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Canonical Form

Many mathematical objects can be represented by several equivalent expressions. A canonical form is a unique representation of an object that is chosen as the standard representation.

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

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

The Compute Engine stores expressions internally in a canonical form to make it easier to work with symbolic expressions.

The canonical form is intended to be “stable”, that is it does not depend on the values of non-constant symbols or on assumptions about a symbol or expression.

The type of symbols can be used during canonicalization of expressions referencing the symbol, as the type can only be narrowed later and thus would not change the result of the canonicalization. The value of variables (non-constant symbols) is never used during canonicalization as it could be changed later.

ce.assign("x", -2);
console.info(ce.parse("\\frac{10}{x}").json);
// ➔ ["Rational", 10, "x"]
// and not `["Rational", -10, ["Negate", "x"]]` or `5`

Future versions of the Compute Engine could have different canonical forms, however a given version of the Compute Engine will always produce the same canonical form for a given expression, given the same type information about symbols and the same dictionary.

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

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

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

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

The ce.box() and ce.parse() functions return a canonical expression by default, which is the desirable behavior in most cases.

To get a non-canonical version of an expression use of ce.parse(s, {canonical: false}) or ce.box(expr, {canonical: false}).

You can further customize the canonical form of an expression by using the ["CanonicalForm"] function or by specifying the form you want to use. See below for more details.

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

console.info(ce.parse('\\frac{30}{-50}').json);
// ➔ ["Rational", -3, 5]
// The canonical version moves the sign to the numerator 
// and reduces the numerator and denominator

console.info(ce.parse('\\frac{30}{-50}', { canonical: false }).json);
// ➔ ["Divide", 30, -50]
// The non-canonical version does not change the arguments,
// so this is interpreted as a regular fraction ("Divide"), 
// not as a rational number.

The value of expr.json (the plain JSON representation of an expression) may not be in canonical form: some "sugaring" is applied to the internal representation before being returned, for example ["Power", "x", 2] is returned as ["Square", "x"].

To customize how an expression is serialized to plain JSON use expr.toMathJson().

const expr = ce.parse("\\frac{3}{5}");
console.log(expr.toMathJson());
// ➔ ["Rational", 3, 5]

console.log(expr.expr.toMathJson({ exclude: ["Rational"] }));
// ➔ ["Divide", 3, 5]
// We have excluded `["Rational"]` expressions, so it 
// is interepreted as a division instead.
const expr = ce.parse("\\frac{10}{30}", { canonical: false });
console.log(expr.json);
// ➔ ["Divide", 10, 30]

console.log(expr.isCanonical);
// ➔ false

console.log(expr.canonical.json);
// ➔ ["Rational", 1, 3]

Canonical Form and Validity​

The canonical form of an expression may not be valid. A canonical expression may include ["Error"] expressions, for example, indicating missing arguments, excess arguments, or arguments of the wrong type.

For example the canonical form of ["Ln"] is ["Ln", ["Error", "'missing'"]] and it is not a valid expression.

To check if an expression is valid use expr.isValid.

To get a list of errors in an expression use expr.errors.

const expr = ce.parse("Ln");
console.log(expr.json);
// ➔ ["Ln", ["Error", "'missing'"]]
// The canonical form of `Ln` is not valid

console.log(expr.isCanonical);
// ➔ true

console.log(expr.isValid);
// ➔ false

console.log(expr.errors);
// ➔ [["Error", "'missing'"]]

Canonical Form Transformations​

The canonical form used by the Compute Engine follows common conventions. However, it is not always "the simplest" way to represent an expression.

Calculating the canonical form of an expression involves applying some rewriting rules to an expression to put sums, products, numbers, roots, etc... in canonical form. In that sense, it is similar to simplifying an expression with expr.simplify(), but it is more conservative in the transformations it applies.

