Higher-order term indexing using substitution trees

We present a higher-order term indexing strategy based on substitution trees for simply typed lambda-terms. There are mainly two problems in adapting first-order indexing techniques. First, many operations used in building an efficient term index and retrieving a set of candidate terms from a large collection are undecidable in general for higher-order terms. Second, the scoping of variables and binders in the higher-order case presents challenges. The approach taken in this article is to reduce the problem to indexing linear higher-order patterns, a decidable fragment of higher-order terms, and delay solving terms outside of this fragment. We present insertion of terms into the index based on computing the most specific linear generalization of two linear higher-order patterns, and retrieval based on matching two linear higher-order patterns. Our theoretical framework maintains that terms are in βη-normal form, thereby eliminating the need to renormalize and raise terms during insertion and retrieval. Finally, we prove correctness of our presented algorithms. This indexing structure is implemented as part of the Twelf system to speed up the execution of the tabled higher-logic programming interpreter.