## Antimirov

### Dedication

This project is named after Valentin Antimirov (1961 - 1995). His work on partial derivatives of regular expressions is fundmental to this project.

### Overview

Antimirov is a Scala package for working with regular expressions.

Antimirov defines an `Rx`

type, which supports the following regular expression combinators:

- Unmatchable regular expressions (
`Rx.Phi`

) - Matching the empty string (
`Rx.Empty`

) - Matching single characters (
`Rx.Letter(c)`

) - Matching sets of characters (
`Rx.Letters(cs)`

) - Alternation (
`Rx.Choice(x, y)`

, i.e.`x + y`

or`x | y`

) - Concatenation (
`Rx.Concat(x, y)`

, i.e.`x * y`

) - Kleene Star (
`Rx.Star(x)`

, i.e.`x.star`

) - Repetition (
`Rx.Repeat(r, m, n)`

, i.e.`r.repeat(m, n)`

)

In addition to the previous combinators which are reified as algebraic data types, `Rx`

supports additional operations:

- Exponentiation (
`x.pow(k)`

) - Optionality (
`x.optional`

) - Intersection (
`x & y`

) - Exclusive-or (XOR, i.e.
`x ^ y`

) - Difference (
`x - y`

) - Complement (
`~x`

) - Equality (
`x === y`

) - Partial-ordering (
`x < y`

,`x partialCompare y`

, etc.) - Derivatives (
`x.deriv(c)`

,`x.partialDeriv(c)`

)

These operations are consistent with the corresponding set operations. What this means is that each `Rx`

value has a corresponding set of strings it accepts, and that these operations produce new `Rx`

values whose sets are consistent with the corresponding set operations.

Finaly, `Rx`

values can be compiled down to an automaton for more efficient matching (either a `Dfa`

or `Nfa`

).

### Getting Antimirov

Antimirov supports Scala 2.13 and 2.12. It is avilable from Sonatype.

To include Antimirov in your projects, you can use the following `build.sbt`

snippet:

```
libraryDependencies +=
"org.spire-math" %% "antimirov-core" % "0.2.4"
libraryDependencies += // optional scalacheck support
"org.spire-math" %% "antimirov-check" % "0.2.4"
libraryDependencies += // optional scalaprops support
"org.spire-math" %% "antimirov-props" % "0.2.4"
```

Antimirov also supports Scala.js. To use Antimirov in your Scala.js projects, include the following `build.sbt`

snippet:

```
libraryDependencies +=
"org.spire-math" %%% "antimirov-core" % "0.2.4"
libraryDependencies += // optional scalacheck support
"org.spire-math" %%% "antimirov-check" % "0.2.4"
libraryDependencies += // optional scalaprops support
"org.spire-math" %%% "antimirov-props" % "0.2.4"
```

### Details

Antimirov provides an algebraic interface for building regular expressions, as well as testing them for equality, subset/superset relationships, and more:

```
import antimirov.Rx
val x: Rx = Rx.parse("[1-9][0-9]*")
x.accepts("0") // false
x.accepts("1") // true
x.accepts("19") // true
x.accepts("09") // false
val y: Rx = Rx.parse("[0-9a-f][0-9a-f]")
y.accepts("af") // true
y.accepts("09") // true
y.accepts("099") // false
// set operations
//
// note that the full Char range is:
// ['\u0000', ..., '/', '0', ... '9', ':', ... '\uffff']
val z1: Rx = x | y // [1-9][0-9]*|[0-9a-f][0-9a-f]
val z2: Rx = x & y // [1-9][0-9]
val z3: Rx = x ^ y // 0[0-9a-f]|[1-9][0-9][0-9][0-9]*|[1-9][a-f]|[1-9]|[a-f][0-9a-f]
val z4: Rx = x - y // [1-9][0-9][0-9][0-9]*|[1-9]
val z5: Rx = ~x // [^1-9].*|[1-9][0-9]*[^0-9].*|
// equality, subset, and superset comparisons
val xx: Rx = Rx.parse("[1-4][0-9]*|[5-9][0-9]*")
x == xx // false
x === xx // true
x <= xx // true
x < xx // false
val U: Rx = Rx.parse(".*")
x == U // false
x === U // false
x <= U // true
x < U // true
```

An `antimirov.Rx`

value can be converted to any of:

`antimirov.Nfa`

`antimirov.Dfa`

`java.util.regex.Pattern`

`scala.util.matching.Regex`

.

Note that unlike many modern regex libraries, Antimirov's regular expressions do not contain non-regular features (such as back-references), and are solely focused on matching, not on searching or extraction.

