Complexity characterizations of attribute grammar languages

Attribute grammars were introduced by Knuth as a mechanism for specifying the semantics of context-free languages (Knuth, 1968). Almost two decades after their introduction, they remain one of the main techniques for specifying semantics of programming languages, and have recently been used in new and quite diverse domains. An attribute grammar is a context-free grammar, in which each nonterminal has been endowed with certain attributes which can take values. Each production is enriched with functions whereby one can compute the values of attributes of nonterminals involved in the production in terms of the values of other such attributes. Also, there may be a predicate associated with a production, specifying that a particular relation must hold between the values of the various attributes. Attribute grammers are usually viewed as translations from strings in the underlying context-free language to attribute values (in the programming language application, from programs to executable code). Recently, it was shown (Engelfriet, 1986) that the ranges of such mappings constitute all languages log-tape reducible to the context-free languages, if we assume that the attribute computations involve only string concatenation (as it is natural to do in the context of compilation). However, it is also useful and instructive to study attribute grammars as language generators, since in many applications of attribute grammars, parsing is the main interest. An atribute grammar AG generates the language consisting of all strings that have a legal parse tree in AG (that is, a parse tree in which all attribute values relate in the prescribed way). Because of the predicates, a parse tree of the original context-free grammar may no longer be a legal parse tree of the attribute grammar, and thus the language accepted by an attribute grammar is in general a subset of the corresponding context-free language. It is clear that any context-free language can be generated by an attribute