++ed by:

4 PAUSE users
1 non-PAUSE user.

Author image Jeffrey Kegler
and 1 contributors


Marpa::Parse_Terms - Standard Parsing Terms used in the Marpa Documents


This document is intended as a reminder of the standard vocabulary of parsing. I put defining uses of terms in boldface, for easy skimming. A reader who feels comfortable with parsing terminology can skip this document entirely. A reader completely new to parsing will find this document too terse and should 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.

All the definitions here are consistent with at least some of the textbook definitions, and are in that sense standard. But no effort is made to cover the full range of standard meaning, or even to give the most common meaning. The focus is on the meaning of the terms as used in the Marpa documentation.

Basic terms

A grammar is a set of rules. The rules describe a set of strings of symbols. A string of symbols is often called a symbol string. The rules of a grammar are often called productions.

Stages of Parsing

A recognizer is a program that determines whether its input is one of the symbol strings in the set described by the rules of a grammar. A parser is a program which finds the structure of the input according to the rules of a grammar.

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

Parsers often use a lexical analyzer to convert raw input, usually input text, into a series of tokens. Each token represents a symbol of the grammar and has a value. The series of symbols represented by the series of tokens becomes the symbol string input seen by the recognizer. The symbol string input is also called the input sentence. A lexical analyzer is often called a lexer or a scanner, and lexical analysis is often called lexing or scanning.


A standard way of describing rules is Backus-Naur Form, or BNF. 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 left hand side and a right hand side. In a 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, 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 rhs and lhs. If the rhs of a production has no symbols, the production is called an empty production or an empty rule.

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


A step of a derivation, or derivation step, is made by taking a symbol string and any production in the grammar whose lhs occurs in that symbol string, and replacing any occurrence of the lhs symbol in the symbol string with 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" derives the symbol string "A B C D".

A derivation is a sequence of derivation steps. The length of a derivation is its length in steps. A symbol string directly derives another if and only if there is a derivation of length 1 from the first symbol string to the second string. Every symbol string is said to derive itself in a derivation of length 0. Such a zero length derivation is a trivial derivation.

If a derivation is not trivial or direct, that is, if it has more than one step, then it is an indirect derivation. A derivation which is not trivial (that is, a derivation which has one or more steps) is a non-trivial derivation.

Where the symbol string beginning a derivation consists of a single symbol, we often say that symbol produces the symbol string which results from the derivation. We say that the beginning symbol trivially, non-trivially, directly or indirectly produces the symbol string if the length of the derivation is respectively, 0, greater than 0, 1, or greater than 1, just as we do when we say a symbol string derives another symbol string. When a symbol produces or derives a symbol string, we also say that the symbol matches the symbol string, or that the symbol string matches the symbol.

In any parse, one symbol is distinguished as the start symbol. The parse of an input is successful if and only if the start symbol produces the input sentence according to the grammar.


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

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

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

Other Special Cases

If any derivation from the start symbol uses a rule, that rule is called reachable or accessible. A rule that is not accessible is called unreachable or inaccessible. If any derivation which results in a sentence uses a rule, that rule is said to be productive. A rule that is not productive is called unproductive. A simple case of an unproductive rule is one whose rhs contains a symbol which is not a terminal and not on the lhs of any other rule. A rule which is inaccessible or unproductive is called a useless rule.

A symbol is reachable if it appears in a reachable production. A symbol is productive if it appears on the lhs of a productive rule, or if it is a nullable symbol. If a symbol is not reachable or not accessible, it is unreachable or inaccessible. If a symbol is not productive, it is unproductive.

If any symbol in the grammar non-trivially produces a symbol string containing itself, the grammar is said to be recursive. If any symbol non-trivially produces a symbol string with itself on the left, the grammar is said to be left-recursive. If any symbol non-trivially produces a symbol string with itself on the right, the grammar is said to be right-recursive.

A cycle is a non-trivial derivation from any symbol to the string of symbols which contains only itself. Since the result of a cycle is exactly the same as the beginning of a cycle, a cycle is the parsing equivalent of an infinite loop. A grammar which contains no cycles is cycle-free.

A grammar which contains a cycle is infinitely ambigious. This is a grammar with a cycle can repeat the cycle in a derivation an arbitrary number of times. Therefore, for some input, there is no limit to the number of parses a grammar with a cycle can produce.

A proper context-free grammar in one which is cycle-free and has no useless rules or empty productions. Marpa does not require the context-free grammars it parses to be proper. Marpa can handles grammars with cycles, useless rules or both.


The structure of a parse can be represented by as a series of derivation steps from the start symbol to the input. Another way to represent structure is as a parse tree. Every symbol used in the parse is represented by a node of the parse tree. Wherever a production is used in the parse, its lhs is represented by a parent node and the rhs symbols are represented by child nodes. The start symbol is the root of the tree. Terminals are leaf nodes. Symbols on the left hand side of empty productions are also leaf nodes. If a node is not a leaf node, it is an inner node. A nulled node is one that represents a nulled symbol.

If, for a given grammar and a given input, more than one derivation tree is possible, we say that parse is ambiguous. If any parse from a grammar is ambiguous, the grammar is ambiguous.

The node at the root of the tree is the start node. Any node with a symbol being used as a terminal is a terminal node. The depth of a node is its distance from the root. The root node is at depth 0. If a parent node is at depth n, its children are at depth n+1.

As with depths in bodies of water, the concepts of higher and lower have the sense opposite to that of the numerical values. The node at depth 0 is higher than any other node. Parent nodes are higher than their children.


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

The tree representation is especially useful when evaluating a parse. The rest of this section describes the traditional textbook method of evaluating parse trees. In practice trees can be transformed and evaluated in many ways, but the traditional method is very much in use and in practical applications the traditional method is usually the first considered, if not the one ultimately used.

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 2007-2009 Jeffrey Kegler, all rights reserved. Marpa is free software under the Perl license. For details see the LICENSE file in the Marpa distribution.