Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

Abstract

In this thesis the theoretical foundations and the practical components for implementing an effective automated theorem proving system for higher-order logic are presented. A primary focus of this thesis is the provision of evidence that a paramodulation-based proof calculus can effectively be employed for performant equational reasoning in Extensional Type Theory (higher-order logic). To that end, a sound and complete paramodulation calculus for extensional higherorder logic with Henkin semantics is presented. The completeness proof hereby unifies and simplifies existing abstract consistency techniques for a formulation of higher-order logic that is based on primitive equality as sole logical connective. In the practically motivated main part of this thesis, the design and architecture of the new higher-order theorem prover Leo-III is presented. Leo-III is based on the above paramodulation calculus and implements additional practically motivated inference rules including equational simplification routines such as heuristic rewriting and support for reasoning with choice. The system encompasses a flexible mechanism for asynchronous cooperation with first-order reasoning systems, a powerful proof search procedure and a sophisticated and efficient set of underlying data structures. Pragmatic and practically significant features of Leo-III are discussed, including its native support for polymorphic higher-order logic and reasoning in higher-order quantified modal logics. An evaluation on a heterogeneous set of benchmark problems confirms that Leo-III is one of the most effective and versatile higher-order automated reasoning systems to date. vii The dissertation project was conducted within the "Leo-III" project funded by the German Research Foundation (DFG) under grant BE 2501/11-1, and the project "Consistent Rational Argumentation in Politics" funded by the Volkswagenstiftung. All source code related to work presented in this thesis is publicly available (under BSD-3 license) at https://github.com/leoprover/Leo-III.

Making Higher-Order Superposition Work

“…As an optimization and to simplify the implementation, Leo-III [40] and Vampire 4.4 [9] (which uses a predecessor of combinatory superposition) compute only a finite subset of the possible conclusions for inferences that require enumerating a CSU. Not only is this a source of incompleteness, but choosing the cardinality of the computed subset is a difficult heuristic choice.…”

“…Cooperation with efficient first-order theorem provers is an essential feature of higher-order theorem provers such as Leo-III [40,Sect. 4.4] and Satallax [11].…”

“…In his thesis [40,Sect. 6.1], Steen extensively evaluates the effects of using different first-order backends on the performance of Leo-III.…”

Making Higher-Order Superposition Work

“…Cooperation with efficient first-order theorem provers is an essential feature of higher-order theorem provers such as Leo-III [40,Sect. 4.4] and Satallax [11].…”

Restricted Combinatory Unification

“…With respect to higher-order unification, some work has been carried out on developing indexing data structures [20] [24], but has gained little acceptance. Steen [28] suggests that the reason for this is the complexity or undecidability of many of the operations required to build and maintain higher-order indexing structures. For example, higher-order unification, anti-unification and matching are all either undecidable or have large complexities.…”