@INPROCEEDINGS{HanusPrehofer99JFP,
author = "Hanus, M. and Prehofer, C.",
title = "Higher-Order Narrowing with Definitional Trees",
journal = "Journal of Functional Programming",
volume = "9",
number = "1",
pages = "33-75",
year = "1999",
abstract = {
Functional logic languages with a sound and complete operational
semantics are mainly based on an inference rule called narrowing.
Narrowing extends functional evaluation by goal solving capabilities
as in logic programming. Due to the huge search space of simple
narrowing, steadily improved narrowing strategies have been
developed in the past. Needed narrowing is currently the best
narrowing strategy for first-order functional logic programs due to
its optimality properties w.r.t.\ the length of derivations and the
number of computed solutions. In this paper, we extend the needed
narrowing strategy to higher-order functions and $\lambda$-terms as
data structures. By the use of definitional trees, our strategy
computes only independent solutions. Thus, it is the first calculus
for higher-order functional logic programming which provides for
such an optimality result. Since we allow higher-order logical
variables denoting $\lambda$-terms, applications go beyond current
functional and logic programming languages.
We show soundness and completeness of our strategy with respect to
LNT reductions, a particular form of higher-order reductions defined via
definitional trees. A general completeness result is only provided
for terminating rewrite systems due to the lack of an overall
theory of higher-order reduction which is outside the scope
of this paper.
}
}