Math Schema

Writing math expressions is a very natural way within education, to learn and solve problems. Now, to make this content exchangeable and understandable, we introduce a schema that describes mathematical structures and their attributes. Humans read and write math as a line of structured text with commonly shared formatting conventions, e.g.:

with the recognized formula: $\left( x^2+2\right) \cdot \left( x^2-2\right) = 0$.

This would result in a MathML expression:

<math xmlns='http://www.w3.org/1998/Math/MathML'>
  <mrow>
    <mo> ( </mo>
      <msup>
        <mrow>
          <mi> x </mi>
        </mrow>
        <mrow>
          <mn> 2 </mn>
        </mrow>
      </msup>
      <mo> + </mo>
      <mn> 2 </mn>
    <mo> ) </mo>
  </mrow>
  <mo> &#x00B7; <!-- middle dot --> </mo>
  <mrow>
    <mo> ( </mo>
      <msup>
        <mrow>
          <mi> x </mi>
        </mrow>
        <mrow>
          <mn> 2 </mn>
        </mrow>
      </msup>
      <mo> - </mo>
      <mn> 2 </mn>
    <mo> ) </mo>
  </mrow>
  <mo> = </mo>
  <mn> 0 </mn>
</math>

or a Latex representation:

\left( x^{2}+2\right) \cdot \left( x^{2}-2\right) = 0

For computer systems (e.g. computer algebra systems, or CAS) - a representation as a tree (expression trees) is better suited. A binary expression tree is a specific kind of binary tree used to represent math expressions. Two common types of expressions that a binary expression tree can represent are algebraic and boolean. These trees can represent expressions that contain both unary and binary operators.

Each node of a binary tree, and hence of a binary expression tree, has zero, one, or two children. This restricted structure simplifies the processing of expression trees.

The math schema will leverage the math structure and it is based on similar terminology as MathML.

Includes

Name	Value
Schema URI	schema will:seg/0.3
Class Count	45
Classes	will:seg/0.3/Root, will:seg/0.3/ContentBlock, will:seg/0.3/TextRegion, will:seg/0.3/TextLine, will:seg/0.3/Word, will:seg/0.3/WordOfStrokes, will:seg/0.3/ExtendedTextRegion, will:seg/0.3/Line, will:seg/0.3/UnlabelledBlock, will:seg/0.3/UnlabelledItemGroup, will:seg/0.3/UnlabelledItem, will:seg/0.3/Drawing, will:seg/0.3/DrawingItemGroup, will:seg/0.3/DrawingItem, will:seg/0.3/ExpressionBlock, will:seg/0.3/GenericExpressionBlock, will:seg/0.3/GenericExpressionItemGroup, will:seg/0.3/GenericExpressionItem, will:seg/0.3/MathBlock, will:seg/0.3/MathItemGroup, will:seg/0.3/MathItem, will:seg/0.3/ChemistryBlock, will:seg/0.3/ChemistryItemGroup, will:seg/0.3/ChemistryItem, will:seg/0.3/PhysicsBlock, will:seg/0.3/PhysicsItemGroup, will:seg/0.3/PhysicsItem, will:seg/0.3/ContentBlockOfStrokeNodes, will:seg/0.3/ChemicalStructure, will:seg/0.3/Signature, will:seg/0.3/Garbage, will:seg/0.3/Unlabeled, will:seg/0.3/Doodle, will:seg/0.3/Diagram, will:seg/0.3/DiagramConnector, will:seg/0.3/DiagramPart, will:seg/0.3/Table, will:seg/0.3/Border, will:seg/0.3/List, will:seg/0.3/ListItem, will:seg/0.3/ListItemBullet, will:seg/0.3/Annotation, will:seg/0.3/Marking, will:seg/0.3/Connector, will:seg/0.3/Correction

Classes

Term

Term is an abstract class covering atomic parts of a math formula.

Name	Value
Class	will:math/0.6/Term [abstract]
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItem
Specialized By	will:math/0.6/Symbol,will:math/0.6/Number

Properties

Property	Attribute	Value
hasChild	Data Type(s)	will:uim/1.1/StrokeNode
	Association Type	composition
	Cardinality	[0, 1] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)

Symbol

A Symbol can include variables, function names, and symbolic constants.

