NAME¶
grammar::peg - Create and manipulate parsing expression grammars
SYNOPSIS¶
package require
Tcl 8.4
package require
snit
package require
grammar::peg ?0.1?
::grammar::peg pegName
?
=|
:=|
<--|
as|
deserialize src?
pegName destroy
pegName clear
pegName = srcPEG
pegName --> dstPEG
pegName serialize
pegName deserialize serialization
pegName is valid
pegName start ?
pe?
pegName nonterminals
pegName nonterminal add nt pe
pegName nonterminal delete nt1 ?
nt2 ...?
pegName nonterminal exists nt
pegName nonterminal rename nt ntnew
pegName nonterminal mode nt ?
mode?
pegName nonterminal rule nt
pegName unknown nonterminals
DESCRIPTION¶
This package provides a container class for
parsing expression grammars
(Short: PEG). It allows the incremental definition of the grammar, its
manipulation and querying of the definition. The package neither provides
complex operations on the grammar, nor has it the ability to execute a grammar
definition for a stream of symbols. Two packages related to this one are
grammar::mengine and
grammar::peg::interpreter. The first of
them defines a general virtual machine for the matching of a character stream,
and the second implements an interpreter for parsing expression grammars on
top of that virtual machine.
TERMS & CONCEPTS¶
PEGs are similar to context-free grammars, but not equivalent; in some cases
PEGs are strictly more powerful than context-free grammars (there exist PEGs
for some non-context-free languages). The formal mathematical definition of
parsing expressions and parsing expression grammars can be found in section
PARSING EXPRESSION GRAMMARS.
In short, we have
terminal symbols, which are the most basic building
blocks for
sentences, and
nonterminal symbols with associated
parsing expressions, defining the grammatical structure of the
sentences. The two sets of symbols are distinctive, and do not overlap. When
speaking about symbols the word "symbol" is often left out. The
union of the sets of terminal and nonterminal symbols is called the set of
symbols.
Here the set of
terminal symbols is not explicitly managed, but
implicitly defined as the set of all characters. Note that this means that we
inherit from Tcl the ability to handle all of Unicode.
A pair of
nonterminal and
parsing expression is also called a
grammatical rule, or
rule for short. In the context of a rule
the nonterminal is often called the left-hand-side (LHS), and the parsing
expression the right-hand-side (RHS).
The
start expression of a grammar is a parsing expression from which all
the sentences contained in the language specified by the grammar are
derived. To make the understanding of this term easier let us assume
for a moment that the RHS of each rule, and the start expression, is either a
sequence of symbols, or a series of alternate parsing expressions. In the
latter case the rule can be seen as a set of rules, each providing one
alternative for the nonterminal. A parsing expression A' is now a derivation
of a parsing expression A if we pick one of the nonterminals N in the
expression, and one of the alternative rules R for N, and then replace the
nonterminal in A with the RHS of the chosen rule. Here we can see why the
terminal symbols are called such. They cannot be expanded any further, thus
terminate the process of deriving new expressions. An example
Rules
(1) A <- a B c
(2a) B <- d B
(2b) B <- e
Some derivations, using starting expression A.
A -/1/-> a B c -/2a/-> a d B c -/2b/-> a d e c
A derived expression containing only terminal symbols is a
sentence. The
set of all sentences which can be derived from the start expression is the
language of the grammar.
Some definitions for nonterminals and expressions:
- [1]
- A nonterminal A is called reachable if it is possible to derive a
parsing expression from the start expression which contains A.
- [2]
- A nonterminal A is called useful if it is possible to derive a
sentence from it.
- [3]
- A nonterminal A is called recursive if it is possible to derive a
parsing expression from it which contains A, again.
- [4]
- The FIRST set of a nonterminal A contains all the symbols which can
occur of as the leftmost symbol in a parsing expression derived from A. If
the FIRST set contains A itself then that nonterminal is called
left-recursive.
- [5]
- The LAST set of a nonterminal A contains all the symbols which can
occur of as the rightmost symbol in a parsing expression derived from A.
If the LAST set contains A itself then that nonterminal is called
right-recursive.
- [6]
- The FOLLOW set of a nonterminal A contains all the symbols which
can occur after A in a parsing expression derived from the start
expression.
- [7]
- A nonterminal (or parsing expression) is called nullable if the
empty sentence can be derived from it.
