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From CFG to NPDA

For any context-free grammar in Greibach Normal Form we can build an equivalent nondeterministic pushdown automaton. This establishes that an npda is at least as powerful as a cfg.

Key idea: Any string of a context-free language has a leftmost derivation. We set up the npda so that the stack contents "correspond" to this sentential form; every move of the npda represents one derivation step.

abba [ABz] The sentential form is:

the characters already read,
PLUS the symbols on the stack
MINUS the final z (the initial stack symbol).

In the npda we will construct, the states are hardly important at all. All the real work is done on the stack. In fact, we will use only the following three states, regardless of the complexity of the grammar:

Start state q₀ just gets things initialized. We use the transition from q₀ to q₁ to put the grammar's start symbol on the stack. (q₀, , z) {(q₁, Sz)}
State q₁ does the bulk of the work. We represent every derivation step as a move from q₁ to q₁.
We use the transition from q₁ to q_f to accept the string. (q₁, , z) {(q_f, z)}