Implementation notes about Henry Spencer's regex library
========================================================

If Henry ever had any internals documentation, he didn't publish it.
So this file is an attempt to reverse-engineer some docs.

General source-file layout
--------------------------

There are six separately-compilable source files, five of which expose
exactly one exported function apiece:
	regcomp.c: pg_regcomp
	regexec.c: pg_regexec
	regerror.c: pg_regerror
	regfree.c: pg_regfree
	regprefix.c: pg_regprefix
(The pg_ prefixes were added by the Postgres project to distinguish this
library version from any similar one that might be present on a particular
system.  They'd need to be removed or replaced in any standalone version
of the library.)

The sixth file, regexport.c, exposes multiple functions that allow extraction
of info about a compiled regex (see regexport.h).

There are additional source files regc_*.c that are #include'd in regcomp,
and similarly additional source files rege_*.c that are #include'd in
regexec.  This was done to avoid exposing internal symbols globally;
all functions not meant to be part of the library API are static.

(Actually the above is a lie in one respect: there are two more global
symbols, pg_set_regex_collation and pg_reg_getcolor in regcomp.  These are
not meant to be part of the API, but they have to be global because both
regcomp and regexec call them.  It'd be better to get rid of
pg_set_regex_collation, as well as the static variables it sets, in favor of
keeping the needed locale state in the regex structs.  We have not done this
yet for lack of a design for how to add application-specific state to the
structs.)

What's where in src/backend/regex/:

regcomp.c		Top-level regex compilation code
regc_color.c		Color map management
regc_cvec.c		Character vector (cvec) management
regc_lex.c		Lexer
regc_nfa.c		NFA handling
regc_locale.c		Application-specific locale code from Tcl project
regc_pg_locale.c	Postgres-added application-specific locale code
regexec.c		Top-level regex execution code
rege_dfa.c		DFA creation and execution
regerror.c		pg_regerror: generate text for a regex error code
regfree.c		pg_regfree: API to free a no-longer-needed regex_t
regexport.c		Functions for extracting info from a regex_t
regprefix.c		Code for extracting a common prefix from a regex_t

The locale-specific code is concerned primarily with case-folding and with
expanding locale-specific character classes, such as [[:alnum:]].  It
really needs refactoring if this is ever to become a standalone library.

The header files for the library are in src/include/regex/:

regcustom.h		Customizes library for particular application
regerrs.h		Error message list
regex.h			Exported API
regexport.h		Exported API for regexport.c
regguts.h		Internals declarations


DFAs, NFAs, and all that
------------------------

This library is a hybrid DFA/NFA regex implementation.  (If you've never
heard either of those terms, get thee to a first-year comp sci textbook.)
It might not be clear at first glance what that really means and how it
relates to what you'll see in the code.  Here's what really happens:

* Initial parsing of a regex generates an NFA representation, with number
of states approximately proportional to the length of the regexp.

* The NFA is then optimized into a "compact NFA" representation, which is
basically the same idea but without fields that are not going to be needed
at runtime.  It is simplified too: the compact format only allows "plain"
and "LACON" arc types.  The cNFA representation is what is passed from
regcomp to regexec.

* Unlike traditional NFA-based regex engines, we do not execute directly
from the NFA representation, as that would require backtracking and so be
very slow in some cases.  Rather, we execute a DFA, which ideally can
process an input string in linear time (O(M) for M characters of input)
without backtracking.  Each state of the DFA corresponds to a set of
states of the NFA, that is all the states that the NFA might have been in
upon reaching the current point in the input string.  Therefore, an NFA
with N states might require as many as 2^N states in the corresponding
DFA, which could easily require unreasonable amounts of memory.  We deal
with this by materializing states of the DFA lazily (only when needed) and
keeping them in a limited-size cache.  The possible need to build the same
state of the DFA repeatedly makes this approach not truly O(M) time, but
in the worst case as much as O(M*N).  That's still far better than the
worst case for a backtracking NFA engine.

If that were the end of it, we'd just say this is a DFA engine, with the
use of NFAs being merely an implementation detail.  However, a DFA engine
cannot handle some important regex features such as capturing parens and
back-references.  If the parser finds that a regex uses these features
(collectively called "messy cases" in the code), then we have to use
NFA-style backtracking search after all.

When using the NFA mode, the representation constructed by the parser
consists of a tree of sub-expressions ("subre"s).  Leaf tree nodes are
either plain regular expressions (which are executed as DFAs in the manner
described above) or back-references (which try to match the input to some
previous substring).  Non-leaf nodes are capture nodes (which save the
location of the substring currently matching their child node),
concatenation, alternation, or iteration nodes.  At execution time, the
executor recursively scans the tree.  At concatenation, alternation, or
iteration nodes, it considers each possible alternative way of matching the
input string, that is each place where the string could be split for a
concatenation or iteration, or each child node for an alternation.  It
tries the next alternative if the match fails according to the child nodes.
This is exactly the sort of backtracking search done by a traditional NFA
regex engine.  If there are many tree levels it can get very slow.

