Many AI-related reasoning problems are based on the problem of satisfiability of propositional formulas with some cardinality-minimality condition. While the complexity of the satisfiability problem (SAT) is well understood when considering systematically all fragments of propositional logic within Schaefer's framework (STOC 1978) this is not the case when such minimality condition is added. We consider the CardMinSat problem, which asks, given a formula F and an atom x, whether x is true in some cardinality-minimal model of F. We completely classify the computational complexity of the CardMinSat problem within Schaefer's framework, thus paving the way for a better understanding of the tractability frontier of many AI-related reasoning problems. To this end we use advanced algebraic tools developed by (Schnoor & Schnoor 2008) and (Lagerkvist 2014).

翻译：许多与大赦国际相关的推理问题是基于具有某种最基本条件的公式的可比较性问题。虽然在系统地考虑Schaefer框架(STOC,1978年)内所有参数逻辑的碎片时,可以清楚地理解可比较性问题的复杂性(SAT),但在增加这种最低条件时,情况并非如此。我们考虑到CardMinSat问题,根据一个公式F和一个原子x,在F的一些最基本模式中x是否属实。我们将CardMinSat问题的计算复杂性完全分类在Schaefer的框架内,从而为更好地了解许多与AI有关的推理问题的可移动性界限铺平了道路。为此,我们使用由(Schnoor & Schnoor,2008年)和(Lagerkvist,2014年)开发的先进的代数工具。</s>