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 is one more global symbol, pg_set_regex_collation in regcomp. It is not meant to be part of the API, but it has to be global because both regcomp and regexec call it. It'd be better to get rid of that, 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. The color map proper (that is, the per-character lookup array) is handled as a multi-level tree, with each tree level indexed by one byte of a character's value. The code arranges to not have more than one copy of bottom-level tree pages that are all-the-same-color. Unfortunately, this design does not seem terribly efficient for common cases such as a tree in which all Unicode letters are colored the same, because there aren't that many places where we get a whole page all the same color, except at the end of the map. (It also strikes me that given PG's current restrictions on the range of Unicode values, we could use a 3-level rather than 4-level tree; but there's not provision for that in regguts.h at the moment.) A bigger problem is that it just doesn't seem very reasonable to have to consider each Unicode letter separately at regex parse time for a regex such as "\w"; more than likely, a huge percentage of those codes will never be seen at runtime. We need to fix things so that locale-based character classes are somehow processed "symbolically" without making a full expansion of their contents at parse time. This would mean that we'd have to be ready to call iswalpha() at runtime, but if that only happens for high-code-value characters, it shouldn't be a big performance hit. 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 "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 "(?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.