Time and tape complexity of pushdown automaton languages

An algorithm is presented which will determine whether any string w in Σ^*, of length n, is contained in a language L ⊆ Σ^* defined by a two-way nondeterministic pushdown automation. This algorithm requires time n³ when implemented on a random access computer. It requires n⁴ time and n² tape when implemented on a multitape Turing machine.

If the pushdown automaton is deterministic, the algorithm requires n² time on a random access computer and n² log n time on a multitape Turing machine.