Parsing Expression Grammars For Lua, version 0.8


LPeg is a new pattern-matching library for Lua, based on Parsing Expression Grammars (PEGs). In this text, I assume you are familiar with PEGs. If you are not, you can get a quick start reading the Wikipedia Entry for PEGs or Section 2 of Parsing Expression Grammars: A Recognition-Based Syntactic Foundation (the section has only one page). The nice thing about PEGs is that it has a formal basis (instead of being an ad-hoc set of features), allows an efficient and simple implementation, and does most things we expect from a pattern-matching library (and more, as we can define entire grammars).

Following the Snobol tradition, LPeg defines patterns as first-class objects. That is, patterns are regular Lua values (represented by userdata). The library offers several functions to create and compose patterns. With the use of metamethods, several of these functions are provided as infix or prefix operators. On the one hand, the result is usually much more verbose than the typical encoding of patterns using the so called regular expressions (which typically are not regular expressions in the formal sense). On the other hand, first-class patterns allow much better documentation (as it is easy to comment the code, to use auxiliary variables to break complex definitions, etc.) and are extensible, as we can define new functions to create and compose patterns.

For a quick glance of the library, the following table summarizes its basic operations for creating patterns:

lpeg.P(string) Matches string literally
lpeg.P(number) Matches exactly number characters
lpeg.S(string) Matches any character in string (set)
lpeg.R("xy") Matches any character between x and y (range)
patt^n Matches at least n repetitions of patt
patt^-n Matches at most n repetitions of patt
patt1 * patt2 Matches patt1 followed by patt2
patt1 + patt2 Matches patt1 or patt2 (ordered choice)
patt1 - patt2 Matches patt1 if patt2 does not match
-patt Equivalent to "" - patt
#patt Matches patt but consumes no input

As a very simple example, lpeg.R("09")^1 matches a non-empty sequence of digits. As a not so simple example, -lpeg.P(1) (which can be written as lpeg.P(-1) or simply -1 for operations expecting a pattern) matches an empty string only if it cannot match a single character; so, it succeeds only at the subject's end.

Those not convinced by the previous syntax can try the re module, which implements patterns following a regular-expression style (e.g., [09]+). (This module is 200 lines of Lua code, and of course uses LPeg to parse regular expressions.)


lpeg.match (pattern, subject [, init])

The matching function. It attempts to match the given pattern against the subject string. If the match succeeds, returns the index in the subject of the first character after the match, or the values of captured values (if the pattern captured any value).

An optional numeric argument init makes the match starts at that position in the subject string. As usual in Lua libraries, a negative value counts from the end.

Unlike typical pattern-matching functions, match works only in anchored mode; that is, it tries to match the pattern with a prefix of the given subject string (at position init), not with an arbitrary substring of the subject. So, if we want to find a pattern anywhere in a string, we must either write a loop in Lua or write a pattern that matches anywhere. This second approach is easy and quite efficient; see examples.

lpeg.type (value)

If the given value is a pattern, returns the string "pattern". Otherwise returns nil.

lpeg.version ()

Returns a string with the running version of LPEG.

Basic Constructions

The following operations build patterns. All operations that expect a pattern as an argument may receive also strings, tables, numbers, booleans, or functions, which are translated to patterns according to the rules of function lpeg.P.

lpeg.P (value)

Converts the given value into a proper pattern, according to the following rules:

  • If the argument is a pattern, it is returned unmodified.

  • If the argument is a string, it is translated to a pattern that matches literally the string.

  • If the argument is a number, it is translated as follows. A non-negative number n gives a pattern that matches exactly n characters; a negative number -n gives a pattern that succeeds only if the input string does not have n characters. It is (as expected) equivalent to the unary minus operation (see below) applied over the absolute value of n.

  • If the argument is a boolean, the result is a pattern that always succeeds or always fails (according to the boolean value), without consuming any input.

  • If the argument is a table, it is interpreted as a grammar (see Grammars).

  • If the argument is a function, returns a pattern equivalent to a match-time capture over the empty string.

