# The MASTON Format

The Math Abstract Syntax Tree Object Notation is a lightweight data interchange format for mathematical notation.

It is human-readable, while being easy for machines to generate and parse.

It is built on the JSON [1] format. Its focus is on interoperability between software programs to facilitate the exchange of mathematical data, as well as the building of complex software through the integration of software components communicating with a common format.

It is not suitable as a visual representation of arbitrary mathematical notations, and as such is not a replacement for LaTeX or MathML.

## Examples

### Euler's Identity

In TeX

e^{\imaginaryI \pi }+1=0

In MASTON:

    {
"fn":"=",
"arg":[{
"fn":"+",
"arg":[
{
"sym":"e",
"sup":{
"fn":"*",
"arg":["ⅈ","π"]
}
},
1
]
},
0]
}

### An approximation of Pi

\frac {63}{25}\times \frac {17+15\sqrt{5}}{7+15\sqrt{5}}
{"fn":"*","arg":[{"fn":"/","arg":[63,25]},{"fn":"/","arg":[{"fn":"+","arg":[17,{"fn":"*","arg":[15,{"fn":"sqrt","arg":5}]}]},{"fn":"+","arg":[7,{"fn":"*","arg":[15,{"fn":"sqrt","arg":5}]}]}]}]}

## Design Goals

### Definitions

• producer software that generates a MASTON data structure
• consumer software that parses and acts on a MASTON data structure

### Goals

• Easy to consume, even if that's at the expense of complexity to generate.
• Extensibility. It should be possible to add information to the data structure that can help its interpretation or its rendition. This information should be optional and can be ignored by any consumer.

### Non-goals

• Be suitable as an internal data structure
• Be suitable as a display format
• Capture complete semantic information with no ambiguity and in a self-sufficient manner.

## Encoding

A MASTON expression is an abstract syntax tree encoded as a JSON object.

The root element is an ⟨expression⟩, with child nodes according to the grammar below.

### Native Numbers

A native number is encoded following the JSON grammar, with two extensions:

• support for arbitrary precision numbers. The number of digits included may be more than supported by consuming software. The software can handle this situation by either reading only as many digits as can be supported internally or by treating it as an error.
• support for NaN and infinity

⟨native-number⟩ := '"NaN"' | ⟨native-infinity⟩ | ['-'] ⟨native-int⟩ [ ⟨native-frac⟩] [ ⟨native-exp⟩ ]

⟨native-infinity⟩ := '"' ['+' | '-'] 'infinity' '"'

⟨native-int⟩ := '0' | [ '1' - '9' ]*

⟨native-frac⟩ := '.' ('0' - '9')*

⟨native-exp⟩ := ['e' | 'E'] ['+' | '-'] ('0' - '9' )*

### Native Strings

Native strings are a sequence of Unicode characters.

As per JSON, any Unicode character may be escaped using a \u escape sequence.

MATSON producing software should not generate character entities in strings.

Whenever applicable, a specific Unicode symbol should be used.

For example, the set of complex numbers should be represented with U+2102 ℂ, not with U+0043 C and a math variant styling attribute.

When used with markup languages—for example, with Mathematical Markup Language (MathML)—the characters are expected to be used directly, instead of indirectly via entity references or by composing them from base letters and style markup.

### Optional keys

All elements may have the following keys:

• comment: A human readable string to annotate an expression, since JSON does not allow comments in its encoding
• error: A human readable string that can be used to indicate a syntax error or other problem when parsing or evaluating an expression.
• latex: A visual representation in LaTeX of the expression. This can be useful to preserve non-semantic details, for example parentheses in an expression.
• mathml: A visual representation in MathML of the expression.
• class: A CSS class to be associated with a representation of this element
• id: A CSS id to be associated with a representation of this element
• style: A CSS style string
• wikidata: A short string indicating an entry in a wikibase. For example, "Q2111"
• wikibase: A base URL for the wikidata key. A full URL can be produced by concatenating this key with the wikidata key. This key applies to this element and all its children. The default value is "https://www.wikidata.org/wiki/"
• openmathcd: A base URL for an OpenMath content dictionary. This key applies to this element and all its children. The default value is "http://www.openmath.org/cd".
• openmathsymbol: A short string indicating an entry in an OpenMath Content Dictionary. For example: arith1/#abs.

