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Safety is Not a Restriction at Level 2 for String Languages ?

Summary: Safety is Not a Restriction at Level 2
for String Languages ?
K. Aehlig ?? , J. G. de Miranda, and C.-H. L. Ong
Oxford University Computing Laboratory
Abstract. Recent work by Knapik, Niwinski and Urzyczyn (in FOS-
SACS 2002) has revived interest in the connexions between higher-order
grammars and higher-order pushdown automata. Both devices can be
viewed as de nitions 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.
1 Introduction
Higher-order pushdown automata and higher-order grammars were originally in-
troduced as de nitional devices for string languages by Maslov [10] and Damm
[4] respectively. Damm de ned an in nite hierarchy of languages, the OI Hi-
erarchy, the nth level of which is generated by level-n grammars that satisfy
a syntactic constraint called safety 1 . Similarly, Maslov de ned an in nite hier-


Source: Aehlig, Klaus T. - Institut für Informatik, Ludwig-Maximilians-Universität München


Collections: Mathematics; Computer Technologies and Information Sciences