# Copyright 2022 Jeffrey Kegler
# This file is part of Marpa::R2.  Marpa::R2 is free software: you can
# redistribute it and/or modify it under the terms of the GNU Lesser
# General Public License as published by the Free Software Foundation,
# either version 3 of the License, or (at your option) any later version.
#
# Marpa::R2 is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser
# General Public License along with Marpa::R2.  If not, see
# http://www.gnu.org/licenses/.

=head1 NAME

Marpa::R2::Vocabulary - Standard parsing terms as used within Marpa

=head1 Description

The definitions in this document are of standard parsing
terms as they are used
in the Marpa context.
I put B<defining uses> of terms in boldface, for easy skimming.
Where a narrow
or specialized sense of the term is the one that applies
within Marpa, that is the only definition given.
Marpa also sometimes uses a standard term
with a definition which is slightly different from
the standard one.
("Ambiguous grammar" is one example, and "grammar" itself is
another.)
When this is the case, it is explicitly pointed out.

A reader totally new to parsing will find this document too
terse to act as a textbook or tutorial.
They will need to look elsewhere first.
As an introduction, I recommend
L<Mark Jason Dominus's
excellent chapter on parsing in the Perl context|Marpa::R2::Advanced::Bibliography/"Dominus 2005">.
It's available on-line.
L<Wikipedia|Marpa::R2::Advanced::Bibliography/"Wikipedia"> is also an excellent place to start.

=head2 Basic

A B<grammar> is a set of rules,
associated with a set of symbols,
one of which is distinguished as the start symbol.
A B<symbol string>, or simply B<string> where the meaning is clear,
is an ordered series of symbols.
The B<length> of a string is the number of symbols in it.

A B<language> is a set of B<symbol strings>.
A grammar defines a B<language>,
as will be described later.

It is important to note that the term language, as it is
used in parsing theory, means something very different
from what it means in ordinary use.
The meaning of the strings is an essential part
of the ordinary idea of what a language is.
In ordinary use, the word "language" means
far more than a unordered list of its sentences.
In parsing terminology, meaning (or B<semantics> as it is
called) is a separate issue.
For parsing theory a
language is exactly a set of strings --
that and nothing more.

The Marpa definition of a grammar differs slightly from the various
standard ones.  Standard definitions usually sharply distinguish terminal symbols
from non-terminals.  Marpa does not.

=head2 Stages of parsing

A B<recognizer> is a program that determines whether its B<input>
is in the language of a grammar and a start symbol.
A B<parser> is a program which finds the structure of that input.

The term B<parsing> is used in a strict and a loose sense.
B<Parsing in the loose sense> is all phases of finding a grammar's structure,
including a separate recognition phase if the parser has one.  (Marpa does.)
If a parser has phases,
B<parsing in the strict sense> refers specifically to the phase that finds the structure of the input.
When the Marpa documents use the term B<parsing> in its strict sense, they will
speak explicitly of "parsing in the strict sense".
Otherwise, B<parsing> will mean parsing in the loose sense.

Parsers often use a
B<lexical analyzer> to convert B<raw input>,
usually B<input text>,
into a B<token stream>,
which is a series of B<tokens>.
Each token represents a B<symbol> of the grammar and has a B<value>.
A lexical analyzer is often called a B<lexer> or a B<scanner>,
and B<lexical analysis> is often called B<lexing> or B<scanning>.

The series of symbols represented by the series of tokens
becomes the B<symbol string input>
seen by the recognizer.
The B<symbol string input> is more often called the B<input sentence>.

By default, Marpa uses the token stream model of input.
Marpa also allows alternative input models.
These are new to Marpa, so that their terminology is
of necessity non-standard.
The terminology needed for alternative input models
is explained in L<the document that introduces
them|Marpa::R2::Advanced::Models>.

=head2 Rules

A standard way of describing rules is Backus-Naur Form, or B<BNF>.
A rule of a grammar is also often called a B<production>.
In one common way of writing BNF, a production looks like this:

=for Marpa::R2::Display:
ignore: 1

    Expression ::= Term Factor

=for Marpa::R2::Display::End

In the production above, C<Expression>, C<Term> and C<Factor> are symbols.
A production consists of a B<left hand side> and a B<right hand side>.
In a B<context-free grammar>,
like those Marpa parses,
the left hand side of a production
is always a symbol string of length 1.
The right hand side of a production is a symbol string of zero or more symbols.
In the example, C<Expression> is the left hand side, and
C<Term> and C<Factor> are right hand side symbols.

Left hand side and right hand side are often abbreviated as B<RHS> and B<LHS>.
If the RHS of a production has no symbols,
the production is called an B<empty production>
or an B<empty rule>.

