PDA and Context-free Languages
Every context-free grammar can be transformed into an equivalent pushdown automaton. The derivation process of the grammar is simulated in a leftmost way. Where the grammar rewrites a nonterminal, the PDA takes the topmost nonterminal from its stack and replaces it by the right-hand part of a grammatical rule (expand). Where the grammar generates a terminal symbol, the PDA reads a symbol from input when it is the topmost symbol on the stack (match). In a sense the stack of the PDA contains the unprocessed data of the grammar, corresponding to a pre-order traversal of a derivation tree.
Technically, given a context-free grammar, the PDA is constructed as follows.
- for each rule (expand)
- for each terminal symbol (match)
As a result we obtain a single state pushdown automaton, the state here is, accepting the context-free language by empty stack. Its initial stack symbol equals the axiom of the context-free grammar.
The converse, finding a grammar for a given PDA, is not that easy. The trick is to code two states of the PDA into the nonterminals of the grammar.
Theorem. For each pushdown automaton one may construct a context-free grammar such that .
Read more about this topic: Pushdown Automaton
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