### Key order

The order of the keys in an element is not significant. That is, all these expressions are equivalent:

   {"fn":"+", "arg":[1, 2]}
{"arg":[1, 2], "fn":"+"}

Howeve, the order of the elements in an array is significant. These two expressions are not equivalent:

   {"fn":"-", "arg":[3, 1]}
{"fn":"-", "arg":[1, 3]}

## Grammar

⟨expression⟩ := ⟨num⟩ | ⟨complex⟩ | ⟨symbol⟩ | ⟨function⟩ | ⟨group⟩ | ⟨range⟩ | ⟨array⟩ | ⟨dictionary⟩ | ⟨text⟩ | ⟨block⟩

An expression is an Abstract Syntax Tree. As such, there is no need to introduce parentheses or to resort to operator precedence in order to parse the expression correctly.

### ⟨num⟩

A native number or an object with the following key

• num: ⟨native-number⟩ or ⟨native-string⟩

Note: When only the num key is present a shortcut may be used by replacing the element with the number. That is, both representations are equivalent:

   {"fn":"+", "arg":[{"num":1}, {"num":2}]}
{"fn":"+", "arg":[1, 2]}

### ⟨complex⟩

• re: ⟨native-number⟩, 0 by default.
• im: ⟨native-number⟩, 0 by default.

One of the keys re or im must be present.

Note that {re:1} is a complex number with a null imaginary part.

### ⟨symbol⟩

A string or an object with the following keys

• sym: ⟨native-string⟩
• type: the data type of the symbol, as a string. See table below.
• index: A 0-based index into a vector or array. An index can be a number or an array of numbers.
• accent: ⟨string⟩, a single unicode character representing the accent to display over the symbol.

### Type

The data type of a symbol can be used to refine the interpretation of operations performed upon it.

Data Type Value Meanings
Scalar scalar scalar number
Complex complex complex number
Vector vector an element composed of n scalars or complex numbers
Matrix matrix an element composed of n vectors
Function function
String string an array of characters
Dictionary dictionary a collection of key/value pairs
Boolean boolean true or false
Table table a two-dimensional array of cells. Each cell can be of a different type.
Date date
Duration duration

### Accent

An accent is a decoration over a symbol that provides the proper context to interpret the symbol or modifies it in some way. For example, an accent can indicate that a symbol is a vector, or to represent the mean, complex conjugate or complement of the symbol.

The following values are recommended:

Accent Value Unicode Possible Meanings
Vector ◌⃗ U+20d7
Bar ◌¯ U+00af Mean, complex conjugate, set complement.
Hat ◌^ U+005e Unit vector, estimator
Dot ◌˙ U+02d9 Derivative with respect to time
Double dot ◌¨ U+00a8 Second derivative with respect to time.
Acute ◌´ U+00b4
Grave ◌ U+0060
Tilde ◌~ U+007e
Breve ◌˘ U+02d8
Check ◌ˇ U+02c7

### ⟨function⟩

• fn: ⟨native-string⟩, the name of the function.
• arg: ⟨expression⟩ | array of ⟨expression⟩, the arguments to the function. If there's a single argument, it should be represented as an expression. If there's more than one, they should be represented as an array of expressions.
• fence: ⟨string⟩, one to three characters indicating the delimiters used for the expression. The first character is the opening delimiter, the second character, if present, is the closing delimiter. The third character, if present, is the delimiters separating the arguments. If no value is provided for this key, the default value (), is used. The character . can be used to indicate the absence of a delimiter, for example ..;.
• sub: ⟨expression⟩
• sup: ⟨expression⟩
• accent: ⟨native-string⟩, a single unicode character representing the accent to display over the function. See the SYMBOL section for more details.

