# 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', '_'];
console.log(ce.match(['Add', 'x', 1], pattern));
// -> { } : 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', '_'];
console.log(ce.match(['Add', 'x', 1], pattern));
// -> { } : the expression matches the pattern
console.log(ce.match(['Add', 1, 'x'], 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
console.log(ce.match(['Add', 'x', 'x'], ['Add', '_a', '_a']));
// -> { "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' }));
// -> ["Add", 1, "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';
console.log(ce.match(['Add', 'x', 1], ['Add', variable, 1]));
// ➔ {}: the two expressions are the same
console.log(ce.match(['Add', 'x', 1], ['Add', 1, 'x']));
// ➔ 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()`

.