*TL;DR: We add variables, let bindings, and explicit recursion via fixed points to classic regular expressions. It turns out that the resulting explicitly recursive, finitely described languages are well suited for analysis and introspection.*

It’s been almost a year since I touched the `kleene`

library, and almost two years since I published it – a good time to write a little about regular expressions.

I like regular expressions very much. They are truly declarative way to write down a grammar… as long as the grammar is expressible in regular expressions.

Matthew Might, David Darais and Daniel Spiewak have written a paper *Functional Pearl: Parsing with Derivatives* published in ICFP ’11 Proceedings [Might2011] in which regular expressions are extended to handle context-free languages. However, they rely on *memoization*, and – as structures are infinite – also on reference equality. In short, their approach is not implementable in idiomatic Haskell.^{1}

There’s another technique that works for a subset of context-free languages. In my opinion, it is very elegant, and it is at least not painfully slow. The result is available on Hackage: the `rere`

library. The idea is to treat regular expressions as a proper programming language, and add a constructions which proper languages should have: *variables* and *recursion*.

This blog post will describe the approach taken by `rere`

in more detail.

### Regular expression recap

The abstract syntax of a regular expression (over the alphabet of unicode characters) is given by the following “constructors”:

- Null regexp:
- Empty string:
- Characters: , etc
- Concatenation:
- Alternation:
- Kleene star:

The above can be translated directly into Haskell:

```
data RE
= Empty
| Eps
| Ch Char
| App RE RE
| Alt RE RE
| Star RE
```

In the `rere`

implementation, instead of bare `Char`

we use a set of characters, `CharSet`

, as recommended by Owens et al. in *Regular-expression derivatives reexamined* [Owens2009]. This makes the implementation more efficient, as a common case of character sets is explicitly taken into account. We write them in curly braces:
.

We can give *declarative* semantics to these constructors. These will look like typing rules. A judgment
denotes that the regular expression
successfully recognises the string
.

For example, the rule for application now looks like:

This rule states that if recognises , and recognises , then the concatenation expression recognises the concatenated string .

For alternation we have two rules, one for each of the alternatives:

The rules resemble the structure of *non-commutative intuitionistic linear logic*, if you are into such stuff. Not only do you have to use everything exactly once; you have to use everything in order, there aren’t any substructural rules, no weakening, no contraction and even no exchange. I will omit the rest of the rules, look them up (and think how rules for Kleene star would look like ‘why not’ exponential `?`

).

It’s a good idea to define smart versions of the constructors, which simplify regular expressions as they are created. For example, in the following `Semigroup`

instance for concatenation, `<>`

is a smart version of `App`

:

```
instance Semigroup RE where
-- Empty annihilates
Empty <> _ = Empty
<> Empty = Empty
_ -- Eps is unit of <>
Eps <> r = r
<> Ep s = r
r -- otherwise use App
<> s = App r s r
```

The smart version of `Alt`

is called `\/`

, and the smart version of `Star`

is called `star`

.

We can check that the simplifications performed by the smart constructors are sound, by using the semantic rules. For example, the simplification `Eps <> r = r`

is justified by the following equivalence of derivation trees:

If string
is matched by
, then “the match” can be constructor only in one way, by applying the
rule. Therefore
is also matched by bare
. If we introduced *proof terms*, we’d have a concrete evidence of the match as terms in this language.

There is, however, a problem: matching using declarative rules is not practical. At several points in these rules, we have to guess. We have to guess whether we should pick left or right branch, or where we should split string to match concatenated regular expression. For a practical implementation, we need a *syntax-directed* approach. Interestingly, we then need just two rules:

In the above rules, we use two operations: The decision procedure “nullable” that tells whether a regular expression can recognise the empty string, and a mapping
that, given a single character
and a regular expression
computes a new regular expression called the *derivative* of
with respect to
.

