# Term Rewriting

## Rule-based rewriting

Rewrite rules match and transform an expression. A rule is written using either the `@rule` macro or the `@acrule` macro. It creates a callable `Rule` object.

### Basics of rule-based term rewriting in SymbolicUtils

Here is a simple rewrite rule, that uses formula for the double angle of the sine function:

``````using SymbolicUtils

@syms w z α::Real β::Real

r1 = @rule sin(2(~x)) => 2sin(~x)*cos(~x)

r1(sin(2z))``````
``````UndefVarError: PatTerm not defined
``````

The `@rule` macro takes a pair of patterns – the matcher and the consequent (`@rule matcher => consequent`). If an expression matches the matcher pattern, it is rewritten to the consequent pattern. `@rule` returns a callable object that applies the rule to an expression.

`~x` in the example is what is a slot variable named `x`. In a matcher pattern, slot variables are placeholders that match exactly one expression. When used on the consequent side, they stand in for the matched expression. If a slot variable appears twice in a matcher pattern, all corresponding matches must be equal (as tested by `Base.isequal` function). Hence this rule says: if you see something added to itself, make it twice of that thing, and works as such.

If you try to apply this rule to an expression with triple angle, it will return `nothing` – this is the way a rule signifies failure to match.

``r1(sin(3z)) === nothing``
``````UndefVarError: r1 not defined
``````

Slot variable (matcher) is not necessary a single variable

``r1(sin(2*(w-z)))``
``````UndefVarError: r1 not defined
``````

but it must be a single expression

``r1(sin(2*(w+z)*(α+β))) === nothing``
``````UndefVarError: r1 not defined
``````

Rules are of course not limited to single slot variable

``````r2 = @rule sin(~x + ~y) => sin(~x)*cos(~y) + cos(~x)*sin(~y);

r2(sin(α+β))``````
``````UndefVarError: PatTerm not defined
``````

If you want to match a variable number of subexpressions at once, you will need a segment variable. `~~xs` in the following example is a segment variable:

``````@syms x y z
@rule(+(~~xs) => ~~xs)(x + y + z)``````
``````UndefVarError: PatTerm not defined
``````

`~~xs` is a vector of subexpressions matched. You can use it to construct something more useful:

``````r3 = @rule ~x * +(~~ys) => sum(map(y-> ~x * y, ~~ys));

r3(2 * (w+w+α+β))``````
``````UndefVarError: PatTerm not defined
``````

Notice that the expression was autosimplified before application of the rule.

``2 * (w+w+α+β)``
``4w + 2α + 2β``

### Predicates for matching

Matcher pattern may contain slot variables with attached predicates, written as `~x::f` where `f` is a function that takes a matched expression and returns a boolean value. Such a slot will be considered a match only if `f` returns true.

Similarly `~~x::g` is a way of attaching a predicate `g` to a segment variable. In the case of segment variables `g` gets a vector of 0 or more expressions and must return a boolean value. If the same slot or segment variable appears twice in the matcher pattern, then at most one of the occurance should have a predicate.

For example,

``````@syms a b c d

r = @rule ~x + ~~y::(ys->iseven(length(ys))) => "odd terms";

@show r(a + b + c + d)
@show r(b + c + d)
@show r(b + c + b)
@show r(a + b)``````
``````UndefVarError: PatSegment not defined
``````

### Associative-Commutative Rules

Given an expression `f(x, f(y, z, u), v, w)`, a `f` is said to be associative if the expression is equivalent to `f(x, y, z, u, v, w)` and commutative if the order of arguments does not matter. SymbolicUtils has a special `@acrule` macro meant for rules on functions which are associate and commutative such as addition and multiplication of real and complex numbers.

``````@syms x y z

acr = @acrule((~a)^(~x) * (~a)^(~y) => (~a)^(~x + ~y))

acr(x^y * x^z)``````
``````UndefVarError: PatTerm not defined
``````

although in case of `Number` it also works the same way with regular `@rule` since autosimplification orders and applies associativity and commutativity to the expression.