Name	Value
Class	will:math/0.6/Symbol
Defined In	will:math/0.6
Specializes	will:math/0.6/Term

Properties

Property	Attribute	Value
hasChild	Data Type(s)	will:uim/1.1/StrokeNode
	Association Type	composition
	Cardinality	[0, 1] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
representation	Data Type(s)	@string
	Description	Contains the representation of the symbol.
	Association Type	composition
	Cardinality	[0, 1] (optional)
symbolType	Data Type(s)	@string
	Description	Defines the type of the symbol.
	Association Type	composition
	Cardinality	[0, 1] (optional)

Number

Numbers represent a 'numeric literal'. In general, a numeric literal is a sequence of digits, perhaps including a decimal point, representing an unsigned integer or real number.

Name	Value
Class	will:math/0.6/Number
Defined In	will:math/0.6
Specializes	will:math/0.6/Term

Properties

Property	Attribute	Value
hasChild	Data Type(s)	will:uim/1.1/StrokeNode
	Association Type	composition
	Cardinality	[0, 1] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
numberType	Data Type(s)	@string
	Description	Defines the type of the number.
	Association Type	composition
	Cardinality	[0, 1] (optional)
representation	Data Type(s)	@float64
	Description	Contains the numerical representation of the number. A symbolic representation such as Π, is using Symbol.
	Association Type	composition
	Cardinality	[0, 1] (optional)

CompoundTerm

Describes entities that are considered atomic and cannot be split further. E.g. 2ab, sin a, f(x), etc.

Name	Value
Class	will:math/0.6/CompoundTerm
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
body	Data Type(s)	will:math/0.6/Term,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/CompoundTerm,will:math/0.6/Indexed,will:math/0.6/Matrix
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory)
	Inverse Of	hasParent
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)

Operator

Describes the operator symbol.

Name	Value
Class	will:math/0.6/Operator [final]
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItem

Properties

Property	Attribute	Value
hasChild	Data Type(s)	will:uim/1.1/StrokeNode
	Association Type	composition
	Cardinality	[0, 1] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
representation	Data Type(s)	@string
	Description	Contains the symbol for the operator.
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)

Relation

Describes the relation symbol.

Name	Value
Class	will:math/0.6/Relation [final]
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItem

Properties

Property	Attribute	Value
hasChild	Data Type(s)	will:uim/1.1/StrokeNode
	Association Type	composition
	Cardinality	[0, 1] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
representation	Data Type(s)	@string
	Description	Contains the symbol for the relation.
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)

FractionLine

Links the strokes that are associated with the fraction line using the hasChild property.

Name	Value
Class	will:math/0.6/FractionLine [final]
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItem

Properties

Property	Attribute	Value
hasChild	Data Type(s)	will:uim/1.1/StrokeNode
	Association Type	composition
	Cardinality	[0, 1] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)

Expression

Flattened representation of the operation. It consists of at least 2 components - operator | relation, n-operands. If there are more operands it would result in, e.g., 5x+3y+3 -> +, 5x, 3y, 3.

Name	Value
Class	will:math/0.6/Expression
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
items	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/Relation,will:math/0.6/Indexed,will:math/0.6/Matrix,will:math/0.6/RootExpression
	Description	If the operator is set with an Operator or a Relation, the items must be considered as operands.
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory) (ordered)
	Inverse Of	hasParent
operator	Data Type(s)	will:math/0.6/Operator,will:math/0.6/Relation
	Description	Operations and Relations need at least one Operator or a Relation, items includes the operands.
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory) (ordered)
	Inverse Of	hasParent

Fenced

A fenced term is a term that will always have opening and closing parentheses, e.g, ((x + 1) * 2).

Name	Value
Class	will:math/0.6/Fenced
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
body	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/Indexed,will:math/0.6/RootExpression
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory)
	Inverse Of	hasParent
closingBracket	Data Type(s)	will:math/0.6/Symbol
	Description	Symbol used for closing bracket, e.g., ), ], or }.
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)
	Inverse Of	hasParent
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
openingBracket	Data Type(s)	will:math/0.6/Symbol
	Description	Symbol used for opening bracket, e.g., (, [, or {.
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)
	Inverse Of	hasParent

System

A system of equations is grouping several lines via equation.