Both operations are quite easy to map to Haskell. The function `nullable`

is defined as a straight-forward recursive function:

```
nullable :: RE -> Bool
Empty = False
nullable Eps = True
nullable Ch _) = False
nullable (App r s) = nullable r && nullable s
nullable (Alt r s) = nullable r || nullable s
nullable (Star _) = True nullable (
```

The *Brzozowski derivative* is best understood by considering the *formal language*
regular expressions represent:

In Haskell terms: `derivative c r`

matches string `str`

if and only if `r`

matches `c : str`

. From this equivalence, we can more or less directly infer an implementation:

```
derivative :: Char -> RE -> RE
Empty = Empty
derivative _ Eps = Empty
derivative _ Ch x)
derivative _ (| c == x = Eps
| otherwise = Empty
App r s)
derivative c (| nullable r = derivative c s \/ derivative c r <> s
| otherwise = derivative c r <> s
Alt r s) = derivative c r \/ derivative c s
derivative c (@(Star r) = derivative c r <> r0 derivative c r0
```

We could try to show that the *declarative* and *syntax directed* systems are equivalent, but I omit it here, because it’s been done often enough in the literature (though probably not in exactly this way and notation).

We can now watch how a regular expression “evolves” while matching a string. For example, if we take the regular expression , which in code looks like

```
ex1 :: RE
= star (Ch 'a' <> Ch 'b') ex1
```

then the following is how `match ex1 "abab"`

proceeds:

We can see that there’s implicitly a small finite state automaton, with two states: an initial state
and secondary state
. This is the approach taken by the `kleene`

package to transform regular expressions into finite state machines. There is an additional *character set* optimization from *Regular-expression derivatives re-examined* [Owens2009] by Owens, Reppy and Turon, but in essence, the approach works as follows: Try all possible derivatives, and in the process collect all the states and construct a transition function.^{2}

The string is accepted as the matching process stops at the state, which is nullable.

### Variables and let-expressions

The first new construct we now add to regular-expressions are *let expressions*. They alone do not add any matching power, but they are prerequisite for allowing recursive expressions.

We already used meta-variables in the rules in the previous section. Let expressions allow us to internalise this notion. The declarative rule for let expressions is:

Here, the notation denotes substituting the variable by the regular expression in the regular expression .

To have let expressions in our implementation, we need to represent *variables* and we must be able to perform substitution. My tool of choice for handling variables and substitution in general is the `bound`

library. But for the sake of keeping the blog post self-contained, we’ll define the needed bits inline. We’ll reproduce a simple variant of `bound`

, which amounts to using de Bruijn indices and polymorphic recursion.

We define our own datatype to represent variables (which is isomorphic to `Maybe`

):

```
data Var a
= B -- ^ bound
| F a -- ^ free
deriving (Eq, Show, Functor, Foldable, Traversable)
```

With this, we can extend regular expressions with let. First we make it a functor, i.e., change `RE`

to `RE a`

, and then also add two new constructors: `Var`

and `Let`

:

```
data RE a
= Empty
| Eps
| Ch Char
| App (RE a) (RE a)
| Alt (RE a) (RE a)
| Star (RE a)
| Var a
| Let (RE a) (RE (Var a))
```

Note that we keep the argument `a`

unchanged in all recursive occurrences of `RE`

, with the exception of the body of the `Let`

, where use use `Var a`

instead, indicating that we can use `B`

in that body to refer to the variable bound by the `Let`

.

In the actual `rere`

library, the `Let`

(and later `Fix`

) constructors additionally have an irrelevant `Name`

field, which allows us to retain the variable names and use them for pretty-printing. I omit them from the presentation in this blog post.

Now, we can write a regular expression with repetitions like instead of ; or in Haskell:

```
ex2 :: RE Void
= Let (star (Ch 'a')) (Var B <> Var B) ex2
```

The use of `Void`

as parameter tells us that expression is *closed*, i.e., doesn’t contain any free variables.

We still need to extend `nullable`

and `derivative`

to work with the new constructors. For `nullable`

, we’ll simply pass a function telling whether variables in context are nullable. The existing constructors just pass a context around:

```
nullable :: RE Void -> Bool
= nullable' absurd
nullable
nullable' :: (a -> Bool) -> RE a -> Bool
Empty = False
nullable' _ Eps = True
nullable' _ Ch _) = False
nullable' _ (App r s) = nullable' f r && nullable' f s
nullable' f (Alt r s) = nullable' f r || nullable' f s
nullable' f (Star _) = True nullable' _ (
```

The cases for `Var`

and `Let`

use and extend the context, respectively:

```
-- Var: look in the context
Var a) = f a
nullable' f (
-- Let: - compute `nullable r`
-- - extend the context
-- - continue with `s`
Let r s) = nullable' (unvar (nullable' f r) f) s nullable' f (
```

The `unvar`

function corresponds to `maybe`

, but transported to our `Var`

type:

```
unvar :: r -> (a -> r) -> Var a -> r
B = b
unvar b _ F x) = f x unvar _ f (
```

How to extend `derivative`

to cover the new cases requires a bit more thinking. The idea is similar: we want to add to the context whatever we need to know about the variables. The key insight is to replace every `Let`

binding by two `Let`

bindings, one copying the original, and one binding to the `derivative`

of the let-bound variable. Because the number of let bindings changes, we have to carefully re-index variables as we go.

