Patterns and Rules

Recognizing patterns and applying rules is a powerful symbolic computing tool to identify and manipulate the structure of expressions.

Wildcards

Wildcard are placeholders identifiers in an expression. They start with a _.

The "_" universal wildcard matches anything that is in the corresponding position in an expression.

The "__" wildcard matches any sequence of 1 or more expressions in its corresponding position. It is useful to capture the arguments of a function.

The "___" wildcard matches any sequence of 0 or more expressions in its corresponding position.

A wildcard symbol may include a name which is used to capture the matching expression, for example _a.

When using a named wildcard, all instances of the named wildcard must match. In contrast, an un-named wildcard (a universal wildcard such as "_" "__" or "___") can be used multiple times to match different values.

Patterns

A pattern is an expression which can include one or more placeholders in the form of wildcards.

Patterns are similar to Regular Expressions in traditional programming languages but they are tailored to deal with MathJSON expressions instead of strings.

Given a pattern and an expression the goal of pattern matching is to find a substitution for all the wildcards such that the pattern becomes the expression.

An expression is said to match a pattern if there exists a set of values such that replacing the wildcards with those values is equal to the expression. This set of values is called a substitution.

For example, the pattern ["Add", 3, "_c"] becomes the expression ["Add", 3, "x"] by replacing the wildcard "_c" with "x". The substitution is therefore {_c : "x"}.

On the other hand, the expression ["Divide", "x", 2] does not match the pattern ["Add", 3, "_c"]: no substitution exists to transform the pattern into the expression by substituting the wildcards.

Matching an Expression to a Pattern

To check if an expression matches a pattern, use the _expression_.match(_pattern_) method.

If there is no match, the method returns null.

If there is a match, a Substitution object literal with keys corresponding to the matching named wildcards is returned.

If no named wildcards are used and there is a match, an empty object literal is returned.

For convenience, the pattern argument can be an unboxed MathJSON expression.

Commutativity

The commutativity of functions is taken into account when matching patterns.

const pattern = ce.box(["Add", "_", "x"]); console.log("x+1 ➔", ce.box(["Add", 1, "x"]).match(pattern)); // ➔ { } : the expression matches the pattern console.log("x+42 ➔", ce.box(["Add", "x", 42]).match(pattern)); // ➔ { } : the expression matches the pattern by commutativity console.log("5*x ➔", ce.box(["Multiply", 5, "x"]).match(pattern)); // ➔ null : the expression does not match the pattern

Exact Matching

By default, the expr.match() method will match some variants of the same expression, for example x+_a and x are considered to match (with the substitution {_a : 0}).

const pattern = ce.box(["Add", "x", "_a"]); const expr = ce.box("x"); console.log("x ➔", expr.match(pattern)); // ➔ { _a: 0 } : the expression matches the pattern

To prevent the matching of variants, set the exact property to true.

const pattern = ce.box(["Add", "x", "_a"]); const expr = ce.box("x"); console.log("exact: x ➔", expr.match(pattern, {exact: true})); // ➔ null : the expression does not match the pattern

The variants can be applied to the whole expression or to sub-expressions.

const pattern = ce.box(["Add", ["Multiply", "_a", "x"], "_b"]); console.log("x ➔", ce.box("x").match(pattern)); // ➔ { _a: 1, _b: 0 } : the expression matches the pattern

Recursive Matching

By default, the expr.match() method does not consider sub-expressions: it is not recursive.

const pattern = ce.box(["Add", "_", "x"]); const expr = ce.box(["Multiply", 2, ["Add", 1, "x"]]); console.log("2(1+x) ➔", expr.match(pattern)); // ➔ null : the expression does not match the pattern

To match sub-expressions recursively, set the recursive property to true.

const pattern = ce.box(["Add", "_", "x"]); const expr = ce.box(["Multiply", 2, ["Add", 1, "x"]]); console.log("recursive: 2(1+x) ➔", expr.match(pattern, {recursive: true})); // ➔ { } : the expression matches the pattern

Repeated Named Wildcards

If a named wildcard is referenced multiple times in a pattern, all its values must match.

console.log(ce.box(["Add", 1, "x"]).match(ce.box(["Add", '_a', '_a']))); // ➔ null console.log(ce.box(["Add", "x", "x"]).match(ce.box(["Add", '_a', '_a']))); // ➔ { _a: "x" }

Capturing the Head of Functions

Wildcards can be used to capture the head of functions:

console.log(ce.box(["Add", 1, "x"]).match(ce.box(["_f", "__args"]))); // ➔ { _f: "Add", __args: ["Sequence", [1, "x"]] }

Substitution

The return value of the expr.match() function is a Substitution object: a mapping from wildcards to expressions.

If there is no match, expr.match() returns null.

To apply a substitution to a pattern, and therefore recover the expression it was derived from, use the subs() function.

const expr = ce.box(["Add", 1, "x"]); const pattern = ce.box(["Add", 1, "_a"]); const subs = expr.match(pattern); console.log(subs); // ➔ { _a: "x" } pattern.subs(subs).print(); // ➔ ["Add", 1, "x"]

Applying Rewrite Rules

A rewrite rule is an object with three properties:

• match: a matching pattern
• replace: a substitution pattern
• condition: an optional condition predicate

To apply a set of rules to an expression, call the expr.replace() function.

Each rule in the ruleset is applied to the expression in turn. If a rule matches, the expression is replaced by the substitution pattern of the rule.

const squareRule = { match: ["Multiply", "_x", "_x"], replace: ["Square", "_x"], }; const expr = ce.box(["Multiply", 7, 7], { canonical: false }); (expr.replace(squareRule) ?? expr).print(); // ➔ ["Square", 7]

The expr.replace() function continues applying all the rules in the ruleset until no rules are applicable.

If expr is not canonical, the result of the replace operation is not canonical either.

Simplifying an Expression

The expr.simplify() function applies a collection of built-in rewrite rules.

You can define your own rules and apply them using expr.replace().

Substituting Symbols

If a pattern does not contain any named wildcards and only symbols, the expr.subs() function can be used to replace all occurrences of matching symbols.

const expr = ce.box(["Add", ["Multiply", "a", "x"], "b"]); expr.replace([ { match: "a", replace: 2 }, { match: "b", replace: 3 } ], { recursive: true } )?.print(); // ➔ 2x + 3 expr.subs({"a": 2, "b": 3}).print(); // ➔ 2x + 3