What is pikevm?
==============

re1 (http://code.google.com/p/re1/) is "toy regular expression implementation"
by Russel Cox, featuring simplicity and minimal code size unheard of in other
implementations. re2 (http://code.google.com/p/re2/) is "an efficient,
principled regular expression library" by the same author. It is robust,
full-featured, and ... bloated, comparing to re1.

This is implementation of pikevm based on re1.5 which adds features required for
minimalistic real-world use, while sticking to the minimal code size and
memory use.
https://github.com/pfalcon/re1.5

Why?
====
Pikevm guarantees that any input regex will scale O(n) with the size of the
string, thus making it the fastest regex implementation. There is no backtracking
that usually expodes to O(n^k) time and space where k is some constant.

Features
========

* UnLike re1.5, here is only pikevm, one file easy to use.
* Unlike re1.5, regexes is compiled to type sized code rather than bytecode,
eliviating the problem of byte overflow in splits/jmps on large regexes.
Currently the type used is int, and every atom in compiled code is aligned
to that.
* Matcher does not take size of string as param, it checks for '\0' instead,
so that the user does not need to waste time taking strlen()
* Highly optimized source code, probably 2x faster than re1.5
* Support for quoted chars in regex. Escapes in brackets.
* Support for ^, $ assertions in regex.
* Support for repetition operator {n} and {n,m} and {n,}.
- Note: cases with 0 are not handled, avoid them, they can easily be replaced.
* Support for Unicode (UTF-8).
* Unlike other engines, the output is byte level offset. (Which is more useful)
* Support for non capture group ?:
* Support for wordend & wordbeg assertions
- Some limitations for word assertions are meta chars like spaces being used
in for expression itself, for example "\< abc" should match " abc" exactly at
that space word boundary but it won't. It's possible to fix this, but it would
require rsplit before word assert, and some dirty logic to check that the character
or class is a space we want to match not assert at. But the code for it was too
dirty and I scrapped it. Syntax for word assertions are like posix C library, not
the pcre "\b" which can be used both in front or back of the word, because there is
no distinction, it makes the implementation potentially even uglier.
* Assert flags like REG_ICASE,REG_NOTEOL,REG_NOTBOL and lookahead inside
negated bracket are implemented here (also shows use case in real world project):
https://github.com/kyx0r/nextvi/blob/master/regex.c

NOTES
=====
The problem described in this paper has been fixed. Ambiguous matching is correct.
HISTORY:
https://re2c.org/2019_borsotti_trofimovich_efficient_posix_submatch_extraction_on_nfa.pdf
"Cox, 2009 (incorrect). Cox came up with the idea of backward POSIX matching,
which is based on the observation that reversing the longest-match rule
simplifies the handling of iteration subexpressions: instead of maximizing
submatch from the first to the last iteration, one needs to maximize the
iterations in reverse order. This means that the disambiguation is always
based on the most recent iteration, removing the need to remember all previous
iterations (except for the backwards-first, i.e.  the last one, which contains
submatch result). The algorithm tracks two pairs of offsets per each submatch
group: the active pair (used for disambiguation) and the result pair. It gives
incorrect results under two conditions: (1) ambiguous matches have equal
offsets on some iteration, and (2) disambiguation happens too late, when
the active offsets have already been updated and the difference between
ambiguous matches is erased. We found that such situations may occur for two
reasons. First, the ε-closure algorithm may compare ambiguous paths after
their join point, when both paths have a common suffix with tagged
transitions. This is the case with the Cox prototype implementation; for
example, it gives incorrect results for (aa|a)* and string aaaaa. Most of such
failures can be repaired by exploring states in topological order, but a
topological order does not exist in the presence of ε-loops. The second reason
is bounded repetition: ambiguous paths may not have an intermediate join point
at all. For example, in the case of (aaaa|aaa|a){3,4} and string aaaaaaaaaa we
have matches (aaaa)(aaaa)(a)(a) and (aaaa)(aaa)(aaa) with a different number
of iterations. Assuming that the bounded repetition is unrolled by chaining
three sub-automata for (aaaa|aaa|a) and an optional fourth one, by the time
ambiguous paths meet both have active offsets (0,4). Despite the flaw, Cox
algorithm is interesting: if somehow the delayed comparison problem was fixed,
it would work.  The algorithm requires O(mt) memory and O(nm^2t) time
(assuming a worst-case optimal closure algorithm), where n is the
length of input, m it the size of RE and t is the number of submatch groups
and subexpressions that contain them."

