Pushdown automaton

A finite control equipped with an unbounded stack. A move reads one input symbol or the empty word, examines the top stack symbol, changes control state and replaces that top by a finite word over the stack alphabet. Nondeterministic choices give finitely many possible moves. With acceptance by final state and the entire input consumed, pushdown automata recognize exactly the context-free languages.

A pushdown automaton accepts a word if some finite run consumes all its input and reaches a designated final control state, regardless of the remaining stack contents. Reaching a final control state while leaving unread input is insufficient.

The control state $q$, unread input word $w$, and stack contents $\alpha$ of a pushdown automaton. A fixed convention must say which end of $\alpha$ is the top. With the top on the left, replacing top $X$ by $\gamma$ changes $X\beta$ into $\gamma \beta$.