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MASTON

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":"equal",
    "arg":[{
        "fn":"add",
            "arg":[{
                "sym":"e",
                    "sup":{
                        "fn":"multiply",
                        "arg":[{"sym": "ⅈ"},{sym:"π"}]
                    }
                },
                {"num": "1"}
            ]
        },
        "num": "0"
    ]
}

An approximation of Pi

\frac {63}{25}\times \frac {17+15\sqrt{5}}{7+15\sqrt{5}}
{"fn":"multiply",
    "arg":[
        {"fn":"divide","arg":[{"num":"63"},{"num":"25"}]},
        {"fn":"divide","arg":[
            {"fn":"add","arg":[
                {"num":"17"},
                {"fn":"multiply","arg":[{"num":"15"},{"fn":"sqrt","arg":[{"num":"5"}]}]}
            ]},
            {"fn":"add","arg":[
                {"num":"7"},
                {"fn":"multiply","arg":[
                    {"num":"15"},
                    {"fn":"sqrt","arg":[
                        {"num":"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.

Basic Number

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

  • support for arbitrary precision numbers. JavaScript and many other languages only support limited precision numbers, generally 52-bit integers (about 15 digits). Therefore to support arbitrary precision, numbers should always be represented as a quoted string.
  • support for NaN and infinity

basic-number := '"NaN"' | infinity | '"' ['-'] int [ frac] [ exp ]'"'

infinity := '"' ['+' | '-'] 'Infinity' '"'

int := '0' | [ '0' - '9' ]*

frac := '.' ('0' - '9')*

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

Native Strings

Native strings are a sequence of Unicode characters.

MATSON producing software should not generate character entities in strings.

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

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.

See Unicode Chapter 22 - Symbols

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 nodes may have the following keys:

  • sub: expression, a subscript
  • sup: expression, a superscript
  • 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 or styling attributes.
  • mathml: A visual representation in MathML of the expression.
  • class: A CSS class to be associated with a representation of this node
  • id: A CSS id to be associated with a representation of this node
  • 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 node 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 node 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 a node is not significant.

All these expressions are equivalent:

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

However, the order of the elements in an array is significant.

These two expressions are not equivalent:

   {"fn":"divide", "arg":[{num:"3"}, {num:"1"}]}
   {"fn":"divide", "arg":[{num:"3"}, {num:"1"}]}

Grammar

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.

The type of each node is indicated by the presence of a specific key, for example sym for the "symbol" node or fn for the "function" node.

There are five types of nodes:

expression := number | symbol | function | group | text |

number

A node with the following key:

  • num: basic-number | complex-number

complex-number := { "re": basic-number, "im": basic-number}

symbol

A node with the following keys

  • sym: native-string
  • type: native-string`, the data type of the symbol. 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 modifier applied to a symbol, such as "hat" or "bar".

The sym key is the only required key.

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: array of expression, the arguments to the function.
  • 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) Arity Comment
add 2+
multiply 2+
subtract 2 Subtract the second from the first.
divide 2 The first argument divided by the second argument
negate 1 Negate the argument
list 1+ comma separated list
list2 1+ semi-colon separated list
Name (and common synonyms) Arity Comment
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
Exponential 1 e to the power of the argument.
Power 2 The first argument to the power of the second argument
Square Root 1
Root 2 The second argument is the degree of the root
Natural log 1
Logarithm 2 The second argument is the base.
abs 1 Absolute value
min 2, n The smallest of the arguments
max 2, n The largest of the arguments
floor 1 The largest integer less than or equal to the argument
ceiling 1 The smallest integer greater than or equal to the argument
gcd 2 Greatest Common Divisor
lcm 2 Least Common Multiple
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 Comment
Equal to equal General purpose equality
Less than lt
Less than or equal to le
Greater than gt
Greater than or equal to ge
Much less than ll x ≪ y
Much greater than gg y ≫ x
Definition/assignment assign Used with a := 5 or f(x) := sin(x)
Identity :=: Used with 1 + 1 :=: 2
Approximately equal to approx Used with π ≈ 3.14
Not equal to ne
Similar to sim 2 ~ 5
Congruent to cong A ≅ B

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 representing 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

When a more specific version cannot be determined from the context, the general purpose equal function should be used.

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":[
        {sym: "i"}, 
        {"fn":"=","arg":["i", 0]},
        {sym: "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

Special Functions

These functions represent higher order data structures and operations common in algebraic manipulation systems.

  • bind first argument is an expression, argument 2n is an identifier, argument 2n+1 is the value the identifier should be replaced with.

Argument 2n+1 can be a number, an expression, an array representing a list of values, or a range. If an array or a range, the result is an array of expression.

For example:

{   "fn":"bind", 
    "arg": [
        {"fn":"+","arg":[{"sym":"x","sup":2},{"fn":"*","arg":[2,"x"]}]},
        "x",
        "1"
    ]
}

would replace the xin the expression with 1.

  • solve first argument is an expression. Return an array of identifier and their value. The first argument can be an array representing a system of equations.

  • range first argument is the start of the range, second argument is the end of the third. The third argument is optional and represent the step between elements of the range. If none is provided, the step is 1.

  • block the arguments are a list of expressions, such as in a system of equations.

  • case the arguments are a list of conditional expressions, such as in a piecewise definition of a function. Arguments 2n-1 are the expression, and 2n are the condition. The last condition is optional and the last value is applied if no other condition match.

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.

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.

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