Therefore, the context for `derivative`

consists of three pieces of information per variable:

- whether the variable is
`nullable`

(we need it for`derivative`

of`App`

), - the variable denoting the derivative of the original variable,
- the re-indexed variable denoting the original value.

The top-level function `derivative :: Char -> RE Void -> RE Void`

now makes use of a local helper function

`derivative' :: (Eq a, Eq b) => (a -> (Bool, b, b)) -> RE a -> RE b`

which takes this context. Note that, as discussed above, `derivative'`

changes the indices of the variables. However, at the top-level, both `a`

and `b`

are `Void`

, and the environment can be trivially instantiated to the function with empty domain.

The `derivative'`

case for `Var`

is simple: we just look up the derivative of the `Var`

in the context.

`Var a) = Var (sndOf3 (f a)) derivative' f (`

The case for `Let`

is quite interesting:

```
Let r s)
derivative' f (= let_ (fmap (trdOf3 . f) r) -- rename variables in r
$ let_ (fmap F (derivative' f r)) -- binding for derivative of r
$ derivative' (\case
B -> (nullable' (fstOf3 . f) r, B, F B)
F x -> bimap (F . F) (F . F) (f x))
$ s
...
```

As a formula it looks like:

For our running example
or `Let (star (Ch 'a')) (Var B <> Var B)`

, we call `derivative'`

recursively with an argument of type `RE (Var a)`

, corresponding to the one variable
, and we get back a `RE (Var (Var b))`

, corresponding to the two variables
and
.

The careful reader will also have noticed the smart constructor `let_`

, which does a number of standard rewritings on the fly (which I explain in a *Do you have a problem? Write a compiler!* talk). These are justified by the properties of substitution:

```
-- let-from-let
let x = (let y = a in b) in c
-- ==>
let y = a; x = b in c
```

```
-- inlining of cheap bindings
let x = a in b
-- ==>
-> a ] -- when a is cheap, i.e. Empty, Eps, Ch or Var b [ x
```

```
-- used once, special case
let x = a in x
-- ==>
a
```

```
-- unused binding
let x = a in b
-- ==>
-- when x is unused in b b
```

And importantly, we employ a quick form common-subexpression-elimination (CSE):

```
let x = a in f x a
-- ==>
let x = a in f x x
```

This form of CSE is easy and fast to implement, as we don’t introduce new `let`

s, only consider what we already bound and try to increase sharing.

It’s time for examples: Recall again `ex2`

which was defined as
or

```
ex2 :: RE Void
= Let "r" (star (Ch 'a')) (Var B <> Var B) ex2
```

Let’s try to observe the match of the string step by step:

As our smart constructors are quite smart, the automaton stays in its single state, the union comes from the `derivative`

of `App`

, as `r`

is nullable, we get `derivative 'a' r \/ derivative 'a' r <> r`

. And as `derivative 'a' r = r`

, we don’t see any additional `let`

bindings.

### Recursion

Now we are ready for the main topic of the post: *recursion*. We add one more constructor to our datatype of regular expressions:

```
data RE a
...
| Fix (RE (Var a))
```

The `Fix`

construct looks similar to `Let`

, except that the bound variable is semantically equivalent to the whole expression. We can *unroll* each
expression by substituting it into itself:

The `Fix`

constructor subsumes the Kleene star, as
can now be expressed as
, which feels like a very natural definition indeed. For example `ex1`

previously defined using Kleene star as
could also be re-defined as
. That looks like

```
ex3 :: RE Void
= Fix "x" (Eps \/ Ch 'a' <> Ch 'b' <> Var B) ex3
```

in code.

The problem is now the same as with `Let`

: How to define `nullable`

and `derivative`

? Fortunately, we have most of the required machinery already in place from the addition of `Var`

and `Let`

.