Any symbol which is allowed to occur
in the symbol string input is called a B<terminal> symbol.
If the symbols in a symbol string are all terminals,
that symbol string is also called a B<sentence>.

=head2 Derivations

A B<step> of a derivation, or B<derivation step>, is a change made to a symbol string
by applying one of the productions from the grammar.
The production must be one of those with a LHS that occurs in the symbol string.
The result of the derivation step is another symbol string,
one in which every occurence of the LHS symbol from the production is
replaced by the RHS of the production.
For example, if C<A>, C<B>, C<C>, C<D>, and C<X> are symbols,
and

=for Marpa::R2::Display:
ignore: 1

    X ::= B C

=for Marpa::R2::Display::End

is a production, then

=for Marpa::R2::Display:
ignore: 1

    A X D -> A B C D

=for Marpa::R2::Display::End

is a derivation step, with "C<A X D>" as its beginning and "C<A B C D>" as its end or result.
We say that the symbol string "C<A X D>"
B<derives> the symbol string
"C<A B C D>".

A B<derivation> is a sequence of derivation steps.
The B<length> of a derivation is its length in steps.

=over

=item *
We say that
a first symbol string B<directly
derives> a second symbol string if and only if there is a derivation of length 1 from the first symbol
string to the second symbol string.

=item *
Every symbol string is said to derive itself in a derivation
of length 0.  A zero length derivation is a B<trivial derivation>.

=item * A derivation which is not trivial
(that is,
a derivation which has one or more steps)
is a B<non-trivial> derivation.

=item *
A non-trivial derivation of a symbol string from itself
is called a B<cycle>.
Grammars which contain cycles are traditionally considered useless,
but Marpa will parse with such grammars.

=item * If a derivation is not trivial or direct, that is, if it has more than one step,
then it is an B<indirect> derivation.

=back

Technically, a symbol C<X> and a string
that consists of only that symbol
are two different things.
But we often say "the symbol C<X>"
as shorthand for "the string
of length 1
whose only symbol is C<X>".
For example, if the string containing only the
symbol C<X> derives a string C<Y>,
we will usually say simply that "C<X> derives C<Y>".

Wherever symbol or string C<X> derives C<Y>,
we may also say C<X> B<produces> C<Y>.
Derivations are often described as symbol matches.
Wherever symbol or string C<X> derives C<Y>,
we may also say that C<Y> B<matches> C<X>
or that C<X> B<matches> C<Y>.
It is particularly common to say that
C<X> matches C<Y> when C<X> or C<Y> is a sentence.

The parse of an input by a grammar is B<successful>
if and only if,
according to the grammar,
the start symbol produces the input sentence.
The set of all input sentences that
a grammar
will successfully parse is
the B<language> of
the grammar.

=head2 Nulling

The zero length symbol string is called the B<empty string>.
The empty string can be considered to be a sentence, in which
case it is the B<empty sentence>.
A string of one or more symbols is B<non-empty>.
A derivation which produces the empty string is a B<null derivation>.
A derivation from the start symbol which produces the empty string
is a B<null parse>.

If in a particular grammar, a symbol has a null derivation,
it is a B<nullable symbol>.
If, in a particular grammar,
the only sentence produced by a symbol is the empty sentence,
it is a B<nulling symbol>.
All nulling symbols are nullable symbols.

If a symbol is not nullable, it is B<non-nullable>.
If a symbol is not nulling, it is B<non-nulling>.
In any instance where a symbol produces the empty string,
it is said to be B<nulled>,
or to be a B<null symbol>.

=head2 Useless rules

If any derivation from the start symbol uses a rule,
that rule is called B<reachable> or B<accessible>.
A rule that is not accessible
is called B<unreachable> or B<inaccessible>.
If any derivation which results in a sentence uses a rule,
that rule is said to be B<productive>.
A rule that is not productive is called B<unproductive>.
For example, a rule is unproductive unless every symbol on
its RHS either is a terminal or is the LHS of some other rule.
A rule which is inaccessible or unproductive is called a
B<useless> rule.
Marpa can handle grammars with useless rules.

A symbol is B<reachable> or B<accessible> if it appears in a reachable production.
If a symbol is not reachable, it is B<unreachable> or B<inaccessible>.
A symbol is B<productive> if it appears on the LHS of a productive rule,
or if it is a nullable symbol.
If a symbol is not productive, it is B<unproductive>.
A symbol which is inaccessible or unproductive is called a
B<useless> symbol.
Marpa can handle grammars with useless symbols.

