# Re: bvar, codition, and domainofapplication (OpenMath - viewpoint)

From: Bill Naylor <Bill.Naylor@mcs.vuw.ac.nz>
Date: Wed, 14 May 2003 13:32:33 +1200 (NZST)
To: Andreas Strotmann <Strotmann@rrz.uni-koeln.de>

Message-ID: <Pine.SOL.4.51.0305141247200.28527@debretts.mcs.vuw.ac.nz>


Hi,

It seems to me that any decisions made on the semantics of these elements
will neccessarily have impactions on the corresponding OpenMath symbols,
in content dictionary fns1.
I just had a look at the domainofapplication symbol, which is defined in
the following way:

"The domainofapplication element denotes the domain over which a given
function is being
applied. It is intended in MathML to be a more general alternative to
specification of this
domain using such quantifier elements as bvar, lowlimit or condition."

This is a bug in OpenMath,

1) it is not a 'domainofapplication element' (in OpenMath)
2) The content dictionaries should say what the OpenMath symbols
are intended to represent not the MathML ones!

I guess this was just cut and pasted from the MathML spec.(?)
As far as Andreass discussion gos, one would hope that the OpenMath
interpretation could be fixed (made constant) by the sts file, I had a
look at this, and think that it is rather flawed also:

domainofapplication

<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMV name="Set"/>
<OMV name="Set"/>
</OMA>
<OMV name="Set"/>
</OMA>
</OMOBJ>

( ( Set >> Set ) >> Set )

personally I would have preferred something along the lines of:

<OMOBJ>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMA>
<OMS name="mapsto" cd="sts"/>
<OMA><OMS name="nary" cd="sts"/>
<OMS name="Object" cd="sts"/>
<OMA>
<OMS name="Object" cd="sts"/>
</OMA>
<OMS name="Object" cd="sts"/>
</OMA>
</OMOBJ>

In fact looking at the entire fns1.sts, I see that almost all the
signatures have this sort of problem!

it would be good to have these ironed out soon. I shall send David a
correct (by my interpretation) version, maybe someone else can check over
that?

Bill

On Tue, 13 May 2003, Andreas Strotmann wrote:

> Date: Tue, 13 May 2003 19:11:36 +0200
> From: Andreas Strotmann <Strotmann@rrz.uni-koeln.de>
> To: Andreas Strotmann <Strotmann@rrz.uni-koeln.de>, www-math@w3.org
> Subject: bvar, codition, and domainofapplication
> Resent-Date: Tue, 13 May 2003 13:11:42 -0400 (EDT)
> Resent-From: www-math@w3.org
>
>
> Hello,
>
> even if these comments may be a bit too late, I feel it would be useful
> to supplement the recommendations I made during the last few days with
> this much more fundamental observation on the use of the main
> MathML-Content qualifiers, as an explanation of the rationale behind
> many of my recommendations.
>
> I'm sorry that this turned out to be such a long comment, but the matter
> feels important to me -- important to get it right in the second
> edition, that is.  I hope this helps!
>
> This of course represents my personal analysis, and though I'm pretty
> confident, I'm not 100% sure I've got it right.  Comments are therefore
> more than welcome.
>
>   -- Andreas
>
>
>   There appears to be some confusion still about how to use the new
> domainofapplication element.  In some places it is clearly marked as a
> qualifier that generalizes the interval qualifier to general sets, but
> in some places (such as in this appendix) it appears to be treated as a
> regular constructor, thus generalizing the interval *constructor*, which
> then brings up what its definition might be.
>
>   It is evident to me that in the minds of those who came up with this
> important addition to MathML, this element was meant to be used
> exclusively as a qualifier (and emphatically not as a constructor) to
> generalize the interval *qualifier* (and in my stated opinion in fact to
> supercede this dangerous dual use of the interval element), and to
> factor out the use of the condition qualifier that allows it to contain
> a set, not just a predicate.
>
>   In the comments that I have been submitting during the past week or
> so, I have therefore assumed that the intention implicit in the choice
> of qualifiers available for general use is as follows:
>
>      bvar - there may be zero(!) or more of these anywhere; they denote
>             bound variables in a particular order.
>
>      uplimit, lowlimit, interval, and domainofapplication - these all
>             represent sets that restrict the domain of a variable. Thus,
>             only the limits (one or both) *or* interval *or*
>             domainofapplication may appear in a sensible apply.
>
>      condition - this one is meant to represent a predicate over the
>             bound variables of the surrounding element which acts as a
>             filter defining the set of combinations of bvar values
>             that the operator ranges over.
>
>   Since domainofapplication is new, condition, interval, and the two
> limits covered its use in earlier versions, which leads to a need to
> deprecate the 'double entendre' usage of these older elements that are
> more clearly represented by domainofapplication.  I therefore recommend
> deprecating the following in order to clean up the spec:
>
>       - the use of a condition qualifier that contains the representation
>         of a set rather than a predicate -- markup that does this should
>         simply replace the condition with the domainofapplication
>         qualifier.
>
>       - the interval qualifier (but not the constructor) -- markup that
>         uses the interval qualifier should replace it with a
>         domainofapplication qualifier that contains an interval
>         constructor element with the same content as the original
>
>         (Incidentally, this deprecation also makes it easier again to
>         understand how an interval *constructor* would need to be
>         allowed to contain a single child element, a condition, because
>         doing the same thing with an interval *qualifier* seemed rather
>         spurious because condition should be used instead.)
>
> With these changes in place, we have a clean separation between the uses
> of the condition and the domainofapplication qualifiers, as follows:
>
>    -  the condition contains a predicate over all the bvars, and may in
>       principle be used with an empty set of bvars (in which case it
>       becomes a predicate over the free variables).  The predicate is
>       in effect the characteristic predicate for a domainofapplication.
>
>    -  the domainofapplication qualifier, by contrast, contains a set.
>       In the current definition, there are two cases that need to be
>       distinguished:
>
>       a) the signature   op: (bvar,dom-of-app,any) -> any
>       b) the signature   op: (dom-of-app,function) -> any
>
>       Of these, a) means the same as
>
>          op(bvar, condition(bvar \in dom-of-app), arg)
>
>       while b) is equivalent to
>
>          op(newvars, dom-of-app, apply(function, newvars))
>
>       Note that in the latter case, the domain of application may
>       actually be a cartesian product in order to assign ranges to
>       several arguments of the argument function simultaneously.
>
>       This is currently not true in a), but it could be specified
>       in the MathML spec that domainofapplication should contain a
>       matching-arity cartesian product in the case of multiple
>       bvars with one domainofapplication (although the fact that it
>       should contain a cartesian product semantically does not mean
>       that it would necessarily contain one syntactically -- but if
>       it does, its meaning is clear).
>
>       Note also that a) means the same as
>
>         op(dom-of-app,lambda(bvar,arg))
>
>       while  a signature of  op(bvars, condition, arg) is equivalent to
>
>         op(
>           dom-of-app(set(bvars,condition,cartesianproduct(bvars)),
>           lambda(bvars,arg)
>          )
>
>       and a signature like   op(bvars,arg)   means the same as
>
>          op(lambda(bvars,arg))
>
> For the signatures in appendix C (and the corresponding text in the main
> body), all of this translates to a few general rules:
>
>   - lowlimit, uplimit, (deprecated interval), and domainofapplication
>     belong in one and the same group of qualifiers, the most general
>     of which is domainofapplication.  They are characterized by not
>     specifying the names of the variables that they constrain.