Name	Value
Class	will:math/0.6/System
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup
Specialized By	will:math/0.6/Cases

Properties

Property	Attribute	Value
equation	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Matrix,will:math/0.6/Indexed,will:math/0.6/Fraction,will:math/0.6/Fenced,will:math/0.6/RootExpression
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory) (ordered)
	Inverse Of	hasParent
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)

Cases

A fenced term with vertical alignment, where there is only one bracket. Cases are added via equation for the System class.

Name	Value
Class	will:math/0.6/Cases
Defined In	will:math/0.6
Specializes	will:math/0.6/System

Properties

Property	Attribute	Value
bracket	Data Type(s)	will:math/0.6/Symbol
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)
	Inverse Of	hasParent
equation	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Matrix,will:math/0.6/Indexed,will:math/0.6/Fraction,will:math/0.6/Fenced,will:math/0.6/RootExpression
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory) (ordered)
	Inverse Of	hasParent
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)

RootExpression

RootExpression can have a symbol n √ x, or a number 3 √ x. If no degree is set, it is considered a squared root.

Name	Value
Class	will:math/0.6/RootExpression
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
body	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)
	Inverse Of	hasParent
degree	Data Type(s)	will:math/0.6/Symbol,will:math/0.6/Number
	Description	Describes the degree of the root expression.
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
rootSign	Data Type(s)	will:math/0.6/Symbol
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)
	Inverse Of	hasParent

Fraction

Fraction expression with numerator and denominator.

Name	Value
Class	will:math/0.6/Fraction
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
denominator	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Denominator part of the fraction (optional for partial fraction).
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent
fractionLine	Data Type(s)	will:math/0.6/FractionLine
	Description	Stroke(s) annotating the fraction line.
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory)
	Inverse Of	hasParent
fractionType	Data Type(s)	@string
	Description	Describes the type of fraction.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
numerator	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Numerator part of the fraction. (optional for partial fraction)
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent

Row

Row of a matrix structure.

Name	Value
Class	will:math/0.6/Row
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
hasCell	Data Type(s)	will:math/0.6/Number,will:math/0.6/Symbol,will:math/0.6/CompoundTerm,will:math/0.6/Fraction,will:math/0.6/Fenced,will:math/0.6/Indexed,will:math/0.6/Expression,will:math/0.6/RootExpression,will:math/0.6/Matrix
	Description	Represents the content a cell within a matrix.
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory) (ordered)
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)

Matrix

Matrix expression, which consists of multiple rows.

Name	Value
Class	will:math/0.6/Matrix
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
closingBracket	Data Type(s)	will:math/0.6/Symbol
	Description	Symbol used for closing bracket, e.g., ), or ].
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
hasRow	Data Type(s)	will:math/0.6/Row
	Description	Rows of the matrix.
	Association Type	aggregation
	Cardinality	[1, inf] (mandatory) (ordered)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
openingBracket	Data Type(s)	will:math/0.6/Symbol
	Description	Symbol used for opening bracket, e.g., ( or [.
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)

Indexed

Generic expression for all indexed terms. There is no mathematical interpretation of the indexed term, thus summations or integrals would be expressed as symbols with an overscript and underscript.

Name	Value
Class	will:math/0.6/Indexed
Defined In	will:math/0.6
Specializes	will:seg/0.3/MathItemGroup

Properties

Property	Attribute	Value
body	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Fenced
	Description	Base expression.
	Association Type	aggregation
	Cardinality	[1, 1] (mandatory)
	Inverse Of	hasParent
hasChild	Data Type(s)
	Association Type	composition
	Cardinality	[0, 0] overridden to [0, inf] (optional)
hasOperandLabel	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasOperandType	Data Type(s)	@string
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
hasParent	Data Type(s)
	Association Type	aggregation
	Cardinality	[1, 1] overridden to [0, 1] (mandatory)
isStructuralItem	Data Type(s)	@bool
	Description	Reserved for debug purposes.
	Association Type	composition
	Cardinality	[0, 1] (optional)
overScript	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Overscript expression / term.
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent
preSubScript	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Pre-subscript expression / term.
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent
preSuperScript	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Pre-superscript expression / term.
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent
subScript	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Subscript expression / term.
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent
superScript	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Superscript expression / term.
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent
underScript	Data Type(s)	will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed
	Description	Underscript expression / term.
	Association Type	composition
	Cardinality	[0, 1] (optional)
	Inverse Of	hasParent

