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Tree Neural Networks in HOL4 [article]

Thibault Gauthier
2020 arXiv   pre-print
We present an implementation of tree neural networks within the proof assistant HOL4. Their architecture makes them naturally suited for approximating functions whose domain is a set of formulas. We measure the performance of our implementation and compare it with other machine learning predictors on the tasks of evaluating arithmetical expressions and estimating the truth of propositional formulas.
arXiv:2009.01827v1 fatcat:vov4ztwwe5a5foplznmbwdpl3y

Matching concepts across HOL libraries [article]

Thibault Gauthier, Cezary Kaliszyk
2014 arXiv   pre-print
Many proof assistant libraries contain formalizations of the same mathematical concepts. The concepts are often introduced (defined) in different ways, but the properties that they have, and are in turn formalized, are the same. For the basic concepts, like natural numbers, matching them between libraries is often straightforward, because of mathematical naming conventions. However, for more advanced concepts, finding similar formalizations in different libraries is a non-trivial task even for
more » ... vial task even for an expert. In this paper we investigate automatic discovery of similar concepts across libraries of proof assistants. We propose an approach for normalizing properties of concepts in formal libraries and a number of similarity measures. We evaluate the approach on HOL based proof assistants HOL4, HOL Light and Isabelle/HOL, discovering 398 pairs of isomorphic constants and types.
arXiv:1405.3906v1 fatcat:t54pk7rj4bfufcccqwul2oqe3e

Sharing HOL4 and HOL Light proof knowledge [article]

Thibault Gauthier, Cezary Kaliszyk
2015 arXiv   pre-print
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.
more » ... other library. Even if exact correspondences are not found, learning-reasoning 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 knowledge-sharing methods by reproving the HOL Light and HOL4 standard libraries. The learning-reasoning 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.
arXiv:1509.03527v1 fatcat:pguhm3mu75bhzi625u63jyaxbi

Self-Learned Formula Synthesis in Set Theory [article]

Chad E. Brown, Thibault Gauthier
2019 arXiv   pre-print
A reinforcement learning algorithm accomplishes the task of synthesizing a set-theoretical formula that evaluates to given truth values for given assignments.
arXiv:1912.01525v1 fatcat:g6cqft7vrbfc7kxdgzmzx2l7ne

Matching Concepts across HOL Libraries [chapter]

Thibault Gauthier, Cezary Kaliszyk
2014 Lecture Notes in Computer Science  
Many proof assistant libraries contain formalizations of the same mathematical concepts. The concepts are often introduced (defined) in different ways, but the properties that they have, and are in turn formalized, are the same. For the basic concepts, like natural numbers, matching them between libraries is often straightforward, because of mathematical naming conventions. However, for more advanced concepts, finding similar formalizations in different libraries is a non-trivial task even for
more » ... vial task even for an expert. In this paper we investigate automatic discovery of similar concepts across libraries of proof assistants. We propose an approach for normalizing properties of concepts in formal libraries and a number of similarity measures. We evaluate the approach on HOL based proof assistants HOL4, HOL Light and Isabelle/HOL, discovering 398 pairs of isomorphic constants and types. ⋆ The final publication is available at http://link.springer.com.
doi:10.1007/978-3-319-08434-3_20 fatcat:nzsczwmuyrd7jnoe3ty5alvv34

Learning to Reason with HOL4 tactics [article]

Thibault Gauthier, Cezary Kaliszyk, Josef Urban
2018 arXiv   pre-print
Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants. Such "hammer" tech- niques complement traditional proof assistant automation as implemented by tactics and decision procedures. In this paper we present a unified proof assistant automation approach which attempts to automate the selection of appropriate tactics and tactic-sequences com- bined with an optimized small-scale hammering approach. We
more » ... ring approach. We implement the technique as a tactic-level automation for HOL4: TacticToe. It implements a modified A*-algorithm directly in HOL4 that explores different tactic-level proof paths, guiding their selection by learning from a large number of previous tactic-level proofs. Unlike the existing hammer methods, TacticToe avoids translation to FOL, working directly on the HOL level. By combining tactic prediction and premise selection, TacticToe is able to re-prove 39 percent of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32 percent in the same amount of time.
arXiv:1804.00595v1 fatcat:p6oehxklwfdalokzeez3v6prga

Learning to Prove with Tactics [article]

