leftmost derivation
Let be a context-free grammar. Recall that a word is generated by if it is
The second condition simply says that there is a finite sequence of derivation steps, starting from , and ending at :
For each derivation step , there is a production in such that
Note that is a non-terminal (an element of ), and are words over .
Definition. Using the notations above, if is the leftmost non-terminal occurring in , then we say the derivation step is leftmost. Dually, is rightmost if is the rightmost non-terminal occurring in .
Equivalently, is leftmost (or rightmost) if (or ) is a word over .
A derivation is said to be leftmost (or rightmost) if each of its derivation steps is leftmost (or rightmost).
Example. Let be the grammar consisting of as terminals, as non-terminals (with as the starting symbol), and , , , , and as productions. is clearly context-free.
The word can be generated by by the following three derivations:
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1.
,
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2.
,
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3.
.
The second is leftmost, the third is rightmost, and the first is neither.
Note that in every derivation, the first and last derivation steps are always leftmost and rightmost.
Remarks.
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One of the main properties of a context-free grammar is that every word generated by is derivable by a leftmost (correspondingly rightmost) derivation, which can be used to show that for every context-free language , there is a pushdown automaton accepting every word in , and conversely that the set of words accepted by a pushdown automaton is context-free.
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Leftmost derivations may be defined for any arbitrary formal grammar satisfying the condition
(*) no terminal symbols occur on the left side of any production.
Given two words , we define if there is a production such that and such that is a terminal word. By taking the reflexive transitive closure of , we have the leftmost derivation . It can be shown that for any grammar satisfying condition (*), the language consisting of terminal words generated by via leftmost derivations is always context-free.
References
- 1 S. Ginsburg The Mathematical Theory of Context-Free Languages, McGraw-Hill, New York (1966).
- 2 D. C. Kozen Automata and Computability, Springer, New York (1997).
Title | leftmost derivation |
---|---|
Canonical name | LeftmostDerivation |
Date of creation | 2013-03-22 18:54:50 |
Last modified on | 2013-03-22 18:54:50 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 7 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 68Q45 |
Classification | msc 68Q42 |
Defines | rightmost derivation |