Re: A proposal for a computer readable mathematical object notation

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On Thu, Mar 22, 2018 at 11:07 AM, Neil Soiffer <soiffer@alum.mit.edu> wrote:
Arno,

I think the situation you describe about interchange is not anywhere near
as bad as you think. Most math programs will except TeX or MathML as input
and can do computation with them (at least Mathematica and Maple do so with
MathML, and I think also TeX). Part of the reason those programs can do so
is because they already have a linear syntax and so inferring a bit of
additional information is not that bad.

As for your proposal, I don't see what it adds that "properly" structured
MathML doesn't already provide. My properly structured MathML, I mean
MathML where the mrows reflect the abstract syntax of the mathematical
expression. E.g, for a binary infix operator, the first element in the mrow
will be an operand, the second the infix operator, and the third an
operand. If the second operator is a relational operator such as '=', then
the lhs is the first element and the rhs is the third element. The elements
might be leaf nodes such as mi's or complicated expressions inside mrows.
MathPlayer does this to infer semantics as does (I think) Volker's Speech
Rule Engine.

As an example,  e^{\imaginaryI \pi }+1=0
<mrow>
   <mrow>
      <msup>
         <mi>&#x2147;</mi>
         <mrow>
           <mi>&#x2148;</mi>
           <mo>&#x2062;</mo>
           <mi>&#x3c0;</mi>
         </mrow>
      </msup>
      <mo>+</mo>
      <mn>1</mn>
   </mrow>
   <mo>=</mo>
   <mn>0</mn>
</mrow>

or in json format:

{  "mrow": {    "msup": {      "mi": "ⅇ",      "mrow": {"mi":
["ⅈ","π"],"mo": "⁢"}    },    "mo": "+",    "mn": "1"  },  "mo": "=",
"mn": "0"}


Note: although many operators such as '=' are formally binary infix
operators, in practice, they are better treated as n-ary operators because
they often are used as "a = b = c" or even "a = b <= c = d" and both layout
and computation are often easier when all are at the same level.

I believe several people have tried at various times to formulate a
"canonical MathML" format that includes properly structured mrows, one of
the canonical Unicode forms, and fixing up pseudo scripts
<https://www.w3.org/TR/MathML3/chapter7.html#chars.pseudo-scripts> among
some other things. Note that proper mrow nesting depends upon guessing
between function call and multiplication if the author didn't indicate
which to use (e.g, in MathML, one should use the invisible operators), so
the parsing can be complicated. Of course, parsing alone doesn't provide
semantics. I believe that Volker adds some attribute that is equivalent to
adding something like a "mathrole" to various elements to specify semantics
-- some "obvious", some derived from other context, etc.

I think it would be helpful to develop a tool that anyone can incorporate
to produce canonical MathML.

    Neil

>
>
> On Monday, March 19, 2018, Arno Gourdol <arno.gourdol@gmail.com> wrote:
>
>> As we’ve previously discussed, a notation to exchange mathematical
>> expressions between computer programs could facilitate the assembly of
>> software components.
>>
>> Consider for example the following category of components:
>> - an equation editor
>> - a numerical computation package
>> - a symbolic algebra package
>> - a graphing package
>>
>> While formats exist to represent what should be displayed to the user in
>> the end (LaTeX, MathML, CSS+HTML), the situation is less encouraging when
>> it comes to representing semantically a mathematical notation so that it
>> can be acted upon by computer programs. As a result it is difficult to
>> impossible to connect software components together.
>>
>> I’m proposing MASTON (Math Abstract Syntax Tree Object Notation), a
>> lightweight, computer readable, notation for math based on JSON. I’ve put
>> together a first draft https://github.com/arnog/mathl
>> ive/blob/master/docs/MASTON.md and I am seeking feedback. Feel free to
>> discuss on this mailing list or on github.
>>
>> Here’s Euler’s identity:
>>
>>         e^{\imaginaryI \pi }+1=0
>>
>>         {"lhs":{"lhs":{"sym":"e","sup":{"lhs":"ⅈ","op":"*","rhs":"π"
>> }},"op":"+","rhs":1},"op":"=","rhs":0}
>>
>> Here’s a formula to convert degrees Fahrenheit to Celsius:
>>
>>         \frac {5}{9}(T_F−32)=T_C
>>
>>         {"lhs":{"lhs":{"num":"5/9"},"op":"*","rhs":{"lhs":{"sym":"T"
>> ,"sub":"F"},"op":"-","rhs":32}},"op":"=","rhs":{"sym":"T","sub":"C"}}
>>
>>
>> To test out the idea, I’ve also implemented MASTON support in MathLive
>> and hooked it up to math.js to perform simple numeric calculations. On
>> mathlive.io, click the </> icon to get a live output of what you type in
>> MASTON, LaTeX and MathML. It’s a good way to get a sense of how it works.
>>
>>
>> Best,
>> Arno.
>>
>>
>>