Reseña: Formalizing Frankl’s conjecture: FC-families

En las CICM 2012 (Conferences on Intelligent Computer Mathematics) se ha presentado un trabajo de razonamiento formalizado en Isabelle/HOL titulado Formalizing Frankl’s conjecture: FC-families.

Sus autores son Filip Marić, Miodrag Živković y Bojan Vučković (del grupo ARGO (Automated Reasoning GrOup) de la Universidad de Belgrado).

Su resumen es

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.

El código con la formalización en Isabelle/HOL se encuentra aquí.