    If the function is called with parameters s and i, its result is valid if it is in the range [i, len(s) + 1].

lpeg.R ({range})

Returns a pattern that matches any single character belonging to one of the given ranges. Each range is a string xy of length 2, representing all characters with code between the codes of x and y (both inclusive).

As an example, the pattern lpeg.R("09") matches any digit, and lpeg.R("az", "AZ") matches any ASCII letter.

lpeg.S (string)

Returns a pattern that matches any single character that appears in the given string. (The S stands for Set.)

As an example, the pattern lpeg.S("+-*/") matches any arithmetic operator.

Note that, if s is a character (that is, a string of length 1), then lpeg.P(s) is equivalent to lpeg.S(s) which is equivalent to lpeg.R(s..s). Note also that both lpeg.S("") and lpeg.R() are patterns that always fail.

lpeg.V (v)

This operation creates a non-terminal (a variable) for a grammar. The created non-terminal refers to the rule indexed by v in the enclosing grammar. (See Grammars for details.)


Returns a pattern equivalent to &patt in the original PEG notation. This is a pattern that matches only if the input string does match patt, but without consuming any input, independently of success or failure.

When it succeeds, #patt produces all captures produced by patt.



Returns a pattern equivalent to !patt in the original PEG notation. This pattern matches only if the input string does not match patt. It does not consume any input, independently of success or failure.

As an example, the pattern -1 matches only the end of string.

This pattern never produces any captures, because either patt fails or -patt fails. (A failing pattern produces no captures.)

patt1 + patt2

Returns a pattern equivalent to an ordered choice of patt1 and patt2. (This is denoted by patt1 / patt2 in the original PEG notation, not to be confused with the / operation in LPeg.) It matches either patt1 or patt2 (with no backtracking once one of them succeeds). The identity element for this operation is the pattern lpeg.P(false), which always fails.

If both patt1 and patt2 are character sets, this operation is equivalent to set union:

lower = lpeg.R("az")
upper = lpeg.R("AZ")
letter = lower + upper

patt1 - patt2

Returns a pattern equivalent to !patt2 patt1. This pattern asserts that the input does not match patt2 and then matches patt1.

If both patt1 and patt2 are character sets, this operation is equivalent to set difference. Note that -patt is equivalent to "" - patt (or 0 - patt). If patt is a character set, 1 - patt is its complement.

patt1 * patt2

Returns a pattern that matches patt1 and then matches patt2, starting where patt1 finished. The identity element for this operation is the pattern lpeg.P(true), which always succeeds.

(LPeg uses the * operator [instead of the more obvious ..] both because it has the right priority and because in formal languages it is common to use a dot for denoting concatenation.)


If n is nonnegative, this pattern is equivalent to pattn patt*. It matches at least n occurrences of patt.

Otherwise, when n is negative, this pattern is equivalent to (patt?)-n. That is, it matches at most -n occurrences of patt.

In particular, patt^0 is equivalent to patt*, patt^1 is equivalent to patt+, and patt^-1 is equivalent to patt? in the original PEG notation.

In all cases, the resulting pattern is greedy with no backtracking. That is, it matches only the longest possible sequence of matches for patt.


With the use of Lua variables, it is possible to define patterns incrementally, with each new pattern using previously defined ones. However, this technique does not allow the definition of recursive patterns. For recursive patterns, we need real grammars.

LPeg represents grammars with tables, where each entry is a rule.

The call lpeg.V(v) creates a pattern that represents the nonterminal (or variable) with index v in a grammar. Because the grammar still does not exist when this function is evaluated, the result is an open reference to the respective rule.

A table is fixed when it is converted to a pattern (either by calling lpeg.P or by using it wherein a pattern is expected). Then every open reference created by lpeg.V(v) is corrected to refer to the rule indexed by v in the table.

When a table is fixed, the result is a pattern that matches its initial rule. The entry with index 1 in the table defines its initial rule. If that entry is a string, it is assumed to be the name of the initial rule. Otherwise, LPeg assumes that the entry 1 itself is the initial rule.

