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

Kinyon, Michael; Veroff, Robert; Vojtěchovský, Petr

doi:10.1007/978-3-642-36675-8_8

Cited by 30 publications

(32 citation statements)

References 43 publications

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“…A finer version of the conjecture is discussed in [12]. Existence of loops (necessarily not associative) Q of nilpotence class 3 with abelian Inn(Q) was first shown by Csörgő [7].…”

Section: Solvability and Nilpotencementioning

confidence: 99%

Abelian Extensions and Solvable Loops

Stanovský

Vojtěchovský

2014

Results. Math.

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Abstract. Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations between nilpotence and solvability for loops and the associated multiplication groups and inner mapping groups.

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“…A finer version of the conjecture is discussed in [12]. Existence of loops (necessarily not associative) Q of nilpotence class 3 with abelian Inn(Q) was first shown by Csörgő [7].…”

Section: Solvability and Nilpotencementioning

confidence: 99%

Abelian Extensions and Solvable Loops

Stanovský

Vojtěchovský

2014

Results. Math.

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“…This technique has been originally developed by Veroff [22] and implemented in Otter [14] and Prover9 [15]. Since then, it has been extensively used in the AIM project [12] for obtaining long and advanced proofs of open algebraic conjectures. The watchlist mechanism is nowadays implemented also in E. All the above implementations use only a single watchlist, as opposed to ProofWatch discussed below.…”

Section: Proofwatch: Proof Guidance By Clause Subsumptionmentioning

confidence: 99%

ENIGMAWatch: ProofWatch Meets ENIGMA

Goertzel

Jakubův

Urban

2019

Lecture Notes in Computer Science

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In this work we describe a new learning-based proof guidance -ENIGMAWatch -for saturation-style first-order theorem provers. ENIGMAWatch combines two guiding approaches for the given-clause selection implemented for the E ATP system: ProofWatch and ENIGMA. ProofWatch is motivated by the watchlist (hints) method and based on symbolic matching of multiple related proofs, while ENIGMA is based on statistical machine learning. The two methods are combined by using the evolving information about symbolic proof matching as an additional information that characterizes the saturation-style proof search for the statistical learning methods. The new system is experimentally evaluated on a large set of problems from the Mizar library. We show that the added proof-matching information is considered important by the statistical machine learners, and that it leads to improvements in E's performance over ProofWatch and ENIGMA.

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“…The main application have so far been problems in equational algebra [15]. A recent example where very long proofs of open conjectures are found thanks to this technique is the project AIMed at characterizing loops with Abelian Inner Mappings groups [9]. A similar technique that extracts and generalizes lemmas from previous proofs and uses them for proof guidance was implemented by Schulz in E prover as a part of his PhD thesis [17].…”

Section: Introductionmentioning

confidence: 99%

“…4 Our very initial experiments (done with Veroff) with hints on large-theory problems have shown that unlike the equational proofs, the proofs of large-theory problems contain many (incompatible) skolem constants and steps depending on the negated conjecture, and thus are harder to re-orient into the strictly-forward proofs [9] from which lemmas derived only from the axioms and containing only known symbols can be extracted. A related issue is that the large-theory proofs seem to be much more heterogeneous than e.g.…”

Section: Introductionmentioning

confidence: 99%

Lemmatization for Stronger Reasoning in Large Theories

Kaliszyk¹,

Urban²,

VyskoăźIl³

2015

Frontiers of Combining Systems

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Abstract. In this work we improve ATP performance in large theories by the reuse of lemmas derived in previous related problems. Given a large set of related problems to solve, we run automated theorem provers on them, extract a large number of lemmas from the proofs found and post-process the lemmas to make them usable in the remaining problems. Then we filter the lemmas by several tools and extract their proof dependencies, and use machine learning on such proof dependencies to add the most promising generated lemmas to the remaining problems.On such enriched problems we run the automated provers again, solving more problems. We describe this method and the techniques we used, and measure the improvement obtained. On the MPTP2078 large-theory benchmark the method yields 6.6% and 6.2% more problems proved in two different evaluation modes.

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

Cited by 30 publications

References 43 publications

Abelian Extensions and Solvable Loops

Abelian Extensions and Solvable Loops

ENIGMAWatch: ProofWatch Meets ENIGMA

Lemmatization for Stronger Reasoning in Large Theories

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