arXiv subjects

Todd Bichoupan

Publications and source records attributed to Todd Bichoupan.

3 recordsLinked to original sources

arXiv2023

Independence of Essential Sets in Finite Implication Bases

A new characterization is given to describe implication bases of a closure system in terms of the system's quasi-closed sets. Using this characterization, it is possible to show that groups of implications corresponding to distinct essential sets are interchangeable across different bases. It follows from this result that the sum of cardinalities of right sides of all implications corresponding to a single essential set in an optimal basis is fixed, solving an open conjecture by K. Adaricheva and J.B. Nation in 2014. These results provider greater insight into the global structure of implication bases.

math.LO↗

arXiv2023

Minimal Posets with Prescribed Maximal Chain Cardinalities

Given a nonempty finite multiset $S$ of positive integers, we wish to find a partially ordered set $P$ of minimal cardinality such that the multiset of cardinalities of all maximal chains in $P$ equals $S$. This paper establishes upper and lower bounds on the size of $P$: $\max(S) + \lceil \log_2 |S| \rceil <= |P| <= \max(S) + |S| - 1$, and both bounds are tight.

math.CO↗

arXiv2022

Complexity Results for Implication Bases of Convex Geometries

A convex geometry is finite zero-closed closure system that satisfies the anti-exchange property. Complexity results are given for two open problems related to representations of convex geometries using implication bases. In particular, the problem of optimizing an implication basis for a convex geometry is shown to be NP-hard by establishing a reduction from the minimum cardinality generator problem for general closure systems. Furthermore, even the problem of deciding whether an implication basis defines a convex geometry is shown to be co-NP-complete by a reduction from the Boolean tautology problem.

cs.CC↗