As an example, the following grammar matches strings of a's and b's that have the same number of a's and b's:

equalcount = lpeg.P{
  "S";   -- initial rule name
  S = "a" * lpeg.V"B" + "b" * lpeg.V"A" + "",
  A = "a" * lpeg.V"S" + "b" * lpeg.V"A" * lpeg.V"A",
  B = "b" * lpeg.V"S" + "a" * lpeg.V"B" * lpeg.V"B",
} * -1


Captures specify what a match operation should return (the so called semantic information). LPeg offers several kinds of captures, which produces values based on matches and combine them to produce new values.

The following table summarizes the basic captures:

OperationWhat it Produces
lpeg.C(patt) the match for patt
lpeg.Cc(values) the given values (matches the empty string)
lpeg.Cp() the current position (matches the empty string)
lpeg.Cb(n) the value produced by the nth previous capture (matches the empty string)
lpeg.Carg(n) the value of the nth extra argument to lpeg.match (matches the empty string)
lpeg.Cs(patt) the match for patt with the values from nested captures replacing their matches
lpeg.Ct(patt) a table with all captures from patt
lpeg.Ca(patt) an accumulation (or folding) of the captures from patt
patt / string string, with some marks replaced by captures of patt
patt / table table[c], where c is the (first) capture of patt
patt / function the returns of function applied to the captures of patt
lpeg.Cmt(patt, function) (see match-time capture)

A capture pattern produces its values every time it succeeds. For instance, a capture inside a loop produces as many values as matched by the loop. A capture produces a value only when it succeeds. For instance, the pattern lpeg.C(lpeg.P"a"^-1) produces the empty string when there is no "a" (because the pattern "a"? succeeds), while the pattern lpeg.C("a")^-1 does not produce any value when there is no "a" (because the pattern "a" fails).

Usually, LPEG evaluates all captures only after (and if) the entire match succeeds. At match time it only gathers enough information to produce the capture values later. As a particularly important consequence, most captures cannot affect the way a pattern matches a subject. The only exception to this rule is the so-called match-time capture. When a match-time capture matches, it forces the immediate evaluation of all its nested captures and then calls its corresponding function, which tells whether the match succeeds and also what values are produced.

lpeg.C (patt)

Creates a simple capture, which captures the substring of the subject that matches patt. The captured value is a string. If patt has other captures, their values are returned after this one.

lpeg.Ca (patt)

Creates an accumulator capture. This capture assumes that patt should produce at least one captured value of any kind, which becomes the initial value of an accumulator. Pattern patt then may produce zero or more function captures. Each of these functions in these captures is called having the accumulator as its first argument (followed by any other arguments provided by its own pattern), and the value returned by the function becomes the new value of the accumulator. The final value of this accumulator is the sole result of the whole capture.

As an example, the following pattern matches a list of numbers separated by commas and returns their addition:

-- matches a numeral and captures its value
local number = lpeg.R"09"^1 / tonumber

-- auxiliary function to add two numbers
local function add (acc, newvalue) return acc + newvalue end

list = lpeg.Ca(number * ("," * number / add)^0)

-- example of use
print(list:match("10,30,43"))   --> 83

lpeg.Carg (n)

Creates an argument capture. This pattern matches the empty string and produces the value given as the nth extra argument given in the call to lpeg.match.

lpeg.Cb (n)

Creates a back capture. This pattern matches the empty string and produces the values produced by the nth previous capture.

Captures are numbered dynamically. So, the first previous capture is the last capture to match before the current one. The numbering includes only complete captures; so, if the back capture is inside another capture, this enclosing capture is ignored (because it is not complete when the back capture is seen). Numbering does not count nested captures. Numbering counts captures, not the values produced by them; it does not matter whether a capture produces zero or many values, it counts as one.

