions, conditions

From: (unknown charset) Andreas Strotmann <strotman@cs.fsu.edu>
Date: Wed, 12 Jul 2000 18:01:52 -0400 (EDT)
To: (unknown charset) Stan Devitt <jsdevitt@radicalflow.com>
cc: (unknown charset) GĂ©rald QUINTANA <quintana@lyon.objectif.fr>, www-math@w3.org
Message-ID: <Pine.GSO.4.10.10007121711170.1848-100000@xi.cs.fsu.edu>
Hi all,

this is actually quite an interesting question, and I don't think I have a
quick answer, either.  Personally, my analysis would deviate somewhat from
Stan's, however.

Off the bat, I would prefer getting the <bvar>s out of the picture, and to
factor the question of representing piecewise functions into the two
"orthogonal" concepts of "functions" and "pieces"/"cases".  There is ample
historical precedence for this approach, of course - LISP's lambda
and cond, plus pretty much all other programming languages do that.

In this sense, sign(x) would be something like:

  sign = lambda(x,  cond( (x<0,-1), (x>0,+1), (x=0,0)))

Note that in this representation, "cond" does not bind variables, and
conditionals can in principle involve arbitrary free variables.

The latter is actually quite useful, I believe, since there are some
well-known scenarios beyond that of piece-wise functions where it is
"parameters" rather than arguments that are used to distinguish cases.
Tables of integrals abound with entries where conditions involving one or
more parameters of the integrand determine the actual closed-form solution
for the integral -- consider

 integral of x^a wrt x      is    x^(a+1)/(a+1)
 unless a=0, in which case it is  ln(x).

for a well-known example;-)

Since parameter a in this example is not a bound variable, MathML's
semantics do not allow the use of <condition> elements here as they
require corresponding <bvar>s.

Thus, the easiest solution with the least amount of impact on MathML
would probably be to represent the sign example above as

  sign = lambda(x, ( x<0 => -1 )  or  ( x>0 => +1 )  or  ( x=0 => 0))

with appropriately extended semantics of <or/>(*) and <implies/>.

Regards,

   Andreas Strotmann

(*) or perhaps <xor/> if at most one case is allowed to return a result.
Multi-branch functions, e.g. the "plus or minus" in solutions to quadratic
equations, might use <or/> to denote admissibility of results from any
conditional branch.

____________________________________________________________
"The act of defending any of the cardinal virtues has today 
all the exhilaration of a vice." -
G.K.Chesterton: A Defense of Humilities, The Defendant, 1901 
                           www.chesterton.org/acs/quotes.htm

On Tue, 11 Jul 2000, Stan Devitt wrote:

> Note that the important thing to do is to capture
> the information.  The essence of a piecewise object is:
> 
>     1.  there is a bound variable
>     2.  there is one or more conditions on that variable
>         bound to outcomes, typically computed using that
>         variable.
>     3.  possibly a default outcome
> 
> MathML does NOT evaluate or simplify its expressions.
> 
> With this in mind, you probably want to choose between
> 
>  piecewise( bvar , (condition, value )*, default )
> 
> and
> 
> piecewise( bvar , default , conditiongroup * )
> 
> where
>         conditiongroup :=   conditiongroup( condition , value )
> 
> Either one will do the job.  Both are easy to encode using
> csymbol, and bvar, at least in MathML 2.0.  (There were some
> funny restrictions in MathML 1.0 that did not allow bvars 
> in new expressions and required function wrappers in the
> first argument to apply.)  In 2.0 the first of these would look
> something like:
> 
>     <apply>
>         <bvar>x</bvar>
>         <csymbol definitionURL="mydefinitionforpiecewise">piecewise</csymbol>
>         <apply><lt/> ....</apply>
>         <apply><times/> ...</apply>
>         <apply></lt/> ... </apply>
>          value 2
>          default value
>     </apply>
> 
> 
> Note that reln is deprecated in MathML 2.0
> Also, generally speaking it is a bad idea to encode essential
> information in comments as sometimes processing strips those
> comments away.
> 
> There is no real need to use the lambda constructs as that information
> would already be contained in the formal definition, and is really more concerned
> with evaluation than representation.
> 
> Stan Devitt
> 
>   ----- Original Message ----- 
>   From: Gérald QUINTANA 
>   To: www-math@w3.org 
>   Sent: Tuesday, July 11, 2000 11:45 AM
>   Subject: Content markups, piecewise functions, conditions
> 
> 
>   Hi,
> 
>   I am thinking about how I could code a piecewise function using
>   content markups. I need content markups because I aim at making a small
>   MathML "parser-compiler". I didn't understood exactly how conditions were
>   working. So as to code the sign function (returns -1 when x<0, 0 when x=0
>   and +1 when x>0) what do you think of this ? How can I tell that those 3
>   definitions belongs to the same function ?
> 
>   <?xml version="1.0" encoding="UTF-8"?>
>   <!DOCTYPE math SYSTEM "file://localhost/S:/java/xerces/mathml/mathml.dtd" >
>   <math>
>     <!--If x<0 then -1-->
>     <lambda>
>       <bvar>
>         <ci>x</ci>
>       </bvar>
>       <apply>
>         <bvar>
>           <ci>x</ci>
>         </bvar>
>         <condition>
>           <reln>
>             <lt/>
>             <ci>x</ci>
>             <cn>0</cn>
>           </reln>
>         </condition>
>         <apply>
>           <minus/>
>           <cn>1</cn>
>         </apply>
>       </apply>
>     </lambda>
>     <!--If x=0 then 0-->
>     <lambda>
>       <bvar>
>         <ci>x</ci>
>       </bvar>
>       <apply>
>         <bvar>
>           <ci>x</ci>
>         </bvar>
>         <condition>
>           <reln>
>             <eq/>
>             <ci>x</ci>
>             <cn>0</cn>
>           </reln>
>         </condition>
>         <cn>0</cn>
>       </apply>
>     </lambda>
>     <!--If x>0 then +1-->
>     <lambda>
>       <bvar>
>         <ci>x</ci>
>       </bvar>
>       <apply>
>         <bvar>
>           <ci>x</ci>
>         </bvar>
>         <condition>
>           <reln>
>             <eq/>
>             <ci>x</ci>
>             <cn>0</cn>
>           </reln>
>         </condition>
>         <cn>1</cn>
>       </apply>
>     </lambda>
>   </math>
> 
>   Thanks for your help,
>   Gerald.
> 
> 
>   ________________________________
>   Gérald QUINTANA
>     gerald.quintana@ecl2000.ec-lyon.fr
>     http://www.multimania.com/gquintana
> 
Received on Wednesday, 12 July 2000 18:01:56 GMT

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