An aspect of the Generalized Phrase Structure Grammar formalism proposed by Gazdar, et al. is the introduction of the notion of "slashed categories" to handle the parsing of structures, such as relative clauses, which involve unbounded dependencies. This has been implemented in Definite Clause Grammars through the technique of gap threading, in which a difference list of extracted noun phrases (gaps) is maintained. However, this technique is cumbersome, and can result in subtle soundness problems in the implemented grammars. Miller and Pareschi have proposed a method implementing gap threading at the logical level in intuitionistic logic. Unfortunately that implementation itself suffered from serious problems, which the authors recognized. This paper builds on work first presented with Miller in which we developed a filler-gap dependency parser in Girard's linear logic. This implementation suffers from non of the pitfalls of either the tradition implementational, or the intuitionistic one. It serves as further demonstration of the usefulness of sub-structural logic in natural language applications.