This is an experimental feature. It probably will be changed or even removed in future releases.

lpeg.Cc ({value})

Creates a constant capture. This pattern matches the empty string and produces all given values as its captured values.

lpeg.Cp ()

Creates a position capture. It matches the empty string and captures the position in the subject where the match occurs. The captured value is a number.

lpeg.Cs (patt)

Creates a substitution capture, which captures the substring of the subject that matches patt, with substitutions. For any capture inside patt, the substring that matched the capture is replaced by the capture value (which should be a string). The capture values from patt are not returned independently (only as substrings in the resulting string).

lpeg.Ct (patt)

Creates a table capture. This capture creates a table and puts all captures made by patt inside this table in successive integer keys, starting at 1.

The captured value is only this table. The captures made by patt are not returned independently (only as table elements).

patt / string

Creates a string capture. It creates a capture string based on string. The captured value is a copy of string, except that the character % works as an escape character: any sequence in string of the form %n, with n between 1 and 9, stands for the match of the n-th capture in patt. (Currently these nested captures can be only simple captures.) The sequence %0 stands for the whole match. The sequence %% stands for a single %.

patt / table

Creates a query capture. It indexes the given table using as key the value of the first capture of patt, or the whole match if patt made no capture. The value at that index is the final value of the capture. If the table does not have that key, there is no captured value. Everything works as if there was no capture.

patt / function

Creates a function capture. It calls the given function passing all captures made by patt as arguments, or the whole match if patt made no capture. The values returned by the function are the final values of the capture. (This capture may create multiple values.) In particular, if function returns no value, there is no captured value; everything works as if there was no capture.

lpeg.Cmt(patt, function)

Creates a match-time capture. Unlike all other captures, this one is evaluated immediately when a match occurs. It forces the immediate evaluation of all its nested captures and then calls function.

The function gets as arguments the entire subject, the current position (after the match of patt), plus any capture values produced by patt.

The first value returned by function defines how the match happens. If the call returns a number, the match succeeds and the returned number becomes the new current position. (Assuming a subject s and current position i, the returned number must be in the range [i, len(s) + 1].) If the call returns false, nil, or no value, the match fails.

Any extra values returned by the function become the values produced by the capture.

Some Examples

Splitting a String

The following code splits a string using a given pattern sep as a separator:

function split (s, sep)
  sep = lpeg.P(sep)
  local elem = lpeg.C((1 - sep)^0)
  local p = elem * (sep * elem)^0
  return lpeg.match(p, s)

First the function ensures that sep is a proper pattern. The pattern elem is a repetition of zero of more arbitrary characters as long as there is not a match against the separator. It also captures its result. The pattern p matches a list of elements separated by sep.

If the split results in too many values, it may overflow the maximum number of values that can be returned by a Lua function. In this case, we should collect these values in a table:

function split (s, sep)
  sep = lpeg.P(sep)
  local elem = lpeg.C((1 - sep)^0)
  local p = lpeg.Ct(elem * (sep * elem)^0)   -- make a table capture
  return lpeg.match(p, s)

Searching for a Pattern

The primitive match works only in anchored mode. If we want to find a pattern anywhere in a string, we must write a pattern that matches anywhere.

Because patterns are composable, we can write a function that, given any arbitrary pattern p, returns a new pattern that searches for p anywhere in a string. There are several ways to do the search. One way is like this:

function anywhere (p)
  return lpeg.P{ p + 1 * lpeg.V(1) }

This grammar has a straight reading: it matches p or skips one character and tries again.

If we want to know where the pattern is in the string (instead of knowing only that it is there somewhere), we can add position captures to the pattern:

local I = lpeg.Cp()
function anywhere (p)
  return lpeg.P{ I * p * I + 1 * lpeg.V(1) }

Another option for the search is like this:

local I = lpeg.Cp()
function anywhere (p)
  return (1 - lpeg.P(p))^0 * I * p * I

Again the pattern has a straight reading: it skips as many characters as possible while not matching p, and then matches p (plus appropriate captures).

