Does one rendering have multiple MathML expressions? - mathml

Question

I am currently working on mathematical expression recognition. I am going through the CROHME dataset and found this:
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mrow>
<mi xml:id="y_1">y</mi>
<mrow>
<mo xml:id="=_1">=</mo>
<mrow>
<mi xml:id="A_1">A</mi>
<mrow>
<mi xml:id="x_1">x</mi>
<mrow>
<mo xml:id="+_1">+</mo>
<msup>
<mi xml:id="A_2">A</mi>
<mn xml:id="2_1">2</mn>
</msup>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
which renders to (with http://www.mathmlcentral.com/Tools/FromMathML.jsp)
The application of mrow seems rather arbitrary to me. Isn't this expression semantically identical to
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mrow>
<mi xml:id="y_1">y</mi>
<mo xml:id="=_1">=</mo>
<mi xml:id="A_1">A</mi>
<mi xml:id="x_1">x</mi>
<mo xml:id="+_1">+</mo>
<msup>
<mi xml:id="A_2">A</mi>
<mn xml:id="2_1">2</mn>
</msup>
</mrow>
</math>
or
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mrow>
<mi xml:id="y_1">y</mi>
<mo xml:id="=_1">=</mo>
<mrow>
<mi xml:id="A_1">A</mi>
<mi xml:id="x_1">x</mi>
</mrow>
<mo xml:id="+_1">+</mo>
<msup>
<mi xml:id="A_2">A</mi>
<mn xml:id="2_1">2</mn>
</msup>
</mrow>
</math>
What is the difference between those three MathML expressions? Should they all render to the same?

Answer 1

mrow exists to allow long expressions inside elements/tags that can one have a limited number of children. For example, the square root of "x + y" must be write as
<math>
<msqrt>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
</msqrt>
</math>
Rendering software must render
<math>
<mi>x</mi>
</math>
and
<math>
<mrow>
<mi>x</mi>
</mrow>
</math>
in the same way.
Backing to your question,
Isn't this expression semantically identical
we are talking about presentation MathML so don't make sense talk about semantic.
What is the difference between those three MathML expressions?
The difference is the MathML tree of the expressions. If you are going to process the expression by some software it can make some assumptions based on MathML tree and have different behaviors based on it.
Should they all render to the same?
Yes.