Alexander Steen

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

Betreuer: C. Benzmüller , Prof. Dr. Marcel Kyas
Abschluss: Master of Science (M.Sc.)
Abgabedatum: 17.10.2014
Homepage des Autors:


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.