From: Stan Devitt <jsdevitt@radicalflow.com>

Date: Wed, 12 Jul 2000 18:49:30 -0400

Message-ID: <001c01bfec53$7131e800$6561a8c0@devitt.local>

To: "Andreas Strotmann" <strotman@cs.fsu.edu>

Cc: Gérald QUINTANA <quintana@lyon.objectif.fr>, <www-math@w3.org>

Date: Wed, 12 Jul 2000 18:49:30 -0400

Message-ID: <001c01bfec53$7131e800$6561a8c0@devitt.local>

To: "Andreas Strotmann" <strotman@cs.fsu.edu>

Cc: Gérald QUINTANA <quintana@lyon.objectif.fr>, <www-math@w3.org>

One quick point. The advantage of a tagged data-structure over a lambda construct is: that can be recognized as being a piecewise definition. In particular, this means that it can be quickly recognized as such and for display purposes, it means that you can invoke a special transformation immmediately instead of having to do sophisticated pattern matching on the arguments to lambda to see if you want a special notation for display. The key approach here is to: 1. Figure out what information is needed to represent piecewise defined functions. 2. Put that information into a tagged data -structure, possibly with explicit pairings as Andre suggests. 3. As a completely separate issue, map the tagged data-structure onto a display. It can't be emphasised too much that the "tagged" data structures are what enable you to re-use the information in an efficient and predictable way. Apart from that, I agree with Andre that some thought should go into the actual structure. Stan. 2. ----- Original Message ----- From: "Andreas Strotmann" <strotman@cs.fsu.edu> To: "Stan Devitt" <jsdevitt@radicalflow.com> Cc: "Gérald QUINTANA" <quintana@lyon.objectif.fr>; <www-math@w3.org> Sent: Wednesday, July 12, 2000 6:01 PM Subject: Re: Content markups, piecewise functions, conditions > 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 18:44:33 GMT

*
This archive was generated by hypermail 2.2.0+W3C-0.50
: Saturday, 20 February 2010 06:12:49 GMT
*