# Patterns and Rules

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

## Wildcards

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

The `"_"` 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. 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 wildcard symbols.

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 match 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 `{"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 expression into the pattern by replacing the wildcards.

## Matching an Expression to a Pattern

To check if an expression matches a pattern, use the `ce.match(<expression>, <pattern>)` function.

If there is a match, `ce.match()` returns a `Substitution` object literal with keys corresponding to the matching named wildcards. If no named wildcards are used and there is a match it returns an empty object literal. If there is no match, it returns `null`.

``````const pattern = ['Add', 'x', '_'];

// -> { } : the expression matches the pattern

console.log(ce.match(['Multiply', 'x', 1], pattern));
// -> null : the expression does not match the pattern
``````

The `ce.match()` function accounts for the commutativity and associativity of functions.

``````const pattern = ['Add', 'x', '_'];

// -> { } : the expression matches the pattern

// -> { } : the expression matches the pattern by commutativity
``````

The `ce.match()` does not consider sub-expressions, it is not recursive.

``````const pattern = ['Add', 'x', '_'];

console.log(ce.match(['Multiply', 2, ['Add', 'x', 1]], pattern));
// -> null : the expression does not match the pattern
``````

If the same named wildcard is used multiple times, all its values must match.

``````console.log(ce.match(['Add', 1, 'x'], ['Add', '_a', '_a']));
// -> null

// -> { "a": "x" }
``````

Wildcards can be used to capture the head of functions:

``````console.log(match(['Add', 1, 'x'], ['_f', 1, 'x']));
// -> { "f": "Add" }
``````

## Substitution

The return value of the `match()` function is a `Substitution` object: a mapping from wildcard names to expressions.

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

``````const expression = ['Add', 1, 'x'];
const pattern = ['Add', 1, '_a'];

console.log(match(expression, pattern));
// -> { a: "x" }

console.log(substitute(pattern, { a: 'x' }));
``````

## Comparing

To compare two expressions, use the `match()` function.

The function returns `null` if the two expressions do not match. It returns an object literal if the expressions do match.

If the second argument included wildcards the resulting object literal indicate the substitutions for those wildcards. If no wildcards were used and the expressions matched, an empty object literal, `{}` is returned. To check if the expressions simply match or not, check if the return value is `null` (indicating not a match) or not (indicating a match).

``````const ce = new ComputeEngine();

const variable = 'x';
// ➔ {}: the two expressions are the same

// ➔ null: the two expressions are the same because `Add` is commutative

console.log(ce.match(parse('2 + 2 + x'), parse('3 + 1 + x')));
// ➔ null: the two expressions are **not** the same: they are not evaluated

console.log(
match(
ce.evaluate(parse('2 + 2 + x')),
ce.evaluate(parse('3 + 1 + x'))
)
);
// ➔ {}: the two expressions are the same once evaluated
``````

## Applying Rewrite Rules

A rewrite rule is a triplet of:

• a left-hand-side pattern, `lhs`
• a right-hand-side pattern, `rhs`
• an optional `condition`

When a rule is applied to an expression `expr`, if `expr` matches `lhs` and the `condition` applies to the resulting substitution, the result of the rule is the substitution applied to the `rhs`.

To apply a set of rules to an expression, use the `ce.replace()` function.

``````const squareRule = [['Multiply', '_x', '_x'], ['Square', '_x']];

ce.replace([squareRule], ['Multiply', 4, 4]);
// -> ['Square', 4]

const sqrtRule =   [
['Sqrt', ['Square',  '_x']],
'_x',
(ce, sub) => ce.isPositive(sub._x)
];
ce.replace([sqrtRule], ['Sqrt', ['Square', 17]]);
// -> 17
``````

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

The `ce.simplify()` method applies a collection of built-in rewrite rules. You can define your own rules and apply them using `ce.replace()`.