As the sibling commenter said, indeed, the regex crate does not use derivatives. If you wouldn't mind, could you share what led you to that conclusion? I'd love to fix it!
I am at least currently not aware of any "production" and "general purpose" regex engine that is built on derivatives. And I'm not really sure how you'd build one. The biggest hurdle you'd have to over come as far as I can tell is that derivatives are usually used to build a DFA. In this case, the OP does matching while taking the derivative simultaneously. My guess is that you'll run into problems doing that which huge character classes, which are easy to get when Unicode is enabled.
Whether "production" and "general purpose" are the same as "practical" is unclear. To put away the vague words, my understanding is that with derivatives, you'll either get slow match times or slow compilation times. (To the point where "slow" becomes enough to notice and be a problem.)
With that said, the world is full of experts saying you can't do something. What I'm trying to say here is that there are some challenges I've faced in the course of building a regex engine that I simply don't know how I'd address with derivatives.
Another thing worth considering here is the match semantics of the regex. I haven't had time yet to try this particular matcher, but when I do, I'd check for how alternations are matched. For example, what does 'samwise|sam' match in the haystack 'samwise'? Either answer is correct, but one is typically found in POSIX engines and the other found in Perl-like engines. Can derivatives implement either?
It's also worth noting that I am not an expert on regex derivatives. I've never actually build a derivative oriented matcher. If I had, I'm sure I could be a lot more specific with my criticisms. :-)
We built the new engine behind .NET's RegexOptions.NonBacktracking with derivatives. We will have a paper at PLDI this year on the work that went into that.
PCRE semantics was indeed the big thing that required new techniques. Basically, you have to modify the derivation function such that the derivatives model what paths through the pattern a backtracking engine would consider before stopping at the current position.
The big thing derivatives buys you is the ability to apply rewrite rules lazily during the construction. For example, when we can prove that a regex R subsumes a regex T, then an alternation R|T can be rewritten to just R, since T is already included in it. These kinds of rewrites often result in the DFA that gets constructed being minimal or close to so. Of course, you do pay somewhat for the machinery to do this, so best-case construction times suffer compared to in traditional NFA+lazy-DFA engines like RE2 or Rust's, but a larger class of patterns get to stay in the DFA world with derivatives.
I hope our work ignites a new interest in regex matching with derivatives. I believe the ability to apply these syntactic rewrites on-the-fly is really powerful and I'd love to see how far folks like you with extensive experience in optimizing regex matchers can take this.
Wow, that's really cool. I can't wait to read the paper. Have y'all written anything else about it?
> For example, when we can prove that a regex R subsumes a regex T, then an alternation R|T can be rewritten to just R, since T is already included in it.
This doesn't compose though, right? For example, if you have `sam|samwise`, then you can do that transformation, but if you have `\b(?:sam|samwise)\b` then you can't.
> but a larger class of patterns get to stay in the DFA world with derivatives.
We have an early tool paper [1] for a previous version of the engine, but that's short and with POSIX semantics, so doesn't include a lot of the interesting stuff. The most relevant bit there is the handling of Unicode.
>This doesn't compose though, right? For example, if you have `sam|samwise`, then you can do that transformation, but if you have `\b(?:sam|samwise)\b` then you can't.
You'd get subsumption when you have something like '(?:sam)?wise|wise' and in fact this kind of subsumption due to a "nullable prefix" is the main one currently detected (because we encountered patterns that motivated it). And you're right that all these rewrites should compose regardless of context so that they can be eagerly applied in the AST constructors.
>> but a larger class of patterns get to stay in the DFA world with derivatives.
>Can you say more about this?
Yeah, the easiest example I can point at is from that tool paper [1], where a subsumption-based rewrite for counted repetitions can turn an exponential blow-up into a linear one. Off the top of my head I think a pattern like '.*a[ab]{0,1000}' would have a 2^1000 blow-up when determinized into a DFA but stays linear in size with derivatives. However, that loop subsumption rule isn't valid as-is under PCRE semantics, so it still needs some work to be ported to the .NET engine.
Before we get that PLDI paper out the best resource is probably just the code under [2]. It's fairly well commented, but of course that's no substitute for a good write-up.
ocaml-re[1] uses a derivative-style construction to lazily build a DFA. The general idea is to use something similar to Owens et al's DFA construction, but doing it inline with some caching, to compile lazily (and building a thomson-like automaton, for group capture). In practice, it is fairly fast, although not as optimized as the Rust crate. :)
Derivatives supports several match semantics very easily (ocaml-re does longest, shortest, greedy, non-greedy, first). It indeed doesn't handle unicode matching though (it's possible, there is a prototype implem, but nobody took the time to push it through).
Note that it's not difficult to (lazily or not) build a NFA using derivatives as well (with Antimirov's construction).
Oh nice! Unicode is definitely something that's on my mind when thinking about derivatives and how to deal with them, but it sounds like ocaml-re is doing pretty well outside of Unicode. I would love to hook it up to my benchmark harness. (It isn't public yet... Hopefully soon. But it supports regexes in any languages. So far I have Rust, C, C++, Python and Go. I hope to add .NET, Perl and Node at least. But this might be a cool addition too.)