Schema Definition

{
    "head": {
        "uri": "will:math/0.6",
        "version": "0.6",
        "description": "Math Schema to formalizing the math expression tree.",
        "classUriTemplate": "will:math/0.6/[class]",
        "propertyUriTemplate": "uim:math/0.6/[class]#[property]",
        "includes": [
            {
                "namespace": "seg",
                "schema": "will:seg/0.3"
            }
        ]
    },
    "body": {
        "extensions": {},
        "classes": {
            "will:math/0.6/Term": {
                "subclassOf": "will:seg/0.3/MathItem",
                "properties": {},
                "isAbstract": true,
                "isFinal": false,
                "@desc": "`Term` is an abstract class covering atomic parts of a math formula."
            },
            "will:math/0.6/Symbol": {
                "subclassOf": "will:math/0.6/Term",
                "properties": {
                    "representation": {
                        "dataType": "@string",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Contains the representation of the symbol."
                    },
                    "symbolType": {
                        "dataType": "@string",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Defines the type of the symbol.",
                        "dataRange": [
                            "Constant,Variable,Function,Numerical"
                        ]
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "A `Symbol` can include variables, function names, and symbolic constants."
            },
            "will:math/0.6/Number": {
                "subclassOf": "will:math/0.6/Term",
                "properties": {
                    "representation": {
                        "dataType": "@float64",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Contains the numerical representation of the number. A symbolic representation such as \u03a0, is using Symbol."
                    },
                    "numberType": {
                        "dataType": "@string",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Defines the type of the number.",
                        "dataRange": [
                            "Integer,Decimal"
                        ]
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "Numbers represents a 'numeric literal'. In general, a numeric literal is a sequence of digits, perhaps including a decimal point, representing an unsigned integer or real number."
            },
            "will:math/0.6/CompoundTerm": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "body": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/CompoundTerm,will:math/0.6/Indexed,will:math/0.6/Matrix",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "inverseOf": "hasParent"
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "Describes entity that is considered atomic and would/could not be split further. E.g. 2ab, sin a, f(x), etc."
            },
            "will:math/0.6/Operator": {
                "subclassOf": "will:seg/0.3/MathItem",
                "properties": {
                    "representation": {
                        "dataType": "@string",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Contains the symbol for the operator."
                    }
                },
                "isFinal": true,
                "@desc": "Describes the operator symbol."
            },
            "will:math/0.6/Relation": {
                "subclassOf": "will:seg/0.3/MathItem",
                "properties": {
                    "representation": {
                        "dataType": "@string",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Contains the symbol for the relation."
                    }
                },
                "isFinal": true,
                "@desc": "Describes the relation symbol."
            },
            "will:math/0.6/FractionLine": {
                "subclassOf": "will:seg/0.3/MathItem",
                "properties": {},
                "isFinal": true,
                "@desc": "Links the strokes that are associated to the fraction line using `hasChild` property"
            },
            "will:math/0.6/Expression": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "operator": {
                        "dataType": "will:math/0.6/Operator,will:math/0.6/Relation",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "isOrdered": true,
                        "inverseOf": "hasParent",
                        "@desc": "Operations and Relations need at least one `Operator` or a `Relation`, items includes the operands."
                    },
                    "items": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/Relation,will:math/0.6/Indexed,will:math/0.6/Matrix,will:math/0.6/RootExpression",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "isOrdered": true,
                        "inverseOf": "hasParent",
                        "@desc": "If the operator is set with an `Operator` or a `Relation`, the items must be considered as operands."
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "Flattened representation of the operation. It consists of at least 2 components - operator | relation, n-operands. If there are more operands it would result in, e.g., 5x+3y+3 -> +, 5x, 3y, 3."
            },
            "will:math/0.6/Fenced": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "openingBracket": {
                        "dataType": "will:math/0.6/Symbol",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Symbol used for opening bracket, e.g., (, [, or {."
                    },
                    "closingBracket": {
                        "dataType": "will:math/0.6/Symbol",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Symbol used for closing bracket, e.g., ), ], or }."
                    },
                    "body": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/Indexed,will:math/0.6/RootExpression",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "inverseOf": "hasParent"
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "A fenced term has always an opening and closing brackets, e.g, ((x + 1) * 2)."
            },