And based on the above definitions for grammars:
- [1]
- A grammar G is recursive if and only if it contains a nonterminal A
which is recursive. The terms left- and right-recursive, and
useful are analogously defined.
- [2]
- A grammar is minimal if it contains only reachable and
useful nonterminals.
- [3]
- A grammar is wellformed if it is not left-recursive. Such grammars
are also complete, which means that they always succeed or fail on
all input sentences. For an incomplete grammar on the other hand input
sentences exist for which an attempt to match them against the grammar
will not terminate.
- [4]
- As we wish to allow ourselves to build a grammar incrementally in a
container object we will encounter stages where the RHS of one or more
rules reference symbols which are not yet known to the container. Such a
grammar we call invalid. We cannot use the term incomplete
as this term is already taken, see the last item.
CONTAINER CLASS API¶
The package exports the API described here.
- ::grammar::peg pegName
?=|:=|<--|as| deserialize
src?
- The command creates a new container object for a parsing expression
grammar and returns the fully qualified name of the object command as its
result. The API the returned command is following is described in the
section CONTAINER OBJECT API. It may be used to invoke various
operations on the container and the grammar within.
The new container, i.e. grammar will be empty if no src is specified.
Otherwise it will contain a copy of the grammar contained in the
src. The src has to be a container object reference for all
operators except deserialize. The deserialize operator
requires src to be the serialization of a parsing expression
grammar instead.
An empty grammar has no nonterminal symbols, and the start expression is the
empty expression, i.e. epsilon. It is valid, but not
useful.
CONTAINER OBJECT API¶
All grammar container objects provide the following methods for the manipulation
of their contents:
- pegName destroy
- Destroys the grammar, including its storage space and associated
command.
- pegName clear
- Clears out the definition of the grammar contained in pegName, but
does not destroy the object.
- pegName = srcPEG
- Assigns the contents of the grammar contained in srcPEG to
pegName, overwriting any existing definition. This is the
assignment operator for grammars. It copies the grammar contained in the
grammar object srcPEG over the grammar definition in
pegName. The old contents of pegName are deleted by this
operation.
This operation is in effect equivalent to
pegName deserialize [srcPEG serialize]
- pegName --> dstPEG
- This is the reverse assignment operator for grammars. It copies the
automation contained in the object pegName over the grammar
definition in the object dstPEG. The old contents of dstPEG
are deleted by this operation.
This operation is in effect equivalent to
dstPEG deserialize [pegName serialize]
- pegName serialize
- This method serializes the grammar stored in pegName. In other
words it returns a tcl value completely describing that grammar.
This allows, for example, the transfer of grammars over arbitrary
channels, persistence, etc. This method is also the basis for both the
copy constructor and the assignment operator.
The result of this method has to be semantically identical over all
implementations of the grammar::peg interface. This is what will
enable us to copy grammars between different implementations of the same
interface.
The result is a list of four elements with the following structure:
- [1]
- The constant string grammar::peg.
- [2]
- A dictionary. Its keys are the names of all known nonterminal symbols, and
their associated values are the parsing expressions describing their
sentennial structure.
- [3]
- A dictionary. Its keys are the names of all known nonterminal symbols, and
their associated values hints to a matcher regarding the semantic values
produced by the symbol.
- [4]
- The last item is a parsing expression, the start expression of the
grammar.
Assuming the following PEG for simple mathematical expressions
Digit <- '0'/'1'/'2'/'3'/'4'/'5'/'6'/'7'/'8'/'9'
Sign <- '+' / '-'
Number <- Sign? Digit+
Expression <- '(' Expression ')' / (Factor (MulOp Factor)*)
MulOp <- '*' / '/'
Factor <- Term (AddOp Term)*
AddOp <- '+'/'-'
Term <- Number
a possible serialization is
grammar::peg \\
{Expression {/ {x ( Expression )} {x Factor {* {x MulOp Factor}}}} \\
Factor {x Term {* {x AddOp Term}}} \\
Term Number \\
MulOp {/ * /} \\
AddOp {/ + -} \\
Number {x {? Sign} {+ Digit}} \\
Sign {/ + -} \\
Digit {/ 0 1 2 3 4 5 6 7 8 9} \\
} \\
{Expression value Factor value \\
Term value MulOp value \\
AddOp value Number value \\
Sign value Digit value \\
}
Expression
A possible one, because the order of the nonterminals in the dictionary is not
relevant.
