Semantic Embedding Examples

Some simple examples of embedded semantic tree. Note that for readability the attributes of the MathML elements have been abbreviated. Usually they have a 'semantic-' prefix.

<mi semantic-type="identifier" semantic-role="latinletter" semantic-id="1" semantic-parent="3">b</mi>

is shortened to

<mi type="identifier" role="latinletter" id="1" parent="3">b</mi>

Single Operation

Original MathML:

<math>
  <mi>a</mi>
  <mo>+</mo>
  <mi>b</mi>
  <mo>+</mo>
  <mi>c</mi>
</math>

Semantic Tree:

<stree>
  <infixop role="addition" id="5">
    +
    <content>
      <operator role="addition" id="1">+</operator>
      <operator role="addition" id="3">+</operator>
    </content>
    <children>
      <identifier role="latinletter" font="italic" id="0">a</identifier>
      <identifier role="latinletter" font="italic" id="2">b</identifier>
      <identifier role="latinletter" font="italic" id="4">c</identifier>
    </children>
  </infixop>
</stree>

Semantically enriched MathML:

<math>
  <mrow type="infixop" role="addition" id="5" content="1,3" children="0,2,4">
    <mi type="identifier" role="latinletter" id="0" parent="5">a</mi>
    <mo type="operator" role="addition" id="1" operator="infixop,+" parent="5">+</mo>
    <mi type="identifier" role="latinletter" id="2" parent="5">b</mi>
    <mo type="operator" role="addition" id="3" operator="infixop,+" parent="5">+</mo>
    <mi type="identifier" role="latinletter" id="4" parent="5">c</mi>
  </mrow>
</math>

Operation and Relation

Original MathML:

<math>
  <mn>5</mn>
  <mo>=</mo>
  <mn>3</mn>
  <mo>+</mo>
  <mn>2</mn>
</math>

Semantically enriched MathML:

<math>
  <mrow type="relseq" role="equality" id="6" content="1" children="0,5">
    <mn type="number" role="integer" id="0" parent="6">5</mn>
    <mo type="relation" role="equality" id="1" operator="relseq,=" parent="6">=</mo>
    <mrow type="infixop" role="addition" id="5" content="3" children="2,4" parent="6">
      <mn type="number" role="integer" id="2" parent="5">3</mn>
      <mo type="operator" role="addition" id="3" operator="infixop,+" parent="5">+</mo>
      <mn type="number" role="integer" id="4" parent="5">2</mn>
    </mrow>
  </mrow>
</math>

Observe that for the semantic interpretation the original MathML tags are pretty irrelevant. E.g., writing numbers as identifiers still yields the same semantic markup.

Original MathML:

<math>
  <mi>5</mi>
  <mo>=</mo>
  <mi>3</mi>
  <mo>+</mo>
  <mi>2</mi>
</math>

Enriched MathML:

<math>
  <mrow type="relseq" role="equality" id="6" content="1" children="0,5">
    <mi type="number" role="integer" id="0" parent="6">5</mi>
    <mo type="relation" role="equality" id="1" operator="relseq,=" parent="6">=</mo>
    <mrow type="infixop" role="addition" id="5" content="3" children="2,4" parent="6">
      <mi type="number" role="integer" id="2" parent="5">3</mi>
      <mo type="operator" role="addition" id="3" operator="infixop,+" parent="5">+</mo>
      <mi type="number" role="integer" id="4" parent="5">2</mi>
    </mrow>
  </mrow>
</math>

Multiple Operations

Original MathML:

<math>
  <mi>a</mi>
  <mo>+</mo>
  <mi>b</mi>
  <mo>-</mo>
  <mi>c</mi>
  <mo>+</mo>
  <mi>d</mi>
</math>

Semantically enriched MathML:

<math>
  <mrow type="infixop" role="addition" id="9" content="5" children="8,6">
    <mrow type="infixop" role="subtraction" id="8" content="3" children="7,4" parent="9">
      <mrow type="infixop" role="addition" id="7" content="1" children="0,2" parent="8">
        <mi type="identifier" role="latinletter" id="0" parent="7">a</mi>
        <mo type="operator" role="addition" id="1" operator="infixop,+" parent="7">+</mo>
        <mi type="identifier" role="latinletter" id="2" parent="7">b</mi>
      </mrow>
      <mo type="operator" role="subtraction" id="3" operator="infixop,-" parent="8">-</mo>
      <mi type="identifier" role="latinletter" id="4" parent="8">c</mi>
    </mrow>
    <mo type="operator" role="addition" id="5" operator="infixop,+" parent="9">+</mo>
    <mi type="identifier" role="latinletter" id="6" parent="9">d</mi>
  </mrow>
</math>

Original MathML:

<math>
  <mi>a</mi>
  <mo>+</mo>
  <mi>b</mi>
  <mo>∘</mo>
  <mi>c</mi>
  <mi>d</mi>
  <mo>+</mo>
  <mi>e</mi>
  <mo>∘</mo>
  <mi>f</mi>
</math>

Semantically enriched MathML:

