Safety Is not a Restriction at Level 2 for String Languages
Recent work by Knapik, Niwiński and Urzyczyn (in FOSSACS 2002) has revived interest in the connexions between higher-order grammars and higher-order pushdown automata. Both devices can be viewed as definitions for term trees as well as string languages. In the latter setting we recall the extensive study by Damm (1982), and Damm and Goerdt (1986). There it was shown that a language is accepted by a level-n pushdown automaton if and only if the language is generated by a safe level-n grammar. We show that at level 2 the safety assumption may be removed. It follows that there are no inherently unsafe string languages at level 2.
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