### Example of applying the rules to simplify expression

Consider expression `(cos(x) + sin(x))^2` that we would like simplify by applying some trigonometric rules. First, we need rule to expand square of `cos(x) + sin(x)`. First we try the simplest rule to expand square of the sum and try it on simple expression

``````using SymbolicUtils

@syms x::Real y::Real

sqexpand = @rule (~x + ~y)^2 => (~x)^2 + (~y)^2 + 2 * ~x * ~y

sqexpand((cos(x) + sin(x))^2)``````
``````UndefVarError: PatTerm not defined
``````

It works. This can be further simplified using Pythagorean identity and check it

``````pyid = @rule sin(~x)^2 + cos(~x)^2 => 1

pyid(cos(x)^2 + sin(x)^2) === nothing``````
``````UndefVarError: PatTerm not defined
``````

Why does it return `nothing`? If we look at the rule, we see that the order of `sin(x)` and `cos(x)` is different. Therefore, in order to work, the rule needs to be associative-commutative.

``````acpyid = @acrule sin(~x)^2 + cos(~x)^2 => 1

acpyid(cos(x)^2 + sin(x)^2 + 2cos(x)*sin(x))``````
``````UndefVarError: PatTerm not defined
``````

It has been some work. Fortunately rules may be chained together into more sophisticated rewirters to avoid manual application of the rules.

## Composing rewriters

A rewriter is any callable object which takes an expression and returns an expression or `nothing`. If `nothing` is returned that means there was no changes applicable to the input expression. The Rules we created above are rewriters.

The `SymbolicUtils.Rewriters` module contains some types which create and transform rewriters.

• `Empty()` is a rewriter which always returns `nothing`

• `Chain(itr)` chain an iterator of rewriters into a single rewriter which applies each chained rewriter in the given order. If a rewriter returns `nothing` this is treated as a no-change.

• `RestartedChain(itr)` like `Chain(itr)` but restarts from the first rewriter once on the first successful application of one of the chained rewriters.

• `IfElse(cond, rw1, rw2)` runs the `cond` function on the input, applies `rw1` if cond returns true, `rw2` if it retuns false

• `If(cond, rw)` is the same as `IfElse(cond, rw, Empty())`

• `Prewalk(rw; threaded=false, thread_cutoff=100)` returns a rewriter which does a pre-order (from top to bottom and from left to right) traversal of a given expression and applies the rewriter `rw`. `threaded=true` will use multi threading for traversal. Note that if `rw` returns `nothing` when a match is not found, then `Prewalk(rw)` will also return nothing unless a match is found at every level of the walk. If you are applying multiple rules, then `Chain` already has the appropriate passthrough behavior. If you only want to apply one rule, then consider using `PassThrough`. `thread_cutoff` is the minimum number of nodes in a subtree which should be walked in a threaded spawn.

• `Postwalk(rw; threaded=false, thread_cutoff=100)` similarly does post-order (from left to right and from bottom to top) traversal.

• `Fixpoint(rw)` returns a rewriter which applies `rw` repeatedly until there are no changes to be made.

• `PassThrough(rw)` returns a rewriter which if `rw(x)` returns `nothing` will instead return `x` otherwise will return `rw(x)`.

### Chaining rewriters

Several rules may be chained to give chain of rules. Chain is an array of rules which are subsequently applied to the expression.

To check that, we will combine rules from previous example into a chain

``````using SymbolicUtils
using SymbolicUtils.Rewriters

sqexpand = @rule (~x + ~y)^2 => (~x)^2 + (~y)^2 + 2 * ~x * ~y
acpyid = @acrule sin(~x)^2 + cos(~x)^2 => 1

csa = Chain([sqexpand, acpyid])

csa((cos(x) + sin(x))^2)``````
``````UndefVarError: PatTerm not defined
``````

Important feature of `Chain` is that it returns the expressiona instead of `nothing` if it doesn't change the expression

``Chain([@acrule sin(~x)^2 + cos(~x)^2 => 1])((cos(x) + sin(x))^2)``
``````UndefVarError: PatTerm not defined
``````

its important to notice, that chain is ordered, so if rules are in different order it wouldn't work the same as in earlier example

``````cas = Chain([acpyid, sqexpand])

cas((cos(x) + sin(x))^2)``````
``````UndefVarError: acpyid not defined
``````

since Pythagorean identity is applied before square expansion, so it is unable to match squares of sine and cosine.

One way to circumvent the problem of order of applying rules in chain is to use `RestartedChain`

``````using SymbolicUtils.Rewriters: RestartedChain

rcas = RestartedChain([acpyid, sqexpand])

rcas((cos(x) + sin(x))^2)``````
``````UndefVarError: acpyid not defined
``````

It restarts the chain after each successful application of a rule, so after `sqexpand` is hit it (re)starts again and successfully applies `acpyid` to resulting expression.

You can also use `Fixpoint` to apply the rules until there are no changes.

``Fixpoint(cas)((cos(x) + sin(x))^2)``
``````UndefVarError: cas not defined
``````