- pegName deserialize serialization
- This is the complement to serialize. It replaces the grammar
definition in pegName with the grammar described by the
serialization value. The old contents of pegName are deleted
by this operation.
- pegName is valid
- A predicate. It tests whether the PEG in pegName is valid.
See section TERMS & CONCEPTS for the definition of this grammar
property. The result is a boolean value. It will be set to true if
the PEG has the tested property, and false otherwise.
- pegName start ?pe?
- This method defines the start expression of the grammar. It
replaces the previously defined start expression with the parsing
expression pe. The method fails and throws an error if pe
does not contain a valid parsing expression as specified in the section
PARSING EXPRESSIONS. In that case the existing start expression is
not changed. The method returns the empty string as its result.
If the method is called without an argument it will return the currently
defined start expression.
- pegName nonterminals
- Returns the set of all nonterminal symbols known to the grammar.
- pegName nonterminal add nt pe
- This method adds the nonterminal nt and its associated parsing
expression pe to the set of nonterminal symbols and rules of the
PEG contained in the object pegName. The method fails and throws an
error if either the string nt is already known as a symbol of the
grammar, or if pe does not contain a valid parsing expression as
specified in the section PARSING EXPRESSIONS. In that case the
current set of nonterminal symbols and rules is not changed. The method
returns the empty string as its result.
- pegName nonterminal delete nt1 ?nt2 ...?
- This method removes the named symbols nt1, nt2 from the set
of nonterminal symbols of the PEG contained in the object pegName.
The method fails and throws an error if any of the strings is not known as
a nonterminal symbol. In that case the current set of nonterminal symbols
is not changed. The method returns the empty string as its result.
The stored grammar becomes invalid if the deleted nonterminals are
referenced by the RHS of still-known rules.
- pegName nonterminal exists nt
- A predicate. It tests whether the nonterminal symbol nt is known to
the PEG in pegName. The result is a boolean value. It will be set
to true if the symbol nt is known, and false
otherwise.
- pegName nonterminal rename nt ntnew
- This method renames the nonterminal symbol nt to ntnew. The
method fails and throws an error if either nt is not known as a
nonterminal, or if ntnew is a known symbol. The method returns the
empty string as its result.
- pegName nonterminal mode nt ?mode?
- This mode returns or sets the semantic mode associated with the
nonterminal symbol nt. If no mode is specified the current
mode of the nonterminal is returned. Otherwise the current mode is set to
mode. The method fails and throws an error if nt is not
known as a nonterminal. The grammar interpreter implemented by the package
grammar::peg::interpreter recognizes the following modes:
- value
- The semantic value of the nonterminal is the abstract syntax tree created
from the AST's of the RHS and a node for the nonterminal itself.
- match
- The semantic value of the nonterminal is an the abstract syntax tree
consisting of single a node for the string matched by the RHS. The ASTs
generated by the RHS are discarded.
- leaf
- The semantic value of the nonterminal is an the abstract syntax tree
consisting of single a node for the nonterminal itself. The ASTs generated
by the RHS are discarded.
- discard
- The nonterminal has no semantic value. The ASTs generated by the RHS are
discarded (as well).
- pegName nonterminal rule nt
- This method returns the parsing expression associated with the nonterminal
nt. The method fails and throws an error if nt is not known
as a nonterminal.
- pegName unknown nonterminals
- This method returns a list containing the names of all nonterminal symbols
which are referenced on the RHS of a grammatical rule, but have no rule
definining their structure. In other words, a list of the nonterminal
symbols which make the grammar invalid. The grammar is valid if this list
is empty.
PARSING EXPRESSIONS¶
Various methods of PEG container objects expect a parsing expression as their
argument, or will return such. This section specifies the format such parsing
expressions are in.
- [1]
- The string epsilon is an atomic parsing expression. It matches the
empty string.
- [2]
- The string alnum is an atomic parsing expression. It matches any
alphanumeric character.
- [3]
- The string alpha is an atomic parsing expression. It matches any
alphabetical character.
- [4]
- The string dot is an atomic parsing expression. It matches any
character.
- [5]
- The expression [list t x] is an atomic parsing expression. It
matches the terminal string x.
- [6]
- The expression [list n A] is an atomic parsing expression. It
matches the nonterminal A.
