Boolean grammars are a promising extension of context-free grammars that supports conjunction and negation. In this paper we give a novel semantics for boolean grammars which applies to all such grammars, independently of their syntax. The key idea of our proposal comes from the area of negation in logic programming, and in particular from the so-called well-founded semantics which is widely accepted in this area to be the ``correct'' approach to negation. We demonstrate that for every boolean grammar there exists a distinguished (three-valued)
language which is a model of the grammar and at the same time the least fixed point of an appropriate operator associated with the grammar. We then demonstrate that every boolean grammar can be transformed into an equivalent (under the new semantics) grammar in normal form. Based on this normal form, we propose an O(n^3)
algorithm for parsing that applies to any such normalized boolean
grammar. In summary, the main contribution of this paper is to
provide a semantics which applies to all boolean grammars
while at the same time retaining the complexity of parsing
associated with this type of grammars.
Key words: boolean grammars, well-founded semantics
