Theoretical Computer Science Elsevier Complexity in Left-associative Grammar

This paper presents a mathematical deenition of Left-Associative Grammar , and describes its formal properties. 1 Conceptually, LA-grammar is based on the notion of possible continuations, in contrast to more traditional systems such as Phrase Structure Grammar and Categorial Grammar , which are linguistically motivated in terms of possible substitutions. It is shown that LA-grammar generates all and only the recursive languages. The Chomsky hierarchy of regular, context-free, and context-sensitive languages is reconstructed in LA-grammar by simulating nite state automata, push-down automata, and linearly bounded automata, respectively. Using alternative restrictions on LA-grammars, the new language hierarchy of A-LAGs, B-LAGs, C-LAGs is proposed. The class of C-LAGs is divided into three subclasses representing diierent degrees of ambiguity and associated computational complexity. The class of C-LAGs without recursive ambiguities (called the C1-LAGs) parses in linear time, and includes all deterministic CF-languages, plus CF-languages with non-recursive ambiguities, e.g., a n b n c m d m a n b m c m d n , plus many context-sensitive languages, such as a n b n c n , a n b n c n d n e n , fa n b n c n g , a 2 i , a k b m c km and a i!. The class of C-LAGs with recursive \single re-turn" ambiguities (called C2-LAGs) parses in n 2 , and includes certain non-deterministic CF-languages such as WW R , plus context-sensitive languages like WW, WWW, WWWWW, and fWWWg. Finally, the class of unrestricted C-LAGs (called C3-LAGs) parses in exponential time and contains CF-languages like Lno and the \hardest context-free language" HCFL, plus context-sensitive languages like NP-complete Subset Sum and SAT.