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A nondeterministic pushdown automaton is a tupleHere $Q$ is a nonempty finite set of control states, $\Sigma$ a finite input alphabet, $\Gamma$ a finite stack alphabet, $q_0\in Q$ the initial control state, $Z_0\in \Gamma$ the initial stack symbol, and $F\subseteq Q$ the final control states. The notation ${\mathcal{P}}_{\mathrm{fin}}$ requires each transition to have only finitely many choices. Since the domain is finite, the whole transition specification is finite.

A pushdown automaton configuration is $(q,w,\alpha )\in Q\times {\Sigma }^{\ast }\times {\Gamma }^{\ast }$, recording the control state, unread input word, and stack contents. Write the top of the stack on the left. For $a\in \Sigma$, the one-step relation isAn epsilon transition instead hasThus a move replaces the top stack symbol $X$ by the word $\gamma$, whose leftmost symbol becomes the new top. It consumes one input symbol or no input symbol. Under this convention there is no move from an empty stack.

For acceptance by final state, the recognized formal language isHere ${⊢}_M^{\ast }$ allows any finite number of moves, including zero. The entire input must have been consumed; the final stack need not be empty. Acceptance requires at least one such run of the nondeterministic pushdown automaton.