<math>
  <mrow type="infixop" role="addition" id="10" content="1,6" children="0,13,14">
    <mi type="identifier" role="latinletter" id="0" parent="10">a</mi>
    <mo type="operator" role="addition" id="1" operator="infixop,+" parent="10">+</mo>
    <mrow type="infixop" role="multiplication" id="13" content="3" children="2,12" parent="10">
      <mi type="identifier" role="latinletter" id="2" parent="13">b</mi>
      <mo type="operator" role="multiplication" id="3" operator="infixop,∘" parent="13">∘</mo>
      <mrow type="infixop" role="implicit" id="12" content="11" children="4,5" parent="13">
        <mi type="identifier" role="latinletter" id="4" parent="12">c</mi>
        <mrow type="operator" role="multiplication" id="11" children="" operator="infixop,⁢" parent="12"/>
        <mi type="identifier" role="latinletter" id="5" parent="12">d</mi>
      </mrow>
    </mrow>
    <mo type="operator" role="addition" id="6" operator="infixop,+" parent="10">+</mo>
    <mrow type="infixop" role="multiplication" id="14" content="8" children="7,9" parent="10">
      <mi type="identifier" role="latinletter" id="7" parent="14">e</mi>
      <mo type="operator" role="multiplication" id="8" operator="infixop,∘" parent="14">∘</mo>
      <mi type="identifier" role="latinletter" id="9" parent="14">f</mi>
    </mrow>
  </mrow>
</math>

The latter contains an implicit operation, resulting in an empty mrow. The question is: What should we do with elements that are introduced by the semantic interpretation but not part of the original MathML. All these are invisible characters: invisible times, invisible comma, function application.

Implicit multiplication

To illustrate the point clearer:

Original MathML:

<math>
  <mi>a</mi>
  <mi>b</mi>
</math>

Semantic Tree:

<stree>
  <infixop role="implicit" id="3">
    ⁢
    <content>
      <operator role="multiplication" id="2">⁢</operator>
    </content>
    <children>
      <identifier role="latinletter" font="italic" id="0">a</identifier>
      <identifier role="latinletter" font="italic" id="1">b</identifier>
    </children>
  </infixop>
</stree>

Semantically enriched MathML:

<math>
  <mrow type="infixop" role="implicit" id="3" content="2" children="0,1">
    <mi type="identifier" role="latinletter" id="0" parent="3">a</mi>
    <mrow type="operator" role="multiplication" id="2" children="" operator="infixop,⁢" parent="3"/>
    <mi type="identifier" role="latinletter" id="1" parent="3">b</mi>
  </mrow>
</math>

While in the semantic tree a new content operator is introduced, there is no equivalent MathML element. Do empty mrows really matter? This is the rendering:

Original MathML: a b

Semantically enriched MathML: a b

The more complex example from above.

Original MathML: a + b ∘ c d + e ∘ f

Semantically enriched MathML: a + b ∘ c d + e ∘ f

Example with sum operator

Original MathML:

<math>
  <msubsup>
    <mi>∑</mi>
    <mi>b</mi>
    <mi>c</mi>
  </msubsup>
  <mi>b</mi>
  <mi>c</mi>
  <mo>+</mo>
  <mi>a</mi>
</math>

Semantic Tree:

<stree>
  <infixop role="addition" id="11">
    +
    <content>
      <operator role="addition" id="6">+</operator>
    </content>
    <children>
      <bigop role="sum" id="10">
        <children>
          <limboth role="sum" id="3">
            <children>
              <largeop role="sum" id="0">∑</largeop>
              <identifier role="latinletter" font="italic" id="1">b</identifier>
              <identifier role="latinletter" font="italic" id="2">c</identifier>
            </children>
          </limboth>
          <infixop role="implicit" id="9">
            ⁢
            <content>
              <operator role="multiplication" id="8">⁢</operator>
            </content>
            <children>
              <identifier role="latinletter" font="italic" id="4">b</identifier>
              <identifier role="latinletter" font="italic" id="5">c</identifier>
            </children>
          </infixop>
        </children>
      </bigop>
      <identifier role="latinletter" font="italic" id="7">a</identifier>
    </children>
  </infixop>
</stree>

Semantically enriched MathML:

<math type="infixop" role="addition" id="11" children="10,7" content="6">
  <mrow type="bigop" role="sum" id="10" children="3,9" parent="11">
    <msubsup type="limboth" role="sum" id="3" children="0,1,2" parent="10">
      <mi type="largeop" role="sum" id="0" parent="3">∑</mi>
      <mi type="identifier" role="latinletter" id="1" parent="3">b</mi>
      <mi type="identifier" role="latinletter" id="2" parent="3">c</mi>
    </msubsup>
    <mrow type="infixop" role="implicit" id="9" children="4,5" content="8" parent="10">
      <mi type="identifier" role="latinletter" id="4" parent="9">b</mi>
      <mrow type="operator" role="multiplication" id="8" children="" operator="infixop,⁢" parent="9"/>
      <mi type="identifier" role="latinletter" id="5" parent="9">c</mi>
    </mrow>
  </mrow>
  <mo type="operator" role="addition" id="6" operator="infixop,+" parent="11">+</mo>
  <mi type="identifier" role="latinletter" id="7" parent="11">a</mi>
</math>

Integral Examples

Big Operator and Function Examples