Canonical Form

Some mathematical objects can be represented by several equivalent expressions.

For example, the expressions in each row below represent the same mathematical object:

\[ 215.3465 \] \[ 2.15346\mathrm{e}2 \] \[ 2.15346 \times 10^2\]
\[ 1 - x \] \[-x + 1 \] \[ 1 + (-x)\]
\[ -2x^{-1}\] \[ -\frac{2}{x} \] \[ \frac{-2}{x} \]

By applying some conventions — for example sorting operands of commutative functions or flattening associative functions — we define a canonical representation.

A canonical representation is somewhat arbitrary, but using it consistently make some operations easier, for example, comparing two expressions for structural equality.

The canonical form used by the Compute Engine follows common (but certainly not universal) conventions in writing mathematical expressions, and expresses them in a way that optimize their computation. It is not always “the simplest” way to represent an expression.

The canonical form of an expression is always the same when used with a given Compute Engine instance. However, do not rely on the canonical form as future versions of the Compute Engine could provide a different result.

The ce.box() and ce.parse() function return a canonical expression by default.

To get a non-canonical version of an experssion set the canonical option of ce.parse() or ce.box() to false.

The non-canonical version will be closer to the literal LaTeX input from a user for example, which may be desirable to compare a “raw” user input with an expected answer.

ce.parse("\\frac{3}{-5}")
// ➔ ["Rational", -3, 5]
// The canonical version moves the sign to the numerator

ce.parse("\\frac{3}{-5}", { canonical: false })
// ➔ ["Divide", 3, -5]
// The non-canonical version does not change the arguments, so this is 
// interpreted as a regular fraction ("Divide"), not a rational.

You can further customize how an expression is interpreted by using ce.jsonSerializationOptions.

ce.parse("\\frac{3}{5}", { canonical: false })
// ➔ ["Rational", 3, 5]
// This is a rational without modifying the arguments, so a `["Rational"]` 
// expression is returned

ce.jsonSerializationOptions = { exclude: ['Rational'] };
ce.parse('\\frac{3}{5}', { canonical: false });
// ➔ ["Divide", 3, 5]
// We've excluded `["Rational"]` expressions, so it is interepreted as a 
// division instead.

The output of expr.simplify(), expr.evaluate() and expr.N() are canonical expressions.

To obtain the canonical representation of an non-canonical expression, use the expr.canonical property.

console.log(ce.box(['Add', 2, 'x', 3]).canonical);
// ➔ ["Add", 5, "x"]

To check if an expression is canonical use expr.isCanonical.

If the expression is already canonical, expr.canonical immediately return expr.

Calculating the canonical form of an expression is applying some rewriting rules to an expression. In that sense, it is similar to simplifying an expression with expr.simplify(), but it is more conservative in the transformations it applies, and it will not take into account any assumptions about symbols.

The default canonical representation applies a series of transformation to put sums, products, numbers, roots, etc… in canonical form. Below is a list of some of the transformations applied to obtain the canonical form:

  • Idempotency: \( f(f(x)) \to f(x) \)
  • Involution: \( f(f(x)) \to x \)
  • Associativity: \( f(a, f(b, c)) \to f(a, b, c) \)
  • Commutativity: sorted arguments
  • Some operations may be substituted with others, for example substraction replaced by addition. \(1 + 2 - 3 \longrightarrow Add(1, 2, -3)\)
  • For Add, literal 0 is removed, small integers and small rationals are added together.
  • For Multiply, literal 1 is removed, small integers and small rations are multiplied together.
  • For Divide, replaced by Multiply / Power
  • For Subtract, replaced by Add
  • For Sqrt and Root, replaced by Power
  • Complex numbers with no imaginary component are replaced with a real number
  • Rational numbers are reduced, the denominator is positive and not 1
  • For Power
    • \[x^{\tilde\infty} \longrightarrow \operatorname{NaN}\]
    • \[x^0 \longrightarrow 1\]
    • \[x^1 \longrightarrow x\]
    • \[(\pm 1)^{-1} \longrightarrow -1\]
    • \[(\pm\infty)^{-1} \longrightarrow 0\]
    • \[0^{\infty} \longrightarrow \tilde\infty\]
    • \[(\pm 1)^{\pm \infty} \longrightarrow \operatorname{NaN}\]
    • \[\infty^{\infty} \longrightarrow \infty\]
    • \[\infty^{-\infty} \longrightarrow 0\]
    • \[(-\infty)^{\pm \infty} \longrightarrow \operatorname{NaN}\]