On the space complexity of turn bounded pushdown automata

A pushdown automaton is said to make a turn at a given instant if it changes at that instant from stack increasing to stack decreasing. Let NPDA-TURN(f(n)) and DPDA-TURN(f(n)) denote the classes of languages accepted by nondeterministic and deterministic pushdown automata respectively that make at most f(n) turns for any input of length n. In this paper the following inclusions that express the space complexity of turn bounded pushdown automata are given: DPDA-TURN(f(n)) ⊆ DSPACE(log f(n) log n), and NPDA-TURN(f(n)) ⊆ NSPACE (log f(n) log n). In particular, it follows that finite-turn pushdown automata are logarithmic space bounded: DPDA-TURN(O(1)) ⊆ DL and NPDA-TURN(O(1)) ⊆ NL, from which two corollaries follow: one is that the class of metalinear context-free languages is complete for NL, and the other is that a more tight inclusion NPDA-TURN(f(n)) ⊆ DSPACE(log² f(n) log n) cannot be derived unless DL = NL, though NPDA-TURN (f(n)) ⊆ DSPACE(log² f(n) log² n) holds.