Thibault Gauthier, Cezary Kaliszyk, Josef Urban, Ramana Kumar, Michael Norrish
2018 arXiv   pre-print
We implement a automated tactical prover TacticToe on top of the HOL4 interactive theorem prover. TacticToe learns from human proofs which mathematical technique is suitable in each proof situation. This knowledge is then used in a Monte Carlo tree search algorithm to explore promising tactic-level proof paths. On a single CPU, with a time limit of 60 seconds, TacticToe proves 66.4 percent of the 7164 theorems in HOL4's standard library, whereas E prover with auto-schedule solves 34.5 percent.
more » ... lves 34.5 percent. The success rate rises to 69.0 percent by combining the results of TacticToe and E prover.
arXiv:1804.00596v1 fatcat:twmo2yoiwrfaxmj2onudf6sw7a

Sharing HOL4 and HOL Light Proof Knowledge [chapter]

Thibault Gauthier, Cezary Kaliszyk
2015 Lecture Notes in Computer Science  
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.
more » ... other library. Even if exact correspondences are not found, learning-reasoning 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 knowledge-sharing methods by reproving the HOL Light and HOL4 standard libraries. The learning-reasoning 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.
doi:10.1007/978-3-662-48899-7_26 fatcat:tdutfexmfzfdbjxla4mx4yvu7q

GRUNGE: A Grand Unified ATP Challenge [article]

Chad E. Brown, Thibault Gauthier, Cezary Kaliszyk, Geoff Sutcliffe, Josef Urban
2019 arXiv   pre-print
This paper describes a large set of related theorem proving problems obtained by translating theorems from the HOL4 standard library into multiple logical formalisms. The formalisms are in higher-order logic (with and without type variables) and first-order logic (possibly with multiple types, and possibly with type variables). The resultant problem sets allow us to run automated theorem provers that support different logical formats on corresponding problems, and compare their performances.
more » ... ir performances. This also results in a new "grand unified" large theory benchmark that emulates the ITP/ATP hammer setting, where systems and metasystems can use multiple ATP formalisms in complementary ways, and jointly learn from the accumulated knowledge.
arXiv:1903.02539v1 fatcat:w2py2275rzc4rh5s3v45fff5vi

Learned Provability Likelihood for Tactical Search

Thibault Gauthier
2021 Electronic Proceedings in Theoretical Computer Science  
To run these functions, one first need to download the file tnn_for_tactictoe from http://grid01. ciirc.cvut.cz/~thibault/ to a desired location path_to_foo.  ... 
doi:10.4204/eptcs.342.7 fatcat:xyp7yj4gond3vbw233kaordqoe

Classification of Alignments Between Concepts of Formal Mathematical Systems [chapter]

Dennis Müller, Thibault Gauthier, Cezary Kaliszyk, Michael Kohlhase, Florian Rabe
2017 Lecture Notes in Computer Science  
Mathematical knowledge is publicly available in dozens of different formats and languages, ranging from informal (e.g. Wikipedia) to formal corpora (e.g., Mizar). Despite an enormous amount of overlap between these corpora, only few machine-actionable connections exist. We speak of alignment if the same concept occurs in different libraries, possibly with slightly different names, notations, or formal definitions. Leveraging these alignments creates a huge potential for knowledge sharing and
more » ... edge sharing and transfer, e.g., integrating theorem provers or reusing services across systems. Notably, even imperfect alignments, i.e. concepts that are very similar rather than identical, can often play very important roles. Specifically, in machine learning techniques for theorem proving and in automation techniques that use these, they allow learning-reasoning based automation for theorem provers to take inspiration from proofs from different formal proof libraries or semi-formal libraries even if the latter is based on a different mathematical foundation. We present a classification of alignments and design a simple format for describing alignments, as well as an infrastructure for sharing them. We propose these as a centralized standard for the community. Finally, we present an initial collection of ≈ 12000 alignments from the different kinds of mathematical corpora, including proof assistant libraries and semi-formal corpora as a public resource.
doi:10.1007/978-3-319-62075-6_7 fatcat:25nchzaurjbarpg4rkav4xdztu

Lyapunov and Minimum-Time Path Planning for Drones

Thibault Maillot, Ugo Boscain, Jean-Paul Gauthier, Ulysse Serres
2014 Journal of dynamical and control systems  
In this paper, we study the problem of controlling an unmanned aerial vehicle (UAV) to provide a target supervision and/or to provide convoy protection to ground vehicles. We first present a control strategy based upon a Lyapunov-LaSalle stabilization method to provide supervision of a stationary target . The UAV is expected to join a fixed target (a point in the space) and turn around tracking a pre-designed admissible circular trajectory. Our strategy is presented for both HALE (High Altitude
more » ... HALE (High Altitude Long Endurance) and MALE (Medium Altitude Long Endurance) types UAVs. A UAV flying at a constant altitude (HALE type) is modeled as a Dubins vehicle (i.e. a planar vehicle with constrained turning radius and constant forward velocity). For a UAV that might change its altitude (MALE type), we use the general kinematic model of a rigid body evolving in R 3 . Both control strategies presented are smooth and unlike what is usually proposed in the literature these strategies asymptotically track a circular trajectory of exact minimum turning radius. We also present the time-optimal control synthesis for tracking a circle by a Dubins vehicle. This optimal strategy, although much simpler than the point-to-point time-optimal strategy obtained by P. Souères and J.-P. Laumond in the 1990s (see [43] ), is very rich. Finally, we propose control strategies to provide supervision of a moving target , that are based upon the previous ones.
doi:10.1007/s10883-014-9222-y fatcat:hxh2omayyjegtgcg5kosybwb2m

