  Constraint logic grammars provide a powerful formalism for expressing complex
logical descriptions of natural language phenomena in exact terms. Describing
some of these phenomena may, however, require some form of graded distinctions
which are not provided by such grammars. Recent approaches to weighted
constraint logic grammars attempt to address this issue by adding numerical
calculation schemata to the deduction scheme of the underlying CLP framework.
Currently, these extralogical extensions are not related to the model-theoretic
counterpart of the operational semantics of CLP, i.e., they do not come with a
formal semantics at all. The aim of this paper is to present a clear formal
semantics for weighted constraint logic grammars, which abstracts away from
specific interpretations of weights, but nevertheless gives insights into the
parsing problem for such weighted grammars. Building on the formalization of
constraint logic grammars in the CLP scheme of Hoehfeld and Smolka 1988, this
formal semantics will be given by a quantitative version of CLP. Such a
quantitative CLP scheme can also be valuable for CLP tasks independent of
grammars.
