In the recent years, the Metis prover based on ordered paramodulation and model elimination has replaced the earlier built-in methods for general-purpose proof automation in HOL4 and Isabelle HOL. In the annual CASC competition, the leanCoP system based on connection tableaux has however performed better than Metis. In this paper we show how the leanCoP's core algorithm can be implemented inside HOLLight. leanCoP's flagship feature, namely its minimalistic core, results in a very simple proof system. This plays a crucial role in extending the MESON proof reconstruction mechanism to connection tableaux proofs, providing an implementation of leanCoP that certifies its proofs. We discuss the differences between our direct implementation using an explicit Prolog stack, to the continuation passing implementation of MESON present in HOLLight and compare their performance on all core HOLLight goals. The resulting prover can be also used as a general purpose TPTP prover. We compare its performance against the resolution based Metis on TPTP and other interesting datasets.
Compact ATP calculi such as @cite_22 and @cite_17 have been used for some time in @cite_27 @cite_12 and s @cite_11 as general first-order automation tactics for discharging goals that are already simple enough. With the arrival of large-theory hammer'' linkups @cite_18 @cite_24 @cite_26 @cite_29 between ITPs, state-of-the-art ATPs such as @cite_14 and @cite_0 , and premise selection methods @cite_8 , such tactics also became used as a relatively cheap method for reconstructing the (minimized) proofs found by the stronger ATPs. In particular, Hurd's has been adopted as the main proof reconstruction tool used by 's linkup @cite_23 @cite_1 , while Harrison's version of could reconstruct in 1 second about 80 proofs found by in the first experiments with the linkup @cite_13 .
@cite_18 @cite_26 @cite_14 @cite_22 @cite_8 @cite_29 @cite_1 @cite_17 @cite_24 @cite_0 @cite_27 @cite_23 @cite_13 @cite_12 @cite_11
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Two complementary AI methods are used to improve the strength of the AI ATP service for proving conjectures over the HOL Light and Flyspeck corpora. First, several schemes for frequency-based feature weighting are explored in combination with distanceweighted k-nearest-neighbor classier. This results in 16 improvement (39.0 to 45.5 Flyspeck problems solved) of the overall strength of the service when using 14 CPUs and 30 seconds. The best premise-selection ATP combination is improved from 24.2 to 31.4 , i.e. by 30 . A smaller improvement is obtained by evolving targetted E prover strategies on two particular premise selections, using the Blind Strategymaker (BliStr) system. This raises the performance of the best AI ATP method from 31.4 to 34.9 , i.e. by 11 , and raises the current 14-CPU power of the service to 46.9 . As a present to Mizar on its 40th anniversary, we develop an AI ATP system that in 30 seconds of real time on a 14-CPU machine automatically proves 40 of the theorems in the latest official version of the Mizar Mathematical Library (MML). This is a considerable improvement over previous performance of large-theory AI ATP methods measured on the whole MML. To achieve that, a large suite of AI ATP methods is employed and further developed. We implement the most useful methods efficiently, to scale them to the 150000 formulas in MML. This reduces the training times over the corpus to 1---3 seconds, allowing a simple practical deployment of the methods in the online automated reasoning service for the Mizar users (Miz AR @math ). The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the Flyspeck proofs, producing an AI system capable of proving a wide range of mathematical conjectures automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39 of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided. Deep learning techniques lie at the heart of several significant AI advances in recent years including object recognition and detection, image captioning, machine translation, speech recognition and synthesis, and playing the game of Go. Automated first-order theorem provers can aid in the formalization and verification of mathematical theorems and play a crucial role in program analysis, theory reasoning, security, interpolation, and system verification. Here we suggest deep learning based guidance in the proof search of the theorem prover E. We train and compare several deep neural network models on the traces of existing ATP proofs of Mizar statements and use them to select processed clauses during proof search. We give experimental evidence that with a hybrid, two-phase approach, deep learning based guidance can significantly reduce the average number of proof search steps while increasing the number of theorems proved. Using a few proof guidance strategies that leverage deep neural networks, we have found first-order proofs of 7.36 of the first-order logic translations of the Mizar Mathematical Library theorems that did not previously have ATP generated proofs. This increases the ratio of statements in the corpus with ATP generated proofs from 56 to 59 .