=head2 Recursion and cycles

If any symbol in the grammar non-trivially produces a symbol string containing itself,
the grammar is said to be B<recursive>.
If any symbol non-trivially produces a symbol string with itself on the left,
the grammar is said to be B<left-recursive>.
If any symbol non-trivially produces a symbol string with itself on the right,
the grammar is said to be B<right-recursive>.
Marpa can handle all recursive grammars,
including
grammars which are left-recursive,
grammars which are right-recursive,
and grammars
which contain both left- and right-recursion.

A B<cycle> is a non-trivial derivation
of a string of symbols from itself.
If it is not possible for any derivation using a grammar
to contain a cycle,
then that grammar is said to be B<cycle-free>.
Traditionally, a grammar is
considered useless
if it is not cycle-free.

The traditional deprecation of cycles is well-founded.
A cycle is
the parsing equivalent of an infinite loop.
Once a cycle appears, it can be
repeated over and over again.
Even a very short input sentence can have an
infinite number of parses when the grammar is not
cycle-free.

For that reason,
a grammar which contains a cycle is also
called B<infinitely ambiguous>.
Marpa can parse with grammars which are not
cycle-free,
and will
even parse inputs that cause cycles.
When a parse is infinitely ambiguous, Marpa
limits cycles to a single loop,
so that only a finite number of parses is returned.

=head2 Parse structure

The structure of a parse can be represented as a series of derivation steps from
the start symbol to the input.
Another way to represent structure is as a B<parse tree>.
Every symbol used in the parse is
represented by a B<node> of the parse tree.
Wherever a production is used in the parse,
its LHS is represented by a B<parent node>
and the RHS symbols are represented by B<child nodes>.
The start symbol is the B<root> of the tree.
The node at the root of the tree is called the B<start node>.

Traditionally,
grammars divide all symbols sharply into terminals
and non-terminals.
A terminal symbol must ALWAYS be used as a terminal.
A non-terminal symbol can NEVER be used as a terminal.

Marpa's use of terminals is non-traditional,
and its terminology is different accordingly.
As in the traditional approach,
Marpa's non-terminals can never be used as terminals.
But Marpa terminals can be used anywhere,
even in places where
the traditional approach requires a
a non-terminal symbol.
In particular, a Marpa terminal can be the LHS
of a rule.

Traditionally, and in Marpa as well,
every node is either a
B<inner node> or a
B<leaf node>.
In Marpa,
B<leaf nodes> are of
two kinds:

=over

=item *

Nodes for nulled symbols.
A node for a nulled symbol is called
a B<nulled node>.

=item *

Nodes for symbols being used as terminals.

=back

=head2 Ambiguity

Marpa allows ambiguous grammars.
Traditionally
we say that a parse is B<ambiguous> if,
for a given grammar and a given input,
more than one derivation tree is possible.
However, Marpa allows ambiguous input tokens,
which the traditional definition does not take
into account.
If Marpa used
the traditional definition,
all grammars would be
ambiguous except those grammars which allowed only
the null parse.

It is easiest if the Marpa definition and the
traditional definition were extensionally
equivalent ---
that is, if Marpa's set of ambiguous grammars
was exactly the same as the set of traditionally
ambiguous grammars.
This can be accomplished by using a slightly
altered definition.
In the Marpa context, a grammar
is B<ambiguous> if and only if,
for some UNAMBIGUOUS stream of input tokens,
that grammar produces
more than one parse tree.

=head2 Semantics

In real life, the structure of a parse is usually a means to an end.
Grammars usually have a B<semantics> associated with them,
and what the user actually wants is the B<value> of the parse
according to the semantics.

The tree representation is especially useful when evaluating a parse.
In the traditional method of evaluating a parse tree,
every node which represents a terminal symbol
has a value associated with it on input.
Non-null inner nodes
take their semantics from the production whose LHS they represent.
Nulled nodes are dealt with as special cases.

The semantics for a production
describe how to calculate the value of the node which represents the LHS
(the parent node)
from the values of zero or more of the nodes which represent the RHS symbols
(child nodes).
Values are computed recursively, bottom-up.
The value of a parse is the value of its start symbol.

=head1 Copyright and License

=for Marpa::R2::Display
ignore: 1

  Copyright 2022 Jeffrey Kegler
  This file is part of Marpa::R2.  Marpa::R2 is free software: you can
  redistribute it and/or modify it under the terms of the GNU Lesser
  General Public License as published by the Free Software Foundation,
  either version 3 of the License, or (at your option) any later version.

  Marpa::R2 is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  Lesser General Public License for more details.

  You should have received a copy of the GNU Lesser
  General Public License along with Marpa::R2.  If not, see
  http://www.gnu.org/licenses/.

=for Marpa::R2::Display::End

=cut

