Object Classes

The fundamental classes that functions are built up from are described below.

Most of these classes are defined in mathics.builtin.base or mathics.core.expression.

Class Diagram for Some of the Classes

Below is a UML 2.5 Class diagram for some of the classes described below:

UML 2.5 Class diagram

A Class name that begins with Base is a Virtual class.

Atom Class

Recall that an Expression to be evaluated is initially a kind of M-expression, an object in the Expression class, where each list item is either itself an Expression or an object in a class derived from Atom.

The Atom class we encountered earlier when describing the nodes that get created intially from a parse. Those were:

  • Number

  • String

  • Symbol

  • Filename

Atoms here is from module mathics.core.parse.ast

However this is converted in mathics.core.parser.convert.Converter.convert() into another kind of expression where Number is replaced by a more specific kind of number like Integer, or Real.

There are a few other kinds of Atoms or fundamental objects like:

  • ByteArray

  • CompiledCode

  • Complex

  • Dispatch

  • Image

In general, atoms are objects might have an underlying internal representation with no user-visable subparts that can be pulled out using Part[].

In Mathics, the function AtomQ[] will tell you if something is an Atom, or can’t be subdivided into subexpressions.

Some examples:

(* Strings and expressions that produce strings are atoms: *)
>> Map[AtomQ, {"x", "x" <> "y", StringReverse["live"]}]
 = {True, True, True}

(* Numeric literals are atoms: *)
>> Map[AtomQ, {2, 2.1, 1/2, 2 + I, 2^^101}]
 = {True, True, True, True, True}

(* So are Mathematical Constants: *)
>> Map[AtomQ, {Pi, E, I, Degree}]
 = {True, True, True, True}

(* A 'Symbol' not bound to a value is an atom too: *)
>> AtomQ[x]
 = True

(* On the other hand, expressions with more than one 'Part' after evaluation, even those resulting in numeric values, aren't atoms: *)
>> AtomQ[2 + Pi]
 = False

(* Similarly any compound 'Expression', even lists of literals, aren't atoms: *)
>> Map[AtomQ, {{}, {1}, {2, 3, 4}}]
 = {False, False, False}

(* Note that evaluation or the binding of "x" to an expression is taken into account: *)
>> x = 2 + Pi; AtomQ[x]
 = False

(* Again, note that the expression evaluation to a number occurs before 'AtomQ' evaluated: *)
>> AtomQ[2 + 3.1415]
 = True

BaseElement Class

A Mathics M-expression is the main data structure which evalution is performed on. An M-expression is, in general, a tree. The nodes of this tree come from the BaseElement class. Note that leaf nodes in addition to being a BaseElement are an Atom as well. In other words, an Atom is a subclass of BaseElement.

The other subclass of BaseElement is an Expression.

Note as the prefix Base implies, a BaseElement is a virtual class.

Builtin class

A number of Mathics variables and functions are loaded when Mathics starts up, thousands of functions even before any Mathics packages are loaded. As with other Mathics objects like Atom and Symbol, Mathics variables and functions are implemented through Python classes.

The reason that we use a class for a Mathics variable or a Mathics function is so that we can give those Mathics object properties and attributes.

At the lowest level of the class hierarchy is Builtin.

Lets look at a simple one:

class Head(Builtin):
        <dd>returns the head of the expression or atom $expr$.

    >> Head[a * b]
     = Times
    >> Head[6]
     = Integer
    >> Head[x]
     = Symbol

    def apply(self, expr, evaluation):

        return expr.get_head()

In the above, we have not defined an evaluation() method explicitly so we get Expressions’s built-in evaluation() method.

A feature of the Builtin class is the convention that its provides a convention by which “apply” methods of the class can be matched using the method’s name which must start with “apply” and a pattern listed in the method’s doc string. This is used in the example above.

Here, Head has one paramater which is called expr. Note that in the Python method there is also expr variable it its method signature which is listed right after the usual self method that you find on all method functions.