Concretely, this means that:

- Patterns are matched against the entire string
- No subgroup extraction is possible
- Zero-width assertions are unsupported

Technically zero-width assertions are still regular, but they are not easily composable. For example, for `(?=x)`

as a zero-width look-ahead assertion on `x`

, and `(?<=y)`

as a zero-width look-behind assertion on `y`

, the following is true:

`[a-z]*(?=[^b])`

is equivalent to`[a-z]*[^b]`

`(?<=[^c])[a-z]*`

is equivalent to`[^c][a-z]*`

`[a-z]*(?=[^b])(?<=[^c])[a-z]*)`

is*NOT*equivalent to`[a-z]*[^b][^c][a-z]*`

- instead it's equivalent to
`[a-z]*[^bc][a-z]*`

These assertions don't play well with the algebraic structure of `Rx`

which is why they are left out (for now at least).

In exchange for giving up these sorts of affordances, Antimirov can do things that most regular expression libraries can't do, such as intersection, exclusive-or, negation, semantic equality checks, subset comparisons (e.g. inclusion), and more.

### ScalaCheck support

The `antimirov-check`

package support ScalaCheck, adding the ability to generate strings according to a regular expression. There are two ways to use it:

```
package demo
import antimirov.Rx
import antimirov.check.Regex
import org.scalacheck.{Prop, Properties}
object Demo extends Properties("Demo") {
property("Arbitrary-based usage") = {
val r1 = Regex("-?(0|[1-9][0-9]*)")
Prop.forAll { (w: r1.Word) =>
val s: String = w.value
// s is guaranteed to be accepted by r1
???
}
}
property("Gen-based usage") = {
val r2 = Rx.parse("-?(0|[1-9][0-9]*)")
val mygen: Gen[String] = Regex.gen(r2)
Prop.forAll(mygen) { s: String =>
// s is guaranteed to be accepted by r2
???
}
}
}
```

One thing to note here is that `antimirov.Rx`

has no direct dependency on ScalaCheck, which is why we introduce `antimirov.check.Regex`

to gain `Arbitrary`

support.

`Regex`

is wrapper type around `Rx`

that adds the path-dependent type `Word`

as well as implementations of `Gen`

and `Arbitrary`

used by ScalaCheck. `Regex`

does not support the full suit of Antimirov operations (such as `&`

) and is really just meant for use with ScalaCheck. (For other usages, prefer its embedded `Rx`

value).

### ScalaProps support

Antimirov also supports ScalaProps via `antimirov-props`

. This package is very similar to `antimirov-check`

:

```
package demo
import antimirov.Rx
import antimirov.props.Regex
import scalaprops.{Gen, Property, Scalaprops}
object Demo extends Scalaprops {
val implicitStyle = {
val r1 = Regex("-?(0|[1-9][0-9]*)")
Property.forAll { (w: r1.Word) =>
val s: String = w.value
// s is guaranteed to be accepted by r1
???
}
}
val explicitStyle = {
val r2 = Rx.parse("-?(0|[1-9][0-9]*)")
val mygen: Gen[String] = Regex.gen(r2)
Property.forAllG(mygen) { s =>
// s is guaranteed to be accepted by r2
???
}
}
}
```

### Web/JS tool

Antimirov has an interactive JS/HTML tool for working with regular expressions.

You can try it out at http://phylactery.org/antimirov/.

To visit this page locally, simply run `sbt web/fullOptJS`

and then visit `web/index.html`

in your web browser.

### Known Issues

The biggest issue with this library is that the problems are exponential in the general case. This means there are plenty of expressions for which Antimirov's operations (equality, inclusion, intersection, and so on) are prohibitively slow.

There are some good strategies for dealing with this complexity through heuristics and optimizations. But some constructions (such as very wide alternations contained within a Kleene star) will probably never perform very well.

Here's a list of other known problems:

- Antimirov doesn't preserve user-specified expression syntax
- Antimirov cannot (yet) check constant expressions at compile-time
- Synthetic operators (e.g.
`+`

and`?`

) are not reified in the AST - There could be more work on simplification/canonicalization

### Future Work

Since the general problem is exponential, there is likely a lot of future work around chipping away at the margins: heuristics that cover most interesting regular expressions users are interested in.

We could add support for other kinds of generators and test frameworks (e.g. Hedgehog, Scalaprops, etc.)

We could add predefined character classes (e.g. `\w`

from PCRE or `[:digit:]`

from POSIX). It's fairly straightforward to add support for reading these, but a bit tricker figuring out when to emit them.

There is room for improvement of the automata implementations (particularly `antimirov.NFA`

). Relatedly, we could add support for searching instead of just matching.

There is major room for improvement for the HTML/JS tool (the current version was optimized for what was easy for the author to produce).

Additionally, we could provide an interactive/command-line tool to help work with regular expressions (potentially using Graal or Scala Native to produce a native executable).

It would be interesting to see how many of Antimirov's features we would have to give up to add support for extracting subgroup matches.

### Copyright and License

All code is available to you under the Apache 2 license, available at https://opensource.org/licenses/Apache-2.0.

Copyright Erik Osheim, 2020.