Below is a list of some of the transformations applied to obtain the canonical form:

  • Literal Numbers
    • Rationals are reduced, e.g. \( \frac64 \to \frac32\)
    • The denominator of rationals is made positive, e.g. \(\frac5-11 \to \frac-511\)
    • A rational with a denominator of 1 is replaced with the numerator, e.g. \(\frac191 \to 19\)
    • Complex numbers with no imaginary component are replaced with the real component
  • Add
    • Literal 0 is removed
    • Sum of a literal and the product of a literal with the imaginary unit are replaced with a complex number.
    • Associativity is applied
    • Arguments are sorted
  • Multiply
    • Literal 1 is removed
    • Product of a literal and the imaginary unit are replaced with a complex number.
    • Literal -1 multiplied by an expression is replaced with the negation of the expression.
    • Signs are simplified: (-x)(-y) -> xy
    • Associativity is applied
    • Arguments are sorted
  • Negate
    • Literal numbers are negated
    • Negate of a negation is removed
  • Power
    • \(x^n)^m \to x^{nm}\)
    • \(x^{\tilde\infty} \to \operatorname{NaN}\)
    • \(x^0 \to 1\)
    • \(x^1 \to x\)
    • \((\pm 1)^{-1} \to -1\)
    • \((\pm\infty)^{-1} \to 0\)
    • \(0^{\infty} \to \tilde\infty\)
    • \((\pm 1)^{\pm \infty} \to \operatorname{NaN}\)
    • \(\infty^{\infty} \to \infty\)
    • \(\infty^{-\infty} \to 0\)
    • \((-\infty)^{\pm \infty} \to \operatorname{NaN}\)
  • Square: ["Power", "x", 2] \(\to\) ["Square", "x"]
  • Sqrt: ["Sqrt", "x"] \(\to\)["Power", "x", "Half"]
  • Root: ["Root", "x", 3] \(\to\) ["Power", "x", ["Rational", 1, 3]]
  • Subtract
    • Replaced with addition, e.g. ["Subtract", "a", "b"] \(\to\) ["Add", ["Negate", "b"], "a"]
  • Other functions:
    • Simplified if idempotent: \( f(f(x)) \to f(x) \)
    • Simplified if an involution: \( f(f(x)) \to x \)
    • Simplified if associative: \( f(a, f(b, c)) \to f(a, b, c) \)

Custom Canonical Forms​

The full canonical form of an expression is not always the most convenient representation for a given application. For example, if you want to check the answers from a quiz, you may want to compare the user input with a canonical form that is closer to the user input.

To get the non-canonical form, use ce.box(expr, { canonical: false }) or ce.parse(s, { canonical: false }).

const expr = ce.parse("2(0+x\\times x-1)", {canonical: false});
console.log(expr.json);
// ➔ ["InvisibleOperator", 
//      2,
//      ["Delimiter",
//        ["Sequence", ["Add", 0, ["Subtract", ["Multiply","x","x"],1]]]
//      ]
//   ]

To get the full canonical form, use ce.box(expr, { canonical: true }) or ce.parse(s, { canonical: true }). The canonical option can be omitted as it is true by default.

const expr = ce.parse("2(0+x\\times x-1)", 
  {canonical: true}
).print();
// ➔ ["Multiply", 2, ["Subtract", ["Square", "x"], 1]]

const expr = ce.parse("2(0+x\\times x-1)").print();
// ➔ ["Multiply", 2, ["Subtract", ["Square", "x"], 1]]

To get a custom canonical form of an expression, use the ["CanonicalForm"] function or specify the form you want to use with the canonical option of ce.box() and ce.parse().

To order the arguments in a canonical order, use ce.box(expr, { canonical: "Order" }) or ce.parse(s, { canonical: "Order" }).

const expr = ce.parse("0+1+x+2+\\sqrt{5}", 
  {canonical: "Order"}
);
expr.print();
// ➔ ["Add", ["Sqrt", 5], "x", 0, 1, 2]

Note in particular that the 0 is preserved in the expression, which is not the case in the full canonical form.

There are other forms that can be used to customize the canonical form of an expression. See the documentation of ["CanonicalForm"] for more details.

For example:

const latex = "2(0+x\\times x-1)";
ce.parse(latex, {canonical: false}).print();
// -> ["InvisibleOperator",2,["Delimiter",["Sequence",["Add",0,["Subtract",["Multiply","x","x"],1]]]]]

ce.parse(latex, {canonical: ["InvisibleOperator"]}).print();
// -> ["Multiply",2,["Add",0,["Subtract",["Multiply","x","x"],1]]]

ce.parse(latex, 
  {canonical: ["InvisibleOperator", "Add", "Order", ]}
);
// -> ["Multiply",2,["Subtract",["Multiply","x","x"],1]]