But all is not lost: we can still be smarter than the average pure NFA
engine.  To do this, each subre node has an associated DFA, which
represents what the node could possibly match insofar as a mathematically
pure regex can describe that, which basically means "no backrefs".
Before we perform any search of possible alternative sub-matches, we run
the DFA to see if it thinks the proposed substring could possibly match.
If not, we can reject the match immediately without iterating through many
possibilities.

As an example, consider the regex "(a[bc]+)\1".  The compiled
representation will have a top-level concatenation subre node.  Its left
child is a capture node, and the child of that is a plain DFA node for
"a[bc]+".  The concatenation's right child is a backref node for \1.
The DFA associated with the concatenation node will be "a[bc]+a[bc]+",
where the backref has been replaced by a copy of the DFA for its referent
expression.  When executed, the concatenation node will have to search for
a possible division of the input string that allows its two child nodes to
each match their part of the string (and although this specific case can
only succeed when the division is at the middle, the code does not know
that, nor would it be true in general).  However, we can first run the DFA
and quickly reject any input that doesn't start with an "a" and contain
one more "a" plus some number of b's and c's.  If the DFA doesn't match,
there is no need to recurse to the two child nodes for each possible
string division point.  In many cases, this prefiltering makes the search
run much faster than a pure NFA engine could do.  It is this behavior that
justifies using the phrase "hybrid DFA/NFA engine" to describe Spencer's
library.


Colors and colormapping
-----------------------

In many common regex patterns, there are large numbers of characters that
can be treated alike by the execution engine.  A simple example is the
pattern "[[:alpha:]][[:alnum:]]*" for an identifier.  Basically the engine
only needs to care whether an input symbol is a letter, a digit, or other.
We could build the NFA or DFA with a separate arc for each possible letter
and digit, but that's very wasteful of space and not so cheap to execute
either, especially when dealing with Unicode which can have thousands of
letters.  Instead, the parser builds a "color map" that maps each possible
input symbol to a "color", or equivalence class.  The NFA or DFA
representation then has arcs labeled with colors, not specific input
symbols.  At execution, the first thing the executor does with each input
symbol is to look up its color in the color map, and then everything else
works from the color only.

To build the colormap, we start by assigning every possible input symbol
the color WHITE, which means "other" (that is, at the end of parsing, the
symbols that are still WHITE are those not explicitly referenced anywhere
in the regex).  When we see a simple literal character or a bracket
expression in the regex, we want to assign that character, or all the
characters represented by the bracket expression, a unique new color that
can be used to label the NFA arc corresponding to the state transition for
matching this character or bracket expression.  The basic idea is:
first, change the color assigned to a character to some new value;
second, run through all the existing arcs in the partially-built NFA,
and for each one referencing the character's old color, add a parallel
arc referencing its new color (this keeps the reassignment from changing
the semantics of what we already built); and third, add a new arc with
the character's new color to the current pair of NFA states, denoting
that seeing this character allows the state transition to be made.

This is complicated a bit by not wanting to create more colors
(equivalence classes) than absolutely necessary.  In particular, if a
bracket expression mentions two characters that had the same color before,
they should still share the same color after we process the bracket, since
there is still not a need to distinguish them.  But we do need to
distinguish them from other characters that previously had the same color
yet are not listed in the bracket expression.  To mechanize this, the code
has a concept of "parent colors" and "subcolors", where a color's subcolor
is the new color that we are giving to any characters of that color while
parsing the current atom.  (The word "parent" is a bit unfortunate here,
because it suggests a long-lived relationship, but a subcolor link really
only lasts for the duration of parsing a single atom.)  In other words,
a subcolor link means that we are in process of splitting the parent color
into two colors (equivalence classes), depending on whether or not each
member character should be included by the current regex atom.