If we want to look for a pattern only at word boundaries, we can use the following transformer:

local wordletter = lpeg.R("AZ", "az")

function atwordboundary (p)
  return lpeg.P{
    [1] = p + wordletter^0 * (1 - wordletter)^1 * lpeg.V(1)

Balanced Parentheses

The following pattern matches only strings with balanced parentheses:

b = lpeg.P{ "(" * ((1 - lpeg.S"()") + lpeg.V(1))^0 * ")" }

Reading the first (and only) rule of the given grammar, we have that a balanced string is an open parenthesis, followed by zero or more repetitions of either a non-parenthesis character or a balanced string (lpeg.V(1)), followed by a closing parenthesis.

Global Substitution

The next example does a job somewhat similar to string.gsub. It receives a pattern and a replacement value, and substitutes the replacement value for all occurrences of the pattern in a given string:

function gsub (s, patt, repl)
  patt = lpeg.P(patt)
  patt = lpeg.Cs((patt / repl + 1)^0)
  return lpeg.match(patt, s)

As in string.gsub, the replacement value can be a string, a function, or a table.

Comma-Separated Values (CSV)

This example breaks a string into comma-separated values, returning all fields:

local field = '"' * lpeg.Cs(((lpeg.P(1) - '"') + lpeg.P'""' / '"')^0) * '"' +
                    lpeg.C((1 - lpeg.S',\n"')^0)

local record = field * (',' * field)^0 * (lpeg.P'\n' + -1)

function csv (s)
  return lpeg.match(record, s)

A field is either a quoted field (which may contain any character except an individual quote, which may be written as two quotes that are replaced by one) or an unquoted field (which cannot contain commas, newlines, or quotes). A record is a list of fields separated by commas, ending with a newline or the string end (-1).

UTF-8 and Latin 1

It is not difficult to use LPeg to convert a string from utf-8 encoding to Latin 1 (ISO 8859-1):

-- convert a two-byte utf8 sequence to a Latin 1 character
local function f2 (s)
  local c1, c2 = string.byte(s, 1, 2)
  return string.char(c1 * 64 + c2 - 12416)

local utf8 = lpeg.R("\0\127")
           + lpeg.R("\194\195") * lpeg.R("\128\191") / f2

local decode_pattern = lpeg.Cs(utf8^0) * -1

In this code, the definition of utf-8 is already restricted to the Latin 1 range (from 0 to 255). Any encoding outside this range (as well as any invalid encoding) will not match that pattern.

As the definition of decode_pattern demands that the pattern matches the whole input (because of the -1 at its end), any invalid string will simply fail to match, without any useful information about the problem. We can improve this situation redefining decode_pattern as follows:

local function er (_, i) error("invalid encoding at position " .. i) end

local decode_pattern = lpeg.Cs(utf8^0) * (-1 + lpeg.P(er))

Now, if the pattern utf8^0 stops before the end of the string, an appropriate error function is called.

UTF-8 and Unicode

We can extend the previous patterns to handle all Unicode code points. Of course, we cannot translate them to Latin 1 or any other one-byte encoding. Instead, our translation results in a array with the code points represented as numbers. The full code is here:

-- decode a two-byte utf8 sequence
local function f2 (s)
  local c1, c2 = string.byte(s, 1, 2)
  return c1 * 64 + c2 - 12416

-- decode a three-byte utf8 sequence
local function f3 (s)
  local c1, c2, c3 = string.byte(s, 1, 3)
  return (c1 * 64 + c2) * 64 + c3 - 925824

-- decode a four-byte utf8 sequence
local function f4 (s)
  local c1, c2, c3, c4 = string.byte(s, 1, 4)
  return ((c1 * 64 + c2) * 64 + c3) * 64 + c4 - 63447168

local cont = lpeg.R("\128\191")   -- continuation byte

local utf8 = lpeg.R("\0\127") / string.byte
           + lpeg.R("\194\223") * cont / f2
           + lpeg.R("\224\239") * cont * cont / f3
           + lpeg.R("\240\244") * cont * cont * cont / f4

local decode_pattern = lpeg.Ct(utf8^0) * -1

Lua's Long Strings

A long string in Lua starts with the pattern [=*[ and ends at the first occurrence of ]=*] with exactly the same number of equal signs. If the opening brackets are followed by a newline, this newline is discharged (that is, it is not part of the string).