How humans fly

Alain Ajami, Jean-Paul Gauthier, Thibault Maillot, Ulysse Serres
2013 E S A I M: Control, Optimisation and Calculus of Variations  
This paper is devoted to the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. This applied question is very similar to the problem of determining what is minimized in human locomotion. These starting points are the reasons for the particular
more » ... the particular classes of control systems and of costs under consideration. To summarize, our conclusion is that in general, inside these classes, three experiments visiting the same values of the control are needed to reconstruct the cost, and two experiments are in general not enough. The method is constructive. The proof of these results is mostly based upon the Thom's transversality theory.
doi:10.1051/cocv/2012043 fatcat:tjpbdkawdbbyxde63bl2g4fjvq

Deep Reinforcement Learning for Synthesizing Functions in Higher-Order Logic [article]

Thibault Gauthier
2020 arXiv   pre-print
The paper describes a deep reinforcement learning framework based on self-supervised learning within the proof assistant HOL4. A close interaction between the machine learning modules and the HOL4 library is achieved by the choice of tree neural networks (TNNs) as machine learning models and the internal use of HOL4 terms to represent tree structures of TNNs. Recursive improvement is possible when a task is expressed as a search problem. In this case, a Monte Carlo Tree Search (MCTS) algorithm
more » ... h (MCTS) algorithm guided by a TNN can be used to explore the search space and produce better examples for training the next TNN. As an illustration, term synthesis tasks on combinators and Diophantine equations are specified and learned. We achieve a success rate of 65% on combinator synthesis problems outperforming state-of-the-art ATPs run with their best general set of strategies. We set a precedent for statistically guided synthesis of Diophantine equations by solving 78.5% of the generated test problems.
arXiv:1910.11797v3 fatcat:bqtdfbrcxrcc5p6e3tzygogxsy

Trophic niche of the invasive gregarious species Crepidula fornicata, in relation to ontogenic changes [article]

Thibault Androuin, Stanislas F Dubois, Cedric Hubas, Gwendoline Lefebvre, Fabienne Le Grand, Gauthier Schaal, Antoine Carlier
2020 bioRxiv   pre-print
Crepidula fornicata is a common and widespread invasive gregarious species along the European coast. Among its life-history traits, well documented ontogenic changes in behavior (i.e., motile male to sessile female) suggest a potential shift in feeding strategy across its life stages. Considering the ecological significance of this species in colonized areas, understanding how conspecifics share the trophic resource is crucial. Using fatty acids (FA) and stable isotopes (SI) as complementary
more » ... as complementary trophic markers, we conducted a field survey between late winter and spring to investigate the trophic niche of three ontogenic stages of C. fornicata that bear different sexual (male/female) and motility (motile/sessile) traits. Potential trophic sources were characterized by their pigment, FA and SI compositions and showed well discriminated compositions over the studied period. We showed that the biofilm covering C. fornicata shells harbored a higher biomass of primary producers (i.e., chlorophytes and diatoms) than the surrounding sediment. Over the studied period, we observed a covariation between the three ontogenic stages for both FA and SI compositions which suggest that the trophic niche of C. fornicata does not change significantly across its benthic life. During periods of low food availability, slipper limpets displayed an opportunistic suspension-feeding behaviour, relying on both fresh and detrital organic matter, likely coming from superficial sedimentary organic matter. However, during high food availability (i.e., spring phytoplankton bloom), all ontogenic stages largely benefited from this fresh supply of organic matter (pelagic diatoms in this case). The three ontogenic stages showed consistent differences in FA composition, and to a lesser extent in SI composition. These differences persist over time, as they originate from ontogenic physiological changes (differential growth rates, metabolic rate or gametogenesis) rather than diet discrepancies. This study revealed that multiple trophic markers allow high complementary to characterize organic matter as well as food partitioning between conspecific organisms.
doi:10.1101/2020.07.30.229021 fatcat:ctajheavvzbktfynqtcwmku4rq
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