Superposition for Lambda-Free Higher-Order Logic

Bentkamp, Alexander; Blanchette, Jasmin Christian; Cruanes, Simon; Waldmann, Uwe

doi:10.1007/978-3-319-94205-6_3

Cited by 29 publications

(52 citation statements)

References 52 publications

(84 reference statements)

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“…It helps us answer questions such as, "Is Lemma 3.13 actually needed, and if so, where?" Indeed, such a question recently arose in the course of Bentkamp's research on superposition [8]. He wanted to understand why the literal selection function S M is defined so that "(ii) S M (C) = S(C), if C is not in K. " He quickly got two replies.…”

Section: A First-order Ordered Resolution Provermentioning

confidence: 99%

“…Bentkamp developed, under Waldmann's and my supervision, a refutationally complete superposition calculus for λ-free higher-order logic and implemented it in a prototype prover developed with Cruanes [8]. He wrote his proofs directly in L A T E X, which was possible only because he is extremely rigorous and could count on two experts to check his proofs-namely, Waldmann and our colleague Tourret.…”

Section: Lambda-free Higher-order Terms Ordersmentioning

confidence: 99%

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Formalizing the metatheory of logical calculi and automatic provers in Isabelle/HOL (invited talk)

Blanchette

2019

Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs

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IsaFoL (Isabelle Formalization of Logic) is an undertaking that aims at developing formal theories about logics, proof systems, and automatic provers, using Isabelle/HOL. At the heart of the project is the conviction that proof assistants have become mature enough to actually help researchers in automated reasoning when they develop new calculi and tools. In this paper, I describe and reflect on three verification subprojects to which I contributed: a first-order resolution prover, an imperative SAT solver, and generalized term orders for λ-free higher-order logic.

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Section: A First-order Ordered Resolution Provermentioning

confidence: 99%

Section: Lambda-free Higher-order Terms Ordersmentioning

confidence: 99%

Formalizing the metatheory of logical calculi and automatic provers in Isabelle/HOL (invited talk)

Blanchette

2019

Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs

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“…adaptions of ordered paramodulation or even weaker forms of ordering constraints. This is also motivated by the observation that considerable effort needs to be invested for constructing a complete superposition calculus for a logic that seems only marginally more expressive than standard first-order logic [BBCW18].…”

Section: Discussionmentioning

confidence: 99%

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

Steen

2019

Künstl Intell

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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.

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“…Our logic corresponds to the intensional λ-free higher-order logic (λfHOL) described by Bentkamp, Blanchette, Cruanes, and Waldmann [7,Sect. 2].…”

Section: Logicmentioning

confidence: 99%

Extending a Brainiac Prover to Lambda-Free Higher-Order Logic

Vukmirović

Blanchette

Cruanes

et al. 2019

Tools and Algorithms for the Construction and Analysis of Systems

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Decades of work have gone into developing efficient proof calculi, data structures, algorithms, and heuristics for first-order automatic theorem proving. Higher-order provers lag behind in terms of efficiency. Instead of developing a new higher-order prover from the ground up, we propose to start with the state-of-the-art superposition-based prover E and gradually enrich it with higher-order features. We explain how to extend the prover's data structures, algorithms, and heuristics to λ-free higher-order logic, a formalism that supports partial application and applied variables. Our extension outperforms the traditional encoding and appears promising as a stepping stone towards full higher-order logic.

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Superposition for Lambda-Free Higher-Order Logic

Cited by 29 publications

References 52 publications

Formalizing the metatheory of logical calculi and automatic provers in Isabelle/HOL (invited talk)

Formalizing the metatheory of logical calculi and automatic provers in Isabelle/HOL (invited talk)

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

Extending a Brainiac Prover to Lambda-Free Higher-Order Logic

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