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Store the start nonterminal symbol on a stack. An epsilon transition nondeterministically replaces its top nonterminal symbol by a production rule right-hand side. A top terminal symbol can be removed only while consuming the same input symbol. With a protected stack bottom marker, enter a final control state when that marker is exposed. Requiring all input to have been read implements acceptance by final state. The stack records the unprocessed suffix of a leftmost derivation, proving equality of the recognized and generated formal languages.

A grammar derivation that always expands the leftmost remaining nonterminal symbol. In a context-free grammar, each parse tree specifies exactly one leftmost derivation. Two distinct leftmost derivations of the same word therefore prove that the context-free grammar is an ambiguous grammar.

Use the standard conversion of a context-free grammar to a pushdown automaton, implementing a leftmost derivation on a stack. Take control states $q_s,q,q_f$, input alphabet $\{a,b\}$, stack alphabet $\{S,T,U,a,b,\perp \}$, initial control state $q_s$, initial stack symbol $\perp$, and sole final control state $q_f$. The symbol $\perp$ is a fresh stack bottom marker. As in part (a), the top of the stack is written on the left.

Initialize withIn control state $q$, replace a top nonterminal symbol by one of its production rule right-hand sides:The replacements are written in the order in which they are to be processed. For example, replacing $S$ by $UST$ leaves $U$ on top; an implementation using individual push operations pushes $T$, then $S$, then $U$.

Match and pop top terminal symbols byFinally includeAll unlisted transitions have no choices; $q_f$ has no outgoing transitions. This is acceptance by final state, leaving the bottom marker on the stack. Taking the last transition too early cannot cause false acceptance, because an unread input suffix remains and $q_f$ cannot consume it.

For correctness, after initialization the stack contains the unprocessed part of a sentential form. Expanding a top nonterminal symbol simulates one step of a leftmost derivation, and popping a matching terminal symbol transfers it to the already-read input prefix. Thus if the input prefix consumed so far is $u$ and the stack above $\perp$ is $\alpha$, then $S{\Rightarrow }_G^{\ast }u\alpha$. Acceptance gives $S{\Rightarrow }_G^{\ast }w$ for the entire input $w$.

Conversely, any leftmost derivation of $w$ specifies choices of these transitions: expand the next nonterminal symbol, and read each resulting leading terminal symbol when it reaches the top. After all symbols of $w$ are consumed, only $\perp$ remains, enabling the move to $q_f$. The nondeterministic pushdown automaton therefore recognizes exactly $L(G)$. The empty word is accepted by choosing $S\to \epsilon$ after initialization.

The word $abb$ has these two leftmost derivations:In the first grammar derivation, $U\to aT$ supplies an extra $b$; the remaining $S$ goes directly to the empty word. In the second grammar derivation, $U\to a$ is used, and the remaining $S$ uses $S\to ST$ before disappearing. Every step expands the leftmost nonterminal symbol, so these are genuinely different leftmost derivations, with different parse trees already distinguished by the production rule at the $U$ node.

A context-free grammar is an ambiguous grammar if some word in its generated formal language has two distinct parse trees. Equivalently, that word has two distinct leftmost derivations from the start nonterminal symbol. A leftmost derivation always expands the leftmost remaining nonterminal symbol.

Two arbitrary grammar derivations of the same word do not by themselves prove ambiguity: expanding independent nonterminal symbols in a different order can describe the same parse tree. Requiring distinct parse trees, or distinct leftmost derivations, avoids that issue.