Automated theorem proving in quasigroup and loop theory

Phillips, J. D.; Stanovský, David

doi:10.3233/aic-2010-0460

Cited by 16 publications

(8 citation statements)

References 38 publications

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“…Following [13], the parameters were set to P T = 55dBm, η = 4 (outdoors), mean X σ = 0, and variance σ = 5. In addition, the network delay was set to 3 seconds according to [13], [26], and [27], and the run time was 300 seconds. For consistency with several real applications, e.g., indoor positioning, geological detection and water quality monitoring, the speeds of the mobile nodes were set to 5 m/s, 10 m/s and 15 m/s.…”

Section: A Settingsmentioning

confidence: 99%

Utilizing Active Sensor Nodes in Smart Environments for Optimal Communication Coverage

Xie

Zhang

et al. 2019

IEEE Access

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Wireless sensor networks are widely used in smart environments to capture and detect the activities of human beings, and achieving reliable transmission between sensor nodes has become one of the main challenges of practical applications. This paper presents a scheme for path planning that is designed to achieve optimal coverage by using active nodes to periodically fill in the blank areas and to replace the failed nodes. This approach can effectively avoid uneven energy consumption while maintaining complete link states. Meanwhile, the curl field of the nodes is used to model the effects of the residual energy and the distance between nodes, thereby effectively relaxing the requirements on the spatial positions of the nodes. Experiments show that in the case of directional transmission, the proposed method demonstrates better performance than other algorithms in terms of the network lifecycle, coverage, and transmission reliability. This method can effectively address the problem of cross-node failure along the transmission paths in complex and dynamic networks.

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Section: A Settingsmentioning

confidence: 99%

Utilizing Active Sensor Nodes in Smart Environments for Optimal Communication Coverage

Xie

Zhang

et al. 2019

IEEE Access

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“…Automated theorem proving and mathematics will benefit from advanced ML integration. One of the mathematical subfields where automated theorem provers are heavily used is the field of quasigroup and loop theory [12]. A quasigroup is similar to a group, but it does not guarantee associativity.…”

Section: Introductionmentioning

confidence: 99%

Learning Equational Theorem Proving

Piepenbrock,

Heskes,

Janota

et al. 2021

Preprint

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We develop Stratified Shortest Solution Imitation Learning (3SIL) to learn equational theorem proving in a deep reinforcement learning (RL) setting. The self-trained models achieve state-of-theart performance in proving problems generated by one of the top open conjectures in quasigroup theory, the Abelian Inner Mapping (AIM) Conjecture. To develop the methods, we first use two simpler arithmetic rewriting tasks that share tree-structured proof states and sparse rewards with the AIM problems. On these tasks, 3SIL is shown to significantly outperform several established RL and imitation learning methods. The final system is then evaluated in a standalone and cooperative mode on the AIM problems. The standalone 3SIL-trained system proves in 60 seconds more theorems (70.2%) than the complex, hand-engineered Waldmeister system (65.5%). In the cooperative mode, the final system is combined with the Prover9 system, proving in 2 seconds what standalone Prover9 proves in 60 seconds.

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“…They appeared already in the milestone paper of Knuth and Bendix [24], and interest in them has continued in the theoremproving community [12] [49]. In more recent years, mathematicians specializing in quasigroups and loops have been making significant use of automated deduction tools [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] [ 36,37,38,39,40,41,42,43,44]. With the exception of [39] and [40], all of the aforementioned references used Bill McCune's Prover9 [31] or its predecessor Otter [30].…”

Section: Introductionmentioning

confidence: 99%

Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction

Kinyon

Veroff

Vojtěchovský

2013

Automated Reasoning and Mathematics

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Abstract. We describe a large-scale project in applied automated deduction concerned with the following problem of considerable interest in loop theory: If Q is a loop with commuting inner mappings, does it follow that Q modulo its center is a group and Q modulo its nucleus is an abelian group? This problem has been answered affirmatively in several varieties of loops. The solution usually involves sophisticated techniques of automated deduction, and the resulting derivations are very long, often with no higher-level human proofs available.

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Automated theorem proving in quasigroup and loop theory

Cited by 16 publications

References 38 publications

Utilizing Active Sensor Nodes in Smart Environments for Optimal Communication Coverage

Utilizing Active Sensor Nodes in Smart Environments for Optimal Communication Coverage

Learning Equational Theorem Proving

Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction

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