The fn key is the only required key.

When using common functions, the following values are recommended:

Name (and common synonyms) Value Arity Comment
Addition + 2
Signum * 1 -1 if the argument is negative, 0 if it is zero, 1 if it is positive; more generally, the intersection of the unit circle with the line from the origin through the argument in the complex plane
Multiplication * 2
Reciprocal / 1 The reciprocal of the argument
Division / 2 The first argument divided by the second argument
Negate - 1 Negate the argument
Substraction - 2 Substract the second from the first.
Exponential ^ 1 e to the power of the argument.
Power ^ 2 The first argument to the power of the second argument
Square Root root 1
Root root 2 The second argument is the degree of the root
Natural log ln 1
Logarithm ln 2 The second argument is the base.
List list n comma separated list
List list2 n semi-colon separated list
Absolute value abs
Floor floor 1 The largest integer less than or equal to the argument
Minimum min 2, n The smallest of the arguments
Ceiling ceiling 1 The smallest integer greater than or equal to the argument
Maximum max 2, n The largest of the arguments
Greatest Common Divisor gcd 2
Least Common Multiple lcm 2
Function composition compose 2, n

#### Complex Arithmetic

Name Value Arity Comment
Conjugate + 1 Conjugate of the argument
Real real 1
Imaginary imaginary 1
Argument arg 1

#### Logic

Name Value Arity Comment
Implies
Logical equivalence
For All (universal quantifier) forall
There Exists (existential quantifier) exists

There Does Not Exists

#### Trigonometry

Note that for inverse functions, no assumptions is made about the branch cuts of those functions. The interpretation is left up to the consuming software.

Name (and common synonyms) Value Arity Comment
Cosine cos 1 angle in radians
Sin sin 1 angle in radians
Tangent (tan, tg) tan 1 angle in radians
Co-tangent (cot, ctg, cotg, ctn) cotangent 1
Secant sec 1
Cosecant csc 1
Arc cosine acos 1 angle in radians
Arc sine asin 1 angle in radians
Arctangent (arctan, arctg) atan 1 angle in radians
Arctangent (arctan, arctg) atan 2 See https://en.wikipedia.org/wiki/Atan2
Arc-cotangent arccot 1
Arc-secant arcsec 1
Arc-cosecant arccsc 1
Hyperbolic tangent (th, tan) tanh 1

#### Relational operators

Operation Value Unicode Comment
Equal to = U+003D
Definition/assignment := U+003D Used with a := 5 or f(x) := sin(x)
Identity :=: U+003D Used with 1 + 1 :=: 2
Approximately equal to ≈ ≈ U+2248
Not equal to ≠ U+2260
Less than < U+003C
Less than or equal to <= ≤ U+2264
Greater than > U+003C
Greater than or equal to >= ≥ U+2265

There are three semantically distinct use for "equal to" which are often all represented with = in mathematical notation:

• conditional equality: the expression is true when the left hand side and the right hand side are equal, for example when defining a curve representin the unit circle: x^2 + y^2 = 1
• definition or assignment: the symbol (or expression) on the left hand side is defined by the expression on the right hand side. For example f(x) := sin x, a = 5
• identity: the right hand side expression is a syntactic derivation from the left hand size expression. For example, 1 + 1 :=: 2

#### Big operators

Big operators, such as ∑, "sum", and ∏, "product", are represented as a function with the following arguments:

• first argument: body of the operation
• second argument (optional): inferior argument of the operation
• third argument (optional): superior argument of the operation

For example:

\sum ^n_{i=0}i
{
"fn":"sum",
"arg":[
"i",
{"fn":"=","arg":["i", 0]},
"n"
]
}

If necessary, an empty argument can be represented by an empty structure.

