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Manual Reference Pages  -  MARPA::PP::VOCABULARY (3)

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Marpa::PP::Vocabulary - Standard Parsing Terms as Used within Marpa



The definitions in this document are of standard parsing terms as they are used in the Marpa context. I put <B>defining usesB> 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 Mark Jason Dominus’s excellent chapter on parsing in the Perl context. It’s available on-line. Wikipedia is also an excellent place to start.


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

A <B>languageB> is a set of <B>symbol stringsB>. A grammar defines a <B>languageB>, 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>semanticsB> 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. In standard definitions, the start symbol is fixed. In Marpa, it is merely a default.

    Stages of Parsing

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

The term <B>parsingB> is used in a strict and a loose sense. <B>Parsing in the loose senseB> 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 senseB> refers specifically to the phase that finds the structure of the input. When the Marpa documents use the term <B>parsingB> in its strict sense, they will speak explicitly of parsing in the strict sense. Otherwise, <B>parsingB> will mean parsing in the loose sense.

Parsers often use a <B>lexical analyzerB> to convert <B>raw inputB>, usually <B>input textB>, into a <B>token streamB>, which is a series of <B>tokensB>. Each token represents a <B>symbolB> of the grammar and has a <B>valueB>. A lexical analyzer is often called a <B>lexerB> or a <B>scannerB>, and <B>lexical analysisB> is often called <B>lexingB> or <B>scanningB>.

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

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 the document that introduces them.


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

    Expression ::= Term Factor

In the production above, Expression, Term and Factor are symbols. A production consists of a <B>left hand sideB> and a <B>right hand sideB>. In a <B>context-free grammarB>, 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, Expression is the left hand side, and Term and Factor are right hand side symbols.

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

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


A <B>stepB> of a derivation, or <B>derivation stepB>, 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 A, B, C, D, and X are symbols, and

    X ::= B C

is a production, then

    A X D -> A B C D

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

A <B>derivationB> is a sequence of derivation steps. The <B>lengthB> of a derivation is its length in steps.
o We say that a first symbol string <B>directly derivesB> 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.
o Every symbol string is said to derive itself in a derivation of length 0. A zero length derivation is a <B>trivial derivationB>.
o A non-trivial derivation of a symbol string from itself is called a <B>cycleB>. Grammars which contain cycles are traditionally considered useless, but Marpa will parse with such grammars.
o If a derivation is not trivial or direct, that is, if it has more than one step, then it is an <B>indirectB> derivation.
o A derivation which is not trivial (that is, a derivation which has one or more steps) is a <B>non-trivialB> derivation.
Technically, a symbol X and a string that consists of only that symbol are two different things. But we often say "the symbol X as shorthand for the string of length 1 whose only symbol is X". For example, if the string containing only the symbol X derives a string Y, we will usually say simply that "X derives Y".

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

In any parse, one symbol is distinguished as the <B>start symbolB>. The parse of an input is <B>successfulB> if and only if the start symbol produces the input sentence according to the grammar. The set of all input sentences that a grammar and a start symbol will successfully parse is their <B>languageB>. Traditionally, the start symbol is part of a grammar’s definition. The <B>languageB> of a grammar is the language of the grammar and its default start symbol.


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

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

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

    Useless Rules

If any derivation from the start symbol uses a rule, that rule is called <B>reachableB> or <B>accessibleB>. A rule that is not accessible is called <B>unreachableB> or <B>inaccessibleB>. If any derivation which results in a sentence uses a rule, that rule is said to be <B>productiveB>. A rule that is not productive is called <B>unproductiveB>. 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>uselessB> rule. Marpa can handle grammars with useless rules.

A symbol is <B>reachableB> or <B>accessibleB> if it appears in a reachable production. If a symbol is not reachable, it is <B>unreachableB> or <B>inaccessibleB>. A symbol is <B>productiveB> 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>unproductiveB>. A symbol which is inaccessible or unproductive is called a <B>uselessB> symbol. Marpa can handle grammars with useless symbols.

    Recursion and Cycles

If any symbol in the grammar non-trivially produces a symbol string containing itself, the grammar is said to be <B>recursiveB>. If any symbol non-trivially produces a symbol string with itself on the left, the grammar is said to be <B>left-recursiveB>. If any symbol non-trivially produces a symbol string with itself on the right, the grammar is said to be <B>right-recursiveB>. 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>cycleB> 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-freeB>. 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 ambiguousB>. 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.

    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 treeB>. Every symbol used in the parse is represented by a <B>nodeB> of the parse tree. Wherever a production is used in the parse, its LHS is represented by a <B>parent nodeB> and the RHS symbols are represented by <B>child nodesB>. The start symbol is the <B>rootB> of the tree. The node at the root of the tree is called the <B>start nodeB>.

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 nodeB> or a <B>leaf nodeB>. In Marpa, <B>leaf nodesB> are of two kinds:
o Nodes for nulled symbols. A node for a nulled symbol is called a <B>nulled nodeB>.
o Nodes for symbols being used as terminals.


Marpa allows ambiguous grammars. Traditionally we say that a parse is <B>ambiguousB> 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. Another complication is that in Marpa the start symbol of a grammar is not fixed.

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 with an associated start symbol is <B>ambiguousB> if and only if, for some UNAMBIGUOUS stream of input tokens, that grammar with that start symbol produces more than one parse result. A Marpa grammar is <B>ambiguousB> if that Marpa grammar with its default start symbol is ambiguous.


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


  Copyright 2012 Jeffrey Kegler
  This file is part of Marpa::PP.  Marpa::PP 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::PP is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  Lesser General Public License for more details.
  You should have received a copy of the GNU Lesser
  General Public License along with Marpa::PP.  If not, see

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