>
>   - the second group of qualifiers is the condition qualifier in its
>     reduced definition that requires a predicate argument and disallows
>     a set argument.  It is characterized by specifically naming the
>     variables it constrains.
>
>   - thus, only one of these groups of qualifiers may be used in an
>     element, but not both.
>
>   - in cases where operators usually take bvar qualifiers, as in
>     int, diff, product, sum, exists, forall, lack of bvars on first
>     sight would automatically mean that the single argument must be a
>     function, as in apply(int,f) and in apply(diff,f).
>     (Incidentally, this rule does not apply to the lambda constructor
>     since it does not have an operator.)
>
>     But in the case of n-ary constructors, this is problematic (this
>     includes the set and matrix constructors and the min and max
>     operators, all of which have versions without any qualifiers
>     with an n-ary regular function interpretation, and a version with
>     qualifiers interpreted like int and sum or lambda.)
>
>     The problem is that in the current MathML spec, operators like min
>     and max (and there are plenty of those) are implicitly 'lifted' to
>     their 'big' version (like \wedge -> \bigwedge , i.e. or -> exists
>     in a well known notation, or max -> sup):
>
>       'small' signature (A*) -> B
>
>        'big' signature  (function) -> B
>                         (domainofapplication, function) -> B
>                         (bvar,A) -> B
>                         (bvar,domainofapplication,A) -> B
>                         (bvar,condition,A) -> B
>
>     In the future, a general-purpose piece of markup for this lifting
>     could be useful, so that one could say, for example
>
>         apply(big(max),my_real_fn)
>
>
>         apply(max,my_real_fn)
>
>     which clearly doesn't do the same thing.
>
>     Fortunately, by supplying an empty domainofapplication, this
>     semantic lifting can be implemented in MathML in a systematic way
>     without introducing the 'big' operator:
>
>         apply(or,domainofapplication(),my_predicate_fn)
>
>      can be defined consistently in MathML to have the required meaning.
>      In other words, we can reconcile the 'big' and 'small' signatures
>      into a single one by providing the following combined signature:
>
>        'combi' signature  (A*) -> B
>                           (domainofapplication,function) -> B
>                           (bvar,A) -> B
>                           (bvar,domainofapplication,A) -> B
>                           (bvar,condition,A) -> B
>
>        Replacing the 'function' with its signature, (A -> B), we get
>
>        'combi' signature  (A*) -> B
>                           (domainofapplication,(A->B)) -> B
>                           (bvar,A) -> B
>                           (bvar,domainofapplication,A) -> B
>                           (bvar,condition,A) -> B
>
>       which is unfortunately a bit different from the
>
>           'big' signature ((A->B)) -> B
>                           (domainofapplication,(A->B)) -> B
>                           (bvar,A) -> B
>                           (bvar,domainofapplication,A) -> B
>                           (bvar,condition,A) -> B
>
>        From the perspective of automatic, semantically correct processing
>        of MathML-Content trees, the crucial question is how to reliably
>        distinguish these two cases when all you have is a user-defined
>        symbol as an operator (say).  Which of these classes does it
>        belong to?  How do you interpret  op(a)  compared with
>        op(bvar(x),b)  safely when you cannot know the signature?
>
>        The practical solution is to always assume the 'combi' signature.
>        This means that apply(product,a) is interpreted the same as
>        apply(plus,a), and that  apply(plus,bvar(x),f)  is interpreted the
>        same as apply(product,bvar(x),f), etc.
>
>        This means that the indefinite integration and differentiation
>        operators are the only ones that violate this pattern; user-
>        defined operators of this sort should always be used with a
>        possibly empty domainofapplication qualifier when they want to
>        express application to a function (as in 'prime'):  the completely
>        qualifier-free use of my_diff_op as in apply(my_diff_op,f) does
>        *not* encode  f^\myprime.  Instead, I would have to use
>        apply(my_diff_op,domainofapplication(),f) to get the desired
>        meaning, unless one is certain that one never would want to
>        use it with a bvar qualifier.  (Note that it is not necessary
>        to change the signatures of the existing MathML operators, because
>        for these, the signatures do exist.)
>
>        This also means that the quantifiers, and the product and sum
>        operators *do* have a reasonable meaning when used without any
>        qualifiers, and can be assigned the appropriate signature copied
>        from the appropriate logical connectives or from times and plus,
>        resp.
>
>        Conversely, it means that all n-ary operators have a reasonable
>        meaning when used with these qualifiers, and their signatures
>        in appendix C should reflect that fact.
>
>        Last not least, it means that apply can be given a general
>        signature, too:
>
>          apply:   ((any*->any), any*)                       -> any
>                   (((any->any)->any),dom-of-app,(any->any)) -> any
>                   (((any->any)->any),bvar,any)              -> any
>                   (((any->any)->any),bvar,dom-of-app,any)   -> any
>                   (((any->any)->any),bvar,condition,any)    -> any
>
>         where
>
>           dom-of-app:: domainofapplication|uplimit|lowlimit
>                           |lowlimit,uplimit
>
>         (leaving out the questionable interval qualifier).
>
>         Still, it might be useful to provide a type attribute
>         in MathML to signal that a csymbol is meant to follow the
>         pattern of the int and diff operators rather that of the
>         max and min operators, just to be on the safe side.
>
>
>
>

Received on Tuesday, 13 May 2003 21:33:08 GMT

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