            "will:math/0.6/System": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "equation": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Matrix,will:math/0.6/Indexed,will:math/0.6/Fraction,will:math/0.6/Fenced,will:math/0.6/RootExpression",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "isOrdered": true,
                        "inverseOf": "hasParent"
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "A system of equations is grouping several lines via `equation`."
            },
            "will:math/0.6/Cases": {
                "subclassOf": "will:math/0.6/System",
                "properties": {
                    "bracket": {
                        "dataType": "will:math/0.6/Symbol",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent"
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "A fenced term with vertical alignment, where there is only one bracket. Cases are added via `equation` for `System` class."
            },
            "will:math/0.6/RootExpression": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "degree": {
                        "dataType": "will:math/0.6/Symbol,will:math/0.6/Number",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Describes the degree of the root expression."
                    },
                    "rootSign": {
                        "dataType": "will:math/0.6/Symbol",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent"
                    },
                    "body": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent"
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "RootExpression can have a symbol  n \u221a x, or a number  3 \u221a x. If no degree is set, it is considered as squared root."
            },
            "will:math/0.6/Fraction": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "fractionLine": {
                        "dataType": "will:math/0.6/FractionLine",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Stroke(s) annotating the fraction line."
                    },
                    "numerator": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Numerator part of the fraction. (optional for partial fraction)"
                    },
                    "denominator": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Denominator part of the fraction (optional for partial fraction)."
                    },
                    "fractionType": {
                        "dataType": "@string",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Describes the type of fraction.",
                        "dataRange": [
                            "Normal,Slanted,Partial"
                        ]
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "Fraction expression with numerator and denominator."
            },
            "will:math/0.6/Row": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "hasCell": {
                        "dataType": "will:math/0.6/Number,will:math/0.6/Symbol,will:math/0.6/CompoundTerm,will:math/0.6/Fraction,will:math/0.6/Fenced,will:math/0.6/Indexed,will:math/0.6/Expression,will:math/0.6/RootExpression,will:math/0.6/Matrix",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "isOrdered": true,
                        "@desc": "Represents the content a cell within a matrix."
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "Row of a matrix structure."
            },
            "will:math/0.6/Matrix": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "hasRow": {
                        "dataType": "will:math/0.6/Row",
                        "minCardinality": 1,
                        "maxCardinality": -1,
                        "isFinal": false,
                        "isOrdered": true,
                        "@desc": "Rows of the matrix."
                    },
                    "openingBracket": {
                        "dataType": "will:math/0.6/Symbol",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Symbol used for opening bracket, e.g., ( or [."
                    },
                    "closingBracket": {
                        "dataType": "will:math/0.6/Symbol",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "@desc": "Symbol used for closing bracket, e.g., ), or ]."
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "Matrix expression, which consists of multiple rows."
            },
            "will:math/0.6/Indexed": {
                "subclassOf": "will:seg/0.3/MathItemGroup",
                "properties": {
                    "body": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Fenced",
                        "minCardinality": 1,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Base expression."
                    },
                    "superScript": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Superscript expression / term."
                    },
                    "subScript": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Subscript expression / term."
                    },
                    "overScript": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Overscript expression / term."
                    },
                    "underScript": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Underscript expression / term."
                    },
                    "preSuperScript": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Pre-superscript expression / term."
                    },
                    "preSubScript": {
                        "dataType": "will:math/0.6/Term,will:math/0.6/CompoundTerm,will:math/0.6/Expression,will:math/0.6/Fenced,will:math/0.6/Fraction,will:math/0.6/RootExpression,will:math/0.6/Indexed",
                        "minCardinality": 0,
                        "maxCardinality": 1,
                        "isFinal": false,
                        "inverseOf": "hasParent",
                        "@desc": "Pre-subscript expression / term."
                    },
                    "hasChild": {
                        "dataType": "",
                        "minCardinality": 0,
                        "maxCardinality": 0,
                        "isFinal": false
                    }
                },
                "isAbstract": false,
                "isFinal": false,
                "@desc": "Generic expression for all indexed terms. There is no mathematical interpretation of the indexed term, thus summations or integrals would be expressed as symbols with an over- and underscript."
            }
        }
    }
}