To match a long string in Lua, the pattern must capture the first repetition of equal signs and then, whenever it finds a candidate for closing the string, check whether it has the same number of equal signs.

open = "[" * lpeg.C(lpeg.P"="^0) * "[" * lpeg.P"\n"^-1
close = "]" * lpeg.C(lpeg.P"="^0) * "]"
closeeq = lpeg.Cmt(close * lpeg.Cb(2), function (s, i, a, b) return a == b end)
string = open * m.C((lpeg.P(1) - closeeq)^0) * close /
  function (o, s) return s end

The open pattern matches [=*[, capturing the repetitions of equal signs for later use; it also discharges an optional newline, if present. The close pattern matches ]=*]. The closeeq pattern first matches close; then it uses a back capture to recover the capture made by the previous open (the immediate previous capture is the one just made by close, so it must get the second previous capture); finally it uses a match-time capture to check whether both captures are equal. The string pattern starts with an open, then it goes as far as possible until matching closeeq, and then matches the final close. The final function capture simply consumes the captures made by open and close and returns only the middle capture, which is the string contents.

Arithmetic Expressions

This example is a complete parser and evaluator for simple arithmetic expressions. We write it in two styles. The first approach first builds a syntax tree and then traverses this tree to compute the expression value:

-- Lexical Elements
local Space = lpeg.S(" \n\t")^0
local Number = lpeg.C(lpeg.P"-"^-1 * lpeg.R("09")^1) * Space
local FactorOp = lpeg.C(lpeg.S("+-")) * Space
local TermOp = lpeg.C(lpeg.S("*/")) * Space
local Open = "(" * Space
local Close = ")" * Space

-- Grammar
local Exp, Term, Factor = lpeg.V"Exp", lpeg.V"Term", lpeg.V"Factor"
G = lpeg.P{ Exp,
  Exp = lpeg.Ct(Factor * (FactorOp * Factor)^0);
  Factor = lpeg.Ct(Term * (TermOp * Term)^0);
  Term = Number + Open * Exp * Close;

G = Space * G * -1

-- Evaluator
function eval (x)
  if type(x) == "string" then
    return tonumber(x)
    local op1 = eval(x[1])
    for i = 2, #x, 2 do
      local op = x[i]
      local op2 = eval(x[i + 1])
      if (op == "+") then op1 = op1 + op2
      elseif (op == "-") then op1 = op1 - op2
      elseif (op == "*") then op1 = op1 * op2
      elseif (op == "/") then op1 = op1 / op2
    return op1

-- Parser/Evaluator
function evalExp (s)
  local t = lpeg.match(G, s)
  if not t then error("syntax error", 2) end
  return eval(t)

-- small example
print(evalExp"3 + 5*9 / (1+1) - 12")

The second style computes the expression value on the fly, without building the syntax tree. The following grammar takes this approach. (It assumes the same lexical elements as before.)

-- Auxiliary function
function eval (v1, op, v2)
  if (op == "+") then return v1 + v2
  elseif (op == "-") then return v1 - v2
  elseif (op == "*") then return v1 * v2
  elseif (op == "/") then return v1 / v2

-- Grammar
local V = lpeg.V
G = lpeg.P{ "Exp",
  Exp = lpeg.Ca(V"Factor" * (FactorOp * V"Factor" / eval)^0);
  Factor = lpeg.Ca(V"Term" * (TermOp * V"Term" / eval)^0);
  Term = Number / tonumber + Open * V"Exp" * Close;

-- small example
print(lpeg.match(G, "3 + 5*9 / (1+1) - 12"))

Note the use of the accumulator capture. To compute the value of an expression, the accumulator starts with the value of the first factor, and then applies eval over the accumulator, the operator, and the new factor for each repetition.


LPeg source code.


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$Id: lpeg.html,v 1.46 2008/03/07 14:20:55 roberto Exp roberto $