A leftmost derivation and a rightmost derivation are two types of derivations commonly used in the field of computational complexity theory, specifically in the study of context-free grammars and languages. Both types of derivations are used to generate strings in a context-free language by applying production rules.
In a leftmost derivation, the leftmost nonterminal symbol in a sentential form is always chosen for expansion. This means that at each step of the derivation, the leftmost nonterminal symbol is replaced by one of its production rules. The process continues until all nonterminal symbols have been replaced by terminal symbols, resulting in a string in the language. The leftmost derivation can be visualized as a tree, where the root represents the start symbol and each node represents a nonterminal symbol that is expanded.
For example, consider the following context-free grammar:
S -> aSb | ε
Using a leftmost derivation, we can derive the string "aabbb" as follows:
S => aSb => aaSbb => aaaSbbb => aabbb
On the other hand, in a rightmost derivation, the rightmost nonterminal symbol in a sentential form is selected for expansion. This means that at each step, the rightmost nonterminal symbol is replaced by one of its production rules. The process continues until all nonterminal symbols have been replaced by terminal symbols. Similar to the leftmost derivation, the rightmost derivation can also be represented as a tree, where the root represents the start symbol and each node represents a nonterminal symbol that is expanded.
Continuing with the same example grammar, we can derive the string "aabbb" using a rightmost derivation as follows:
S => aSb => aaSbb => aaSbbb => aabbb
As we can see, the resulting string is the same as the one obtained from the leftmost derivation. However, the order in which the nonterminal symbols are expanded differs. In the leftmost derivation, the leftmost nonterminal symbol is expanded first, while in the rightmost derivation, the rightmost nonterminal symbol is expanded first.
The main difference between a leftmost derivation and a rightmost derivation lies in the order in which nonterminal symbols are expanded. A leftmost derivation always chooses the leftmost nonterminal symbol for expansion, while a rightmost derivation chooses the rightmost nonterminal symbol for expansion. Both types of derivations can be represented as trees, with the root representing the start symbol and each node representing a nonterminal symbol that is expanded.
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