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Re: content for characters

From: Andreas Strotmann <strotman@cs.fsu.edu>
Date: Thu, 21 Jun 2001 09:59:44 -0400 (EDT)
To: William Arthur Naylor <bill@scl.csd.uwo.ca>
cc: Robert Miner <RobertM@dessci.com>, <dreamthnk@hotmail.com>, <www-math@w3.org>
Message-ID: <Pine.LNX.4.30.0106210916120.8787-100000@quake.cs.fsu.edu>
Personally, I think this is an excellent question that covers a wide range
of problems, e.g. the multiple values of a "root" or those of asin ...

Just like "root" and "asin", "plus/minus" deserves its own OpenMath
Content Dictionary entry, making it available to MathML via the csymbol
construct.

In OpenMath, it could be defined using a construct that I believe I
proposed some years ago for this general purpose, namely, the "one_of"
operator, which takes a set or list as an argument, and produces an
arbitrary element of the set or list.  Using this, the semantics of +/-
can be expressed as

 for all x, +/-( x ) := one_of ({+x, -x}) .

The "one_of" operator would replace the "or" in a previous post, and is
arguably better suited for this purpose.

Note that you have to be very careful with such constructs, as we pointed
out in the OpenMath Objectives paper years ago.  There, we give an example
of the formula for the roots of a univariate complex polynomial of degree
three in terms of such a "+/-" expression, where *all* such "one_of"
choices have to make the same decision for the formula to be correct;
thus, one_of expressions (including plus/minus ones) would need to be
lambda-bound if they appear more than once to preserve semantic
correctness.

The "one_of" construct fits nicely into the basic set of set-theoretic
operators, as it closely corresponds to the use of the axiom of choice.

Note that this solution side-steps several problems.  On the one hand, it
produces the single-valued function that is required to make it work with
any other arithmetic operations; on the other hand, CA systems tend to
understand how to lift or lower the "one_of" expressions along embeddings
of expressions -- the indeterminacy is also maintained.

Robert Miner's second reading would alternatively use an "all_of"
construct.  This would take an indeterminate single-valued function, and
return a set or sequence of all possible values of the function.

 -- Andreas

____________________________________________________________
"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
Received on Thursday, 21 June 2001 09:59:53 GMT

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