|Title:||Formalizing Frankl's conjecture: FC-families||Journal:||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)||Volume:||7362 LNAI||First page:||248||Last page:||263||Conference:||Int. Conf. on Artificial Intelligence and Symbolic Computation, AISC 2012, Symp. on the Integration of Symbolic Computation and Mechanized Reasoning, Calculemus 2012, DML 2012, Conf. on Mathematical Knowledge Management, MKM 2012, Part of CICM 2012; Bremen; Germany; 8 July 2012 through 13 July 2012||Issue Date:||17-Aug-2012||Rank:||M33||ISBN:||978-3-642-31373-8||ISSN:||0302-9743||DOI:||10.1007/978-3-642-31374-5_17||Abstract:||
The Frankl's conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. FC-families are families for which it is proved that every union-closed family containing them satisfies the Frankl's condition (e.g., in every union-closed family that contains a one-element set a, the element a is contained in at least half of the sets, so families of the form a are the simplest FC-families). FC-families play an important role in attacking the Frankl's conjecture, since they enable significant search space pruning. We present a formalization of the computer assisted approach for proving that a family is an FC-family. Proof-by-computation paradigm is used and the proof assistant Isabelle/HOL is used both to check mathematical content, and to perform (verified) combinatorial searches on which the proofs rely. FC-families known in the literature are confirmed, and a new FC-family is discovered.
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