As an example, suppose we have the regex "a\d\wx".  Initially all possible
character codes are labeled WHITE (color 0).  To parse the atom "a", we
create a new color (1), update "a"'s color map entry to 1, and create an
arc labeled 1 between the first two states of the NFA.  Now we see \d,
which is really a bracket expression containing the digits "0"-"9".
First we process "0", which is currently WHITE, so we create a new color
(2), update "0"'s color map entry to 2, and create an arc labeled 2
between the second and third states of the NFA.  We also mark color WHITE
as having the subcolor 2, which means that future relabelings of WHITE
characters should also select 2 as the new color.  Thus, when we process
"1", we won't create a new color but re-use 2.  We update "1"'s color map
entry to 2, and then find that we don't need a new arc because there is
already one labeled 2 between the second and third states of the NFA.
Similarly for the other 8 digits, so there will be only one arc labeled 2
between NFA states 2 and 3 for all members of this bracket expression.
At completion of processing of the bracket expression, we call okcolors()
which breaks all the existing parent/subcolor links; there is no longer a
marker saying that WHITE characters should be relabeled 2.  (Note:
actually, we did the same creation and clearing of a subcolor link for the
primitive atom "a", but it didn't do anything very interesting.)  Now we
come to the "\w" bracket expression, which for simplicity assume expands
to just "[a-z0-9]".  We process "a", but observe that it is already the
sole member of its color 1.  This means there is no need to subdivide that
equivalence class more finely, so we do not create any new color.  We just
make an arc labeled 1 between the third and fourth NFA states.  Next we
process "b", which is WHITE and far from the only WHITE character, so we
create a new color (3), link that as WHITE's subcolor, relabel "b" as
color 3, and make an arc labeled 3.  As we process "c" through "z", each
is relabeled from WHITE to 3, but no new arc is needed.  Now we come to
"0", which is not the only member of its color 2, so we suppose that a new
color is needed and create color 4.  We link 4 as subcolor of 2, relabel
"0" as color 4 in the map, and add an arc for color 4.  Next "1" through
"9" are similarly relabeled as color 4, with no additional arcs needed.
Having finished the bracket expression, we call okcolors(), which breaks
the subcolor links.  okcolors() further observes that we have removed
every member of color 2 (the previous color of the digit characters).
Therefore, it runs through the partial NFA built so far and relabels arcs
labeled 2 to color 4; in particular the arc from NFA state 2 to state 3 is
relabeled color 4.  Then it frees up color 2, since we have no more use
for that color.  We now have an NFA in which transitions for digits are
consistently labeled with color 4.  Last, we come to the atom "x".
"x" is currently labeled with color 3, and it's not the only member of
that color, so we realize that we now need to distinguish "x" from other
letters when we did not before.  We create a new color, which might have
been 5 but instead we recycle the unused color 2.  "x" is relabeled 2 in
the color map and 2 is linked as the subcolor of 3, and we add an arc for
2 between states 4 and 5 of the NFA.  Now we call okcolors(), which breaks
the subcolor link between colors 3 and 2 and notices that both colors are
nonempty.  Therefore, it also runs through the existing NFA arcs and adds
an additional arc labeled 2 wherever there is an arc labeled 3; this
action ensures that characters of color 2 (i.e., "x") will still be
considered as allowing any transitions they did before.  We are now done
parsing the regex, and we have these final color assignments:
	color 1: "a"
	color 2: "x"
	color 3: other letters
	color 4: digits
and the NFA has these arcs:
	states 1 -> 2 on color 1 (hence, "a" only)
	states 2 -> 3 on color 4 (digits)
	states 3 -> 4 on colors 1, 3, 4, and 2 (covering all \w characters)
	states 4 -> 5 on color 2 ("x" only)
which can be seen to be a correct representation of the regex.

Given this summary, we can see we need the following operations for
colors:

* A fast way to look up the current color assignment for any character
  code.  (This is needed during both parsing and execution, while the
  remaining operations are needed only during parsing.)
* A way to alter the color assignment for any given character code.
* We must track the number of characters currently assigned to each
  color, so that we can detect empty and singleton colors.
* We must track all existing NFA arcs of a given color, so that we
  can relabel them at need, or add parallel arcs of a new color when
  an existing color has to be subdivided.

The last two of these are handled with the "struct colordesc" array and
the "colorchain" links in NFA arc structs.

Ideally, we'd do the first two operations using a simple linear array
storing the current color assignment for each character code.
Unfortunately, that's not terribly workable for large charsets such as
Unicode.  Our solution is to divide the color map into two parts.  A simple
linear array is used for character codes up to MAX_SIMPLE_CHR, which can be
chosen large enough to include all popular characters (so that the
significantly-slower code paths about to be described are seldom invoked).
Characters above that need be considered at compile time only if they
appear explicitly in the regex pattern.  We store each such mentioned
character or character range as an entry in the "colormaprange" array in
the colormap.  (Overlapping ranges are split into unique subranges, so that
each range in the finished list needs only a single color that describes
all its characters.)  When mapping a character above MAX_SIMPLE_CHR to a
color at runtime, we search this list of ranges explicitly.