At the end is an evaluation parameter and this contains definitions and the context if the method needs to evaluate expressions.

Definition Class

A Definition is a collection of Rules and attributes which are associated with a Symbol.

A Rule is internally organized in terms of the context of application in

  • OwnValues,

  • UpValues,

  • Downvalues,

  • Subvalues,

  • FormatValues, etc.

Definitions Class

The Definitions class hold state of one instance of the Mathics interpreter is stored in this object.

The state is then stored as Definition object of the different symbols defined during the runtime.

In the current implementation, the Definitions object stores Definition s in four dictionaries:

  • builtins: stores the defintions of the Builtin symbols

  • pymathics: stores the definitions of the Builtin symbols added from pymathics modules.

  • user: stores the definitions created during the runtime.

  • definition_cache: keep definitions obtained by merging builtins, pymathics, and user definitions associated to the same symbol.

Expression Class

An Expression object the main object that we evaluate over. It represents an M-expression formed from input.

Although objects derived from Atom, e.g. symbols and integers, are valid expressions, this class describes compound expressions, or expressions that are more than a single atom/leaf. So in contrast to an object of type Atom, an Expression object is some sort of structured node that as in Mathics itself, has a Head (function designator) and a Rest (or arguments) component.

Predefined Class

Just above Builtin in the Mathics object class hierarchy is Predefined.

Some Mathics values like True are derived from Predefined. For example:

class True_(Predefined):
      <dd>represents the Boolean true value.

    name = "True"

In the above, note that the class name has an underscore (_) appended it. We do this so as not to conflict with the Python value True. The class variable name is used to associate the Mathics name.

A number of Mathics variables like $ByteOrdering are also derived directly from the Predefined class. Since Python class names cannot start with a dollar sign ($), we drop off the leading $, in the class name, and that gives us: ByteOrdering.

As with the True example shown above, the Mathics name is set using class variable name defined in the ByteOrdering class. For example:

class ByteOrdering(Predefined):
     <dd>returns the native ordering of bytes in binary data on your computer system.
   name = "$ByteOrdering"

 def evaluate(self, evaluation) -> Integer:
     return Integer(1 if sys.byteorder == "big" else -1)

The evaluate() function above is called to get the value of variable $ByteOrdering.

Symbol Class

Just above the Atom class is the Symbol which is an atomic element of an Expression. See Atomic Elements of Expressions.

As born from the parser, Symbols start off like Lisp Symbols. Following WL, Mathics has about a thousand named characters, some common ones like “+”, “-”, and some pretty obscure ones. After parsing, each of these can be incorporated into a Symbol object. But in the evaluation process in conjuction with the Definitions object that is in the evaluation object, these symbols get bound to values in a scope, and then they act more like a programming language variable. The Symbol class described here has fields and properties that you of the kind that you’d expect a variable in a programming language to have.


PrefixOperator and PostFixOperator

BinaryOperator and UnaryOperator

SympyConstant, MPMathConstant, and NumpyConstant

SympyFunction and MPMathFunction

Which Class should be used for a Mathics Object?

  • To define a Mathics constant based on a Sympy constant, e.g. Infinity use SympyConstant

  • To define Mathics constants based on a mpmath constant, e.g. Glaisure, use MPMathConstant

  • To define a Mathics constant based on a numpy constant, use NumpyConstant

  • To define a Mathics functions based on a Sympy function, e.g. Sqrt, use SympyFunction

  • To define a Mathics operator use UnaryOperator, PrefixOperator, PostfixOperator, or BinaryOperator depending on the type of operator that is being defined

  • To define a Mathics function which returns a Boolean value e.g. MatchQ use Test

  • To define a Mathics function that doesn’t fall into a category above, e.g. Attributes use Builtin

  • To define a Mathics variable e.g. $TimeZone or Mathics Symbols, e.g. True use Predefined

  • To define a Mathics atomic expression, e.g. ImageAtom use AtomicBuiltin