- [7]
- For parsing expressions e1, e2, ... the result of [list /
e1 e2 ... ] is a parsing expression as well. This is the
ordered choice, aka prioritized choice.
- [8]
- For parsing expressions e1, e2, ... the result of [list x
e1 e2 ... ] is a parsing expression as well. This is the
sequence.
- [9]
- For a parsing expression e the result of [list * e] is a
parsing expression as well. This is the kleene closure, describing
zero or more repetitions.
- [10]
- For a parsing expression e the result of [list + e] is a
parsing expression as well. This is the positive kleene closure,
describing one or more repetitions.
- [11]
- For a parsing expression e the result of [list & e] is a
parsing expression as well. This is the and lookahead
predicate.
- [12]
- For a parsing expression e the result of [list ! e] is a
parsing expression as well. This is the not lookahead
predicate.
- [13]
- For a parsing expression e the result of [list ? e] is a
parsing expression as well. This is the optional input.
Examples of parsing expressions where already shown, in the description of the
method
serialize.
PARSING EXPRESSION GRAMMARS¶
For the mathematically inclined, a PEG is a 4-tuple (VN,VT,R,eS) where
- •
- VN is a set of nonterminal symbols,
- •
- VT is a set of terminal symbols,
- •
- R is a finite set of rules, where each rule is a pair (A,e), A in VN, and
e a parsing expression.
- •
- eS is a parsing expression, the start expression.
Further constraints are
- •
- The intersection of VN and VT is empty.
- •
- For all A in VT exists exactly one pair (A,e) in R. In other words, R is a
function from nonterminal symbols to parsing expressions.
Parsing expression are inductively defined via
- •
- The empty string (epsilon) is a parsing expression.
- •
- A terminal symbol a is a parsing expression.
- •
- A nonterminal symbol A is a parsing expression.
- •
- e1e2 is a parsing expression for parsing expressions
e1 and 2. This is called sequence.
- •
- e1/e2 is a parsing expression for parsing expressions
e1 and 2. This is called ordered choice.
- •
- e* is a parsing expression for parsing expression e. This is
called zero-or-more repetitions, also known as kleene
closure.
- •
- e+ is a parsing expression for parsing expression e. This is
called one-or-more repetitions, also known as positive kleene
closure.
- •
- !e is a parsing expression for parsing expression e1. This
is called a not lookahead predicate.
- •
- &e is a parsing expression for parsing expression e1.
This is called an and lookahead predicate.
PEGs are used to define a grammatical structure for streams of symbols over VT.
They are a modern phrasing of older formalisms invented by Alexander Birham.
These formalisms were called TS (TMG recognition scheme), and gTS (generalized
TS). Later they were renamed to TPDL (Top-Down Parsing Languages) and gTPDL
(generalized TPDL).
They can be easily implemented by recursive descent parsers with backtracking.
This makes them relatives of LL(k) Context-Free Grammars.
REFERENCES¶
- [1]
- The Packrat Parsing and Parsing Expression Grammars Page
[http://www.pdos.lcs.mit.edu/~baford/packrat/], by Bryan Ford,
Massachusetts Institute of Technology. This is the main entry page to
PEGs, and their realization through Packrat Parsers.
- [2]
- Parsing Techniques - A Practical Guide
[http://www.cs.vu.nl/~dick/PTAPG.html], an online book offering a clear,
accessible, and thorough discussion of many different parsing techniques
with their interrelations and applicabilities, including error recovery
techniques.
- [3]
- Compilers and Compiler Generators
[http://scifac.ru.ac.za/compilers/], an online book using CoCo/R, a
generator for recursive descent parsers.
BUGS, IDEAS, FEEDBACK¶
This document, and the package it describes, will undoubtedly contain bugs and
other problems. Please report such in the category
grammar_peg of the
Tcllib Trackers [
http://core.tcl.tk/tcllib/reportlist]. Please also
report any ideas for enhancements you may have for either package and/or
documentation.
KEYWORDS¶
LL(k), TDPL, context-free languages, expression, grammar, parsing, parsing
expression, parsing expression grammar, push down automaton, recursive
descent, state, top-down parsing languages, transducer
CATEGORY¶
Grammars and finite automata
COPYRIGHT¶
Copyright (c) 2005 Andreas Kupries <andreas_kupries@users.sourceforge.net>