That's still not quite enough, though, because of locale-dependent
character classes such as [[:alpha:]].  In Unicode locales these classes
may have thousands of entries that are above MAX_SIMPLE_CHR, and we
certainly don't want to be searching large colormaprange arrays at runtime.
Nor do we even want to spend the time to initialize cvec structures that
exhaustively describe all of those characters.  Our solution is to compute
exact per-character colors at regex compile time only up to MAX_SIMPLE_CHR.
For characters above that, we apply the <ctype.h> or <wctype.h> lookup
functions at runtime for each locale-dependent character class used in the
regex pattern, constructing a bitmap that describes which classes the
runtime character belongs to.  The per-character-range data structure
mentioned above actually holds, for each range, a separate color entry
for each possible combination of character class properties.  That is,
the color map for characters above MAX_SIMPLE_CHR is really a 2-D array,
whose rows correspond to high characters or character ranges that are
explicitly mentioned in the regex pattern, and whose columns correspond
to sets of the locale-dependent character classes that are used in the
regex.

As an example, given the pattern '\w\u1234[\U0001D100-\U0001D1FF]'
(and supposing that MAX_SIMPLE_CHR is less than 0x1234), we will need
a high color map with three rows.  One row is for the single character
U+1234 (represented as a single-element range), one is for the range
U+1D100..U+1D1FF, and the other row represents all remaining high
characters.  The color map has two columns, one for characters that
satisfy iswalnum() and one for those that don't.

We build this color map in parallel with scanning the regex.  Each time
we detect a new explicit high character (or range) or a locale-dependent
character class, we split existing entry(s) in the high color map so that
characters we need to be able to distinguish will have distinct entries
that can be given separate colors.  Often, though, single entries in the
high color map will represent very large sets of characters.

If there are both explicit high characters/ranges and locale-dependent
character classes, we may have entries in the high color map array that
have non-WHITE colors but don't actually represent any real characters.
(For example, in a row representing a singleton range, only one of the
columns could possibly be a live entry; it's the one matching the actual
locale properties for that single character.)  We don't currently make
any effort to reclaim such colors.  In principle it could be done, but
it's not clear that it's worth the trouble.


Detailed semantics of an NFA
----------------------------

When trying to read dumped-out NFAs, it's helpful to know these facts:

State 0 (additionally marked with "@" in dumpnfa's output) is always the
goal state, and state 1 (additionally marked with ">") is the start state.
(The code refers to these as the post state and pre state respectively.)

The possible arc types are:

    PLAIN arcs, which specify matching of any character of a given "color"
    (see above).  These are dumped as "[color_number]->to_state".

    EMPTY arcs, which specify a no-op transition to another state.  These
    are dumped as "->to_state".

    AHEAD constraints, which represent a "next character must be of this
    color" constraint.  AHEAD differs from a PLAIN arc in that the input
    character is not consumed when crossing the arc.  These are dumped as
    ">color_number>->to_state".

    BEHIND constraints, which represent a "previous character must be of
    this color" constraint, which likewise consumes no input.  These are
    dumped as "<color_number<->to_state".

    '^' arcs, which specify a beginning-of-input constraint.  These are
    dumped as "^0->to_state" or "^1->to_state" for beginning-of-string and
    beginning-of-line constraints respectively.

    '$' arcs, which specify an end-of-input constraint.  These are dumped
    as "$0->to_state" or "$1->to_state" for end-of-string and end-of-line
    constraints respectively.

    LACON constraints, which represent "(?=re)", "(?!re)", "(?<=re)", and
    "(?<!re)" constraints, i.e. the input starting/ending at this point must
    match (or not match) a given sub-RE, but the matching input is not
    consumed.  These are dumped as ":subtree_number:->to_state".

If you see anything else (especially any question marks) in the display of
an arc, it's dumpnfa() trying to tell you that there's something fishy
about the arc; see the source code.

The regex executor can only handle PLAIN and LACON transitions.  The regex
optimize() function is responsible for transforming the parser's output
to get rid of all the other arc types.  In particular, ^ and $ arcs that
are not dropped as impossible will always end up adjacent to the pre or
post state respectively, and then will be converted into PLAIN arcs that
mention the special "colors" for BOS, BOL, EOS, or EOL.

To decide whether a thus-transformed NFA matches a given substring of the
input string, the executor essentially follows these rules:
1. Start the NFA "looking at" the character *before* the given substring,
or if the substring is at the start of the input, prepend an imaginary BOS
character instead.
2. Run the NFA until it has consumed the character *after* the given
substring, or an imaginary following EOS character if the substring is at
the end of the input.
3. If the NFA is (or can be) in the goal state at this point, it matches.

So one can mentally execute an untransformed NFA by taking ^ and $ as
ordinary constraints that match at start and end of input; but plain
arcs out of the start state should be taken as matches for the character
before the target substring, and similarly, plain arcs leading to the
post state are matches for the character after the target substring.
This definition is necessary to support regexes that begin or end with
constraints such as \m and \M, which imply requirements on the adjacent
character if any.  NFAs for simple unanchored patterns will usually have
pre-state outarcs for all possible character colors as well as BOS and
BOL, and post-state inarcs for all possible character colors as well as
EOS and EOL, so that the executor's behavior will work.