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Hammering towards QED
"... This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “onestroke ” tool for discharging difficult lemmas without the need for careful and detailed manu ..."
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This paper surveys the emerging methods to automate reasoning over large libraries developed with formal proof assistants. We call these methods hammers. They give the authors of formal proofs a strong “onestroke ” tool for discharging difficult lemmas without the need for careful and detailed manual programming of proof search. The main ingredients underlying this approach are efficient automatic theorem provers that can cope with hundreds of axioms, suitable translations of the proof assistant’s logic to the logic of the automatic provers, heuristic and learning methods that select relevant facts from large libraries, and methods that reconstruct the automatically found proofs inside the proof assistants. We outline the history of these methods, explain the main issues and techniques, and show their strength on several large benchmarks. We also discuss the relation of this technology to the QED Manifesto and consider its implications for QEDlike efforts.
Premise selection and external provers for HOL4
 In Certified Programs and Proofs (CPP’15), Lecture Notes in Computer Science
, 2015
"... Learningassisted automated reasoning has recently gained popularity among the users of Isabelle/HOL, HOL Light, and Mizar. In this paper, we present an addon to the HOL4 proof assistant and an adaptation of the HOL(y)Hammer system that provides machine learningbased premise selection and automate ..."
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Learningassisted automated reasoning has recently gained popularity among the users of Isabelle/HOL, HOL Light, and Mizar. In this paper, we present an addon to the HOL4 proof assistant and an adaptation of the HOL(y)Hammer system that provides machine learningbased premise selection and automated reasoning also for HOL4. We efficiently record the HOL4 dependencies and extract features from the theorem statements, which form a basis for premise selection. HOL(y)Hammer transforms the HOL4 statements in the various TPTPATP proof formats, which are then processed by the ATPs. We discuss the different evaluation settings: ATPs, accessible lemmas, and premise numbers. We measure the performance of HOL(y)Hammer on the HOL4 standard library. The results are combined accordingly and compared with the HOL Light experiments, showing a comparably high quality of predictions. The system directly benefits HOL4 users by automatically finding proofs dependencies that can be reconstructed by Metis.
FEMaLeCoP: Fairly Efficient Machine Learning Connection Prover
"... Abstract. FEMaLeCoP is a connection tableau theorem prover based on leanCoP which uses efficient implementation of internal learningbased guidance for extension steps. Despite the fact that exhaustive use of such internal guidance can incur a significant slowdown of the raw inferencing process, FEM ..."
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Abstract. FEMaLeCoP is a connection tableau theorem prover based on leanCoP which uses efficient implementation of internal learningbased guidance for extension steps. Despite the fact that exhaustive use of such internal guidance can incur a significant slowdown of the raw inferencing process, FEMaLeCoP trained on related proofs can prove many problems that cannot be solved by leanCoP. In particular on the MPTP2078 benchmark, FEMaLeCoP adds 90 (15.7%) more problems to the 574 problems that are provable by leanCoP. FEMaLeCoP is thus the first AI/ATP system convincingly demonstrating that guiding the internal inference algorithms of theorem provers by knowledge learned from previous proofs can significantly improve the performance of the provers. This paper describes the system, discusses the technology developed, and evaluates the system.
Sharing HOL4 and HOL Light proof knowledge
"... Abstract. New proof assistant developments often involve concepts similar to already formalized ones. When proving their properties, a human can often take inspiration from the existing formalized proofs available in other provers or libraries. In this paper we propose and evaluate a number of met ..."
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Abstract. New proof assistant developments often involve concepts similar to already formalized ones. When proving their properties, a human can often take inspiration from the existing formalized proofs available in other provers or libraries. In this paper we propose and evaluate a number of methods, which strengthen proof automation by learning from proof libraries of different provers. Certain conjectures can be proved directly from the dependencies induced by similar proofs in the other library. Even if exact correspondences are not found, learningreasoning systems can make use of the association between proved theorems and their characteristics to predict the relevant premises. Such external help can be further combined with internal advice. We evaluate the proposed knowledgesharing methods by reproving the HOL Light and HOL4 standard libraries. The learningreasoning system HOL(y)Hammer, whose single best strategy could automatically find proofs for 30 % of the HOL Light problems, can prove 40 % with the knowledge from HOL4. 1
Initial Experiments with Statistical Conjecturing over Large Formal Corpora
"... Abstract. A critical part of mathematicians' work is the process of conjecturemaking. This involves observing patterns in and between mathematical objects, and transforming such patterns into conjectures that describe or better explain the behavior of the objects. Computer scientists have sin ..."
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Abstract. A critical part of mathematicians' work is the process of conjecturemaking. This involves observing patterns in and between mathematical objects, and transforming such patterns into conjectures that describe or better explain the behavior of the objects. Computer scientists have since long tried to reproduce this process automatically, but most of the methods were typically restricted to specific domains or based on bruteforce enumeration methods. In this work we propose and implement methods for generating conjectures by using statistical analogies extracted from large formal libraries, and provide their initial evaluation.