Simplifying Grammars
Explore methods to simplify context-free grammars by identifying and eliminating useless variables that are unreachable or nonterminating. Learn techniques like back substitution and using reachability graphs to reduce grammar complexity while preserving language generation. This lesson helps you understand how to streamline grammar structures for better modeling of formal languages.
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Reducing the number of variables in a grammar
Under certain circumstances, the number of variables in a CFG can be reduced. Consider the variable in the grammar below:
Since the replacement rules for do not directly refer to itself, we can easily substitute its two rules where appears in the third rule for :
This grammar now has only one variable. This type of back substitution is possible when a rule is not directly recursive. A variable is directly recursive if it appears anywhere in any of its right-hand side rules.
When we convert PDAs to CFGs, some rules generated by the conversion procedure don’t make sense. For example, a variable other than the start variable might never appear on the right-hand side of any rule. Such a variable is unreachable in ...