If anyone wants to add this Ocaml engine to the harness (or any other engine), please email me at jamslam@gmail.com and I'll give access to the repo. The only reason it isn't public yet is because I'm still working on the initial release and iterating. But it's close enough where other people could submit benchmark programs for other regex engines.
I don't think you should be worried about Unicode in particular. Although the derivation formula on paper is parameterized by a character, you don't have to compute the derivative for every character separately.
It's actually easy to compute classes of characters that have the same derivative (it's done in the linked "Regular-expression derivative re-examined" paper, although their particular implementation favors simplicity over efficiency), and it's not even necessary when using Antimirov's partial derivatives.
Actually, the complexity of the derivation is independent of the size of the alphabet. You could even define derivation on an arbitrary semi-lattice, not necessarily a set of characters. (Or a boolean algebra if you care about negation/complementation).
The difficulty in handling unicode has more to do with the efficiency of the automaton representation and manipulation rather than in turning the RE in an NFA or DFA.
Does there exist a regex engine I can try that uses derivatives and supports large Unicode classes and purports to be usable for others? :-)
It has been a long time since I read the "Regular-expression derivative re-examined" derivative paper. Mostly the only thing I remember at this point is that I came away thinking that it would be difficult to adapt in practice for large Unicode classes. But I don't remember the details.
It is honestly very difficult for me to translate your comment here into an actionable implementation strategy. But that's probably just my inexperience with derivatives talking.
> Does there exist a regex engine I can try that uses derivatives and supports large Unicode classes and purports to be usable for others? :-)
I don't know any besides ocaml-re that Drup already linked, sorry :).
And sorry that my comment is hard to decipher. I think the core point is that the "character set" can be an abstract type from the point of view of the derivation algorithm.
So it doesn't matter how they are represented, nor "how big" a character set is.
With Antimirov's derivative (which produces an NFA), there is no constraint on this type.
With Brzozowski's derivative, you need at least the ability to intersect two character sets. So the type should implement a trait with an intersection function (in Rust syntax, `trait Intersect fn intersect(self, Self) -> Self`). That's necessary for any implementation generating a DFA anyway.
And if you also want to deal with complementation, then a second method `fn negate(self) -> Self` is necessary.
Thanks! You might be right. I'm probably at a point where I'd have to actually go out and try it to understand it better.
I do wonder if there is some room for derivatives in a meta regex engine (like RE2 or the regex crate). For example, if it let you build a DFA more quickly (in practice, not necessarily in theory), then you might be able to use it for a big subset of cases. It's tricky to make that case over the lazy DFA, however, a full DFA has more optimization opportunities. For example, identifying states with very few outgoing transitions and "accelerating" them by running memchr (or memchr2 or memchr3) on those outgoing transitions instead of continuing to walk the automaton. It's really hard to do that with a lazy DFA because you don't really compute entire states up front.
I think what you suggest is possible, derivation might even be well suited for this application, however I can't tell if it would be better than existing approaches.
There are some chances that it might be interesting in practice, since it seems that this application of derivatives has not been much studied, but that's highly speculative.
Having a good quality and curated regex benchmarks would be quite useful! I hope you plan on having several features, and engines that can only have partial support. That would make for very interesting comparisons.
It does. And more. The only thing you have to do is provide a short program that parses the description of the benchmark on stdin, and then output a list of samples that consist of the time it took to run a single iteration and the "result" of the benchmark for verification. The harness takes over from there. There's no need to have any Unicode support at all. I even have a program for benchmarking `memmem`, which is of course not a regex engine at all.
I am at least currently not aware of any "production" and "general purpose" regex engine that is built on derivatives. And I'm not really sure how you'd build one. The biggest hurdle you'd have to over come as far as I can tell is that derivatives are usually used to build a DFA. In this case, the OP does matching while taking the derivative simultaneously. My guess is that you'll run into problems doing that which huge character classes, which are easy to get when Unicode is enabled.
Whether "production" and "general purpose" are the same as "practical" is unclear. To put away the vague words, my understanding is that with derivatives, you'll either get slow match times or slow compilation times. (To the point where "slow" becomes enough to notice and be a problem.)
With that said, the world is full of experts saying you can't do something. What I'm trying to say here is that there are some challenges I've faced in the course of building a regex engine that I simply don't know how I'd address with derivatives.
Another thing worth considering here is the match semantics of the regex. I haven't had time yet to try this particular matcher, but when I do, I'd check for how alternations are matched. For example, what does 'samwise|sam' match in the haystack 'samwise'? Either answer is correct, but one is typically found in POSIX engines and the other found in Perl-like engines. Can derivatives implement either?
It's also worth noting that I am not an expert on regex derivatives. I've never actually build a derivative oriented matcher. If I had, I'm sure I could be a lot more specific with my criticisms. :-)