Nullability of `Fix`

relies on Kleene’s theorem to compute the least fixed point of a monotonic recursive definition, like in *Parsing with Derivatives*. The idea is to unroll `Fix`

once, and to pretend that the nullability of the recursive occurrence of the bound variable in `Fix`

is `False`

:

```
nullable' :: (a -> Bool) -> RE a -> Bool
...
Fix _ r) = nullable' (unvar False f) r nullable' f (
```

In other words, we literally assume that the nullability of new binding is `False`

, and see what comes out. We don’t need to iterate more then once, as `False`

will flip to `True`

right away, or will never do so even with further unrollings.

Following a similar idea, our smart constructor `fix_`

is capable of recognising a `Empty`

fixed point by substituting `Empty`

for the recursive occurrence in the unrolling:

```
fix_ :: RE (Var a) -> RE a
| (r >>>= unvar Empty Var) == Empty = Empty
fix_ r ...
```

This works because `Empty`

is a bottom of the language-inclusion lattice (just as `False`

is a bottom of the `Bool`

lattice).

The extension of `derivative`

is again a bit more involved, but it resembles what we did for `Let`

: As the body
of a
contains self references
, the derivative of a
will also be a
. Thus, when we need to compute the derivative of
, we’ll use
. It is important that not all occurrences of
in the body of a
will turn into references to its derivative (e.g., if they appear to the right of an `App`

, or in a `Star`

), so we *need to save the value of
in a let binding* – how fortunate that we just introduced those … Schematically, the transformation looks as follows:

In the rest, we will use a shorthand notation for a let binding to a
, as in
. We will write such a binding more succinctly as
with the
subscript indicating that the binding is recursive. We prefer this notation over introducing
, because in a cascade of
expressions, we can have individual bindings being recursive, but we still *cannot* forward-reference to later bindings.

Applying the abbreviation to our derivation rule above yields

Let’s compare this to the let case, rearranged slightly, to establish the similarity:

Consequently, the implementation in Haskell also looks similar to the `Let`

case:

```
@(Fix r)
derivative' f r0= let_ (fmap (trdOf3 . f) r0)
$ fix_
$ derivative' (\case
B -> (nullable' (fstOf3 . f) r0, B, F B)
F x -> bimap (F . F) (F . F) (f x))
$ r
```

Let’s see how it works in practice. We observe the step-by-step matching of `ex3`

on `abab`

, which was `ex1`

defined using a fixed point rather than the Kleene star:

We couldn’t wish for a better outcome. We see the same two-state ping-pong behavior as we got using the Kleene star.

### More examples

The
/ `Fix`

is a much more powerful construction than the Kleene star. Let’s look at some examples …

#### a^{n}b^{n}

Probably the simplest non-regular language is some amount of
s followed by the *same* amount of
s:

We can describe that language using our library, thanks to the presence of fixed points: (note the variable in between the literal symbols). Transcribed to Haskell code, this is:

```
ex4 :: RE Void
= Fix (Eps \/ Ch 'a' <> Var B <> Ch 'b') ex4
```

And we can test the expression on a string in the language, for example `"aaaabbbb"`

:

Now things become more interesting. We can see how in the trace of this not-so-regular expression, we obtain let bindings resembling the stack of a pushdown automaton.

From the trace one can relatively easily see that if we “forget” one `b`

at the end of the input string, then the “state” `b`

isn’t `nullable`

, so the string won’t be recognized.

#### Left recursion

Previously in this post, we have rewritten as . But another option is to use recursion on the left, i.e., to write instead:

```
ex5 :: RE Void
= Fix "x" (Eps \/ Var B <> Ch 'a' <> Ch 'b') ex5
```

This automaton works as well. In fact, in some sense it works better than the right-recursive one: we can see (as an artifact of variable naming), that we get the derivatives as output of each step. We do save the original expression in a , but as it is unused in the result, our smart constructors will drop it:

#### Arithmetic expressions

Another go-to example of context free grammars is arithmetic expressions:

The Haskell version is slightly more inconvenient to write due to the use of de Bruijn indices, but otherwise straight-forward:

```
ex6 :: RE Void
= let_ (Ch "0123456789")
ex6 $ let_ (Var B <> star_ (Var B))
$ fix_
$ ch_ '(' <> Var B <> ch_ ')'
/ Var (F B)
\/ Var B <> ch_ '+' <> Var B
\/ Var B <> ch_ '*' <> Var B \
```

Here is an (abbreviated) trace of matching the input string :