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:

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:

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.