Research has shown that it is possible to disambiguate NFA in polynomial time
but it brings serious performance issues on non ambiguous inputs.  See the
branch "disambiguate_paths" on this repo shows what is being done to solve it
and the potential performance costs. In short it requires tracking the parent
of every state added on nlist from clist.  If the state from nlist matches
the consumer, the alternative clist state related to that nlist state gets
discarded and the nsub ref can be decremented (freed). The reason why this
problem does not exist for non ambiguous regexes is because the alternative
clist state will never match due to the next state having a different consumer
. There is no need for any extra handling it gets freed normally.  I decided
to not apply this solution here because I think most use cases for regex are
not ambiguious like say regex: "a{10000}". If you try matching 10000 'a'
characters in a row like that you will have a problem where the stack usage
will jump up to 10000*(subsize) but it will never exceed the size of regex
though, but the number of NFA states will also increase by the same amount,
so at the charater 9999 you will find 9999 redundant nlist states, that will
degrade performance linearly, however it will be very slow compared to
uplimited regex like a+. The cost of this solution is somewhere around 2%
general performance decrease (broadly), but a magnitude of complexity
decrease for ambiguous cases, for example matching 64 characters went down
from 30 to 9 microseconds.  Another solution to this problem can be to
determine the ambiguous paths at compile time and flag the inner states as
ambiguous ahead of time, still this can't avoid having a loop though the alt
states as their positioning in clist can't be precomputed due to the dynamic
changes.
(Comment about O(mt) memory complexity)
This worst case scenario can only happen on ambiguous input. Ambiguous
consumers (char, class, any) assuming t is 1. In practice there is almost
never a situation where someone wants to search using regex this large. Most
of the time memory usage is very low and the space complexity for non
ambigious regex is O(nt) where n is the number of currently considering
alternate paths in the regex and t is the number of submatch groups.

This pikevm implementation features an improved submatch extraction algorithm
based on Russ Cox's original design.  I - Kyryl Melekhin have found a way to
optimize the tracking properly of 1st number in the submatch pair. Based on
simple observation of how the NFA is constructed I noticed that there is no
way for addthread1() to ever reach inner SAVE instructions in the regex, so
that leaves tracking 2nd pairs by addthread1() irrelevant to the final
results (except the need to initialize the sub after allocation). This
improved the overall performance by 25% which is massive considering that at
the time there was nothing else left to can be done to make it faster.

What are on##list macros?
Redundant state inside nlist can happen in couple of ways, and has to do with 
the (closure) a* (star) operations and also +. Due to the automata machine 
design split happens to be above the next consumed instruction and if that 
state gets added onto the list we may segfault or give wrong submatch result. 
Rsplit does not have this problem because it is generated below the consumer 
instruction, but it can still add redundant states. Overall this is extremely 
difficult to understand or explain, but this is just something we have to 
check for. We checked for this using extra int inside the split instructions, 
so this left some global state inside the machine insts. Most of the time we 
just added to the next gen number and kept incrementing it forever. This 
leaves a small chance of overflowing the int and getting a run on a false 
state left from previous use of the regex. Though if overflow never happens 
there is no chance of getting a false state. Overflows like this pose a high 
security threat, if the hacker knows how many cycles he needs to overflow the 
gen variable and get inconsistent result. It is possible to reset the marks 
if we near the overflow, but as you may guess that does not come for free.

Currently I removed all dynamic global state from the instructions fixing any 
overlow issue utilizing a sparse set datastructure trick which abuses the 
uninitialized varibles. This allows the redundant states to be excluded in
O(1) operation. That said, don't run valgrind on pikevm as it will go crazy, or 
find a way to surpress errors from pikevm.

Further reading
===============
https://research.swtch.com/sparse
https://swtch.com/~rsc/regexp/regexp1.html

Author and License
==================
licensed under BSD license, just as the original re1.