The following values should be used to represent these common big operators:

Operation Value Comment
Sum sum ∑ U+2211
Product product ∏ U+220f
Intersection intersection ⋂ U+22c2
Union union ⋃ U+22c3
Integral integral ∫ U+222b
Double integral integral2 ∬ U+222c
Triple integral integral3 ∭ U+222d
Contour integral contour_integral ∮ U+222e
Circle Plus circle_plus U+2a01
Circle Times circle_times U+2a02
And n_and U+22c1
Or n_or U+22c0
Coproduct coproduct ∐ U+2210
Square cup square_cup U+2a06
U plus union_plus U+2a04
O dot odot U+2a00

#### Other functions

Operation Value Comment
Factorial factorial !
Double factorial factorial2 !!

Additional functions can be specified using an OpenMath content dictionary. For example, Euler's gamma function:

{
fn: "gamma",
openmathsymbol: "hypergeo0#gamma",
arg: 1
}

If an openmathsymbol key is present it overrides the value of the fn key as far as the semantic of the operation is concerned. However, the fn key can still be used to display information about this expression to a user. For example:

{
fn: "\u0393",
openmathsymbol: "hypergeo0#gamma",
arg: 1
}

where \u0393 is the Unicode character GREEK CAPITAL LETTER GAMMA Γ.

### ⟨group⟩

• group: ⟨expression⟩
• sup: ⟨expression⟩
• sub: ⟨expression⟩
• accent: ⟨string⟩

The group key is the only one required.

This element is used when a sup, sub or accent needs to be applied to an expression, as in (x+1)^2.

### ⟨range⟩

• range_start: ⟨expression⟩
• range_end: ⟨expression⟩
• range_step: ⟨expression⟩
• closure: "open" | "closed" | "open-closed" | "closed-open", default "closed"

The range_start key is the only one required. If absent, range_end is assumed to be infinity. If absent, range_step is assumed to be 1.

### ⟨array⟩

• rows: array of ⟨expression⟩
• fence: ⟨native-string⟩
• index: A 0-based index into the vector or array. An index can be a number or an array of numbers.

The rows key is the only one required.

### ⟨dictionary⟩

• keys: object mapping keys to values

Example:

{keys:{"a":1, "b":"one"}}

defines the following dictionary:

Key Value
a 1
b "one"

### ⟨text⟩

• text: ⟨native-string⟩
• format: "plain" | "markdown" | "html". This key is optional and its default value is plain

The text key is the only one required.

### ⟨block⟩

• block: array of ⟨expression⟩
• conditions: array of ⟨expression⟩

A sequence of expressions, such as in a system of equations or a piecewise definition.

The block key is the only one required.

Example: piecewise definition of absolute value.

\begin{cases}x & \mbox{if }x\ge 0 \\ -x & \mbox{if }x<0 \end{cases}
{"block":[
"x",
{fn:"-",arg:"x"}
],
"conditions":[
{fn:">=",arg:["x",0]},
{fn:"<",arg:["x",0]}
]}

Design note: having the block and conditions in separate keys is britle. Consider using if nodes, e.g. {if:{fn:">=",arg:["x",0]}, then:"x"}

### OPEN QUESTIONS

1. How should exponents be represented? I.e. x^2 or A^\dagger. They could literally be represented with a sup attribute, or as an explicit function, i.e. fn:'pow' or fn:'transjugate'
2. Clarify how to represent variants for multiplications, e.g. a \times b, a . b, a * b, ab, etc...
3. How to encode logarithm and exponential (see 1.)
4. What should the effect of n-ary versions of divide, substract? One option is to apply a left-reduce to the arguments.
5. How should accents (i.e. arrow over symbol) be encoded? As an additional property? As a function? How about other stylistic variant (i.e. bold symbol, fraktur, blackboard, etc...)
6. Should there be a node type to represent conditions, i.e. expressions whose value is a boolean.
7. For functions defined with an openmath identifier, the value of the fn key could be the openmath identifier, i.e. fn:"hypergeo0#gamma"`

### REFERENCES

1. https://www.json.org/
2. http://www.openmath.org/cd