Towards a Substitution Tree Based Index for Higher-order Resolution Theorem Provers

Abstract One of the reasons that forward search methods, like resolution, are efficient in practice is their ability to utilize many optimization techniques. One such technique is subsumption and one way of utilizing subsumption efficiently is by indexing terms using substitution trees. In this paper we describe an attempt to extend such indexes for the use of higher-order resolution theorem provers. Our attempt tries to handle two difficulties which arise when extending the indexes to higher-order. The first difficulty is the need for higher-order anti-unification. The second difficulty is the closure of clauses under associativity and commutativity. We present some techniques which attempt to solve these two problems

Towards a Substitution Tree Based Index for Higher-order Resolution Theorem Provers

International audienceOne of the reasons that forward search methods, like resolution, are efficient in practice is their ability to utilize many optimization techniques. One such technique is subsumption and one way of utilizing subsumption efficiently is by indexing terms using substitution trees. In this paper we describe an attempt to extend such indexes for the use of higher-order resolution theorem provers. Our attempt tries to handle two difficulties which arise when extending the indexes to higher-order. The first difficulty is the need for higher-order anti-unification. The second difficulty is the closure of clauses under associativity and com-mutativity. We present some techniques which attempt to solve these two problems

Efficient Data Structures for Automated Theorem Proving in Expressive Higher-Order Logics

Church's Simple Theory of Types (STT), also referred to as classical higher-order logik, is an elegant and expressive formal system built on top of the simply typed λ-calculus. Its mechanisms of explicit binding and quantification over arbitrary sets and functions allow the representation of complex mathematical concepts and formulae in a concise and unambiguous manner. Higher-order automated theorem proving (ATP) has recently made major progress and several sophisticated ATP systems for higher-order logic have been developed, including Satallax, Osabelle/HOL and LEO-II. Still, higher-order theorem proving is not as mature as its first-order counterpart, and robust implementation techniques for efficient data structures are scarce. In this thesis, a higher-order term representation based upon the polymorphically typed λ-calculus is presented. This term representation employs spine notation, explicit substitutions and perfect term sharing for efficient term traversal, fast β-normalization and reuse of already constructed terms, respectively. An evaluation of the term representation is performed on the basis of a heterogeneous benchmark set. It shows that while the presented term data structure performs quite well in general, the normalization results indicate that a context dependent choice of reduction strategies is beneficial. A term indexing data structure for fast term retrieval based on various low-level criteria is presented and discussed. It supports symbol-based term retrieval, indexing of terms via structural properties, and subterm indexing