We define a new class of predicates called equilevel predicates on a distributive lattice which eases the analysis of parallel algorithms. Many combinatorial problems such as the vertex cover problem, the bipartite matching problem, and the minimum spanning tree problem can be modeled as detecting an equilevel predicate. The problem of detecting an equilevel problem is NP-complete, but equilevel predicates with the helpful property can be detected in polynomial time in an online manner. An equilevel predicate has the helpful property with a polynomial time algorithm if the algorithm can return a nonempty set of indices such that advancing on any of them can be used to detect the predicate. Furthermore, the refined independently helpful property allows online parallel detection of such predicates in NC. When the independently helpful property holds, advancing on all the specified indices in parallel can be used to detect the predicate in polylogarithmic time. We also define a special class of equilevel predicates called solitary predicates. Unless NP = RP, this class of predicate also does not admit efficient algorithms. Earlier work has shown that solitary predicates with the efficient advancement can be detected in polynomial time. We introduce two properties called the antimonotone advancement and the efficient rejection which yield the detection of solitary predicates in NC. Finally, we identify the minimum spanning tree, the shortest path, and the conjunctive predicate detection as problems satisfying such properties, giving alternative certifications of their NC memberships as a result.

翻译：我们定义了一类新的谓词，称为分配格上的等层谓词，它简化了并行算法的分析。许多组合问题，如顶点覆盖问题、二分图匹配问题和最小生成树问题，都可以建模为检测等层谓词。检测等层问题属于NP完全问题，但具有有益性质的等层谓词可以在多项式时间内以在线方式检测。若存在一个多项式时间算法能返回非空索引集合，使得对这些索引中的任意一个进行推进即可用于检测该谓词，则该等层谓词具有有益性质。进一步地，精炼的独立有益性质允许在NC类中对此类谓词进行在线并行检测。当独立有益性质成立时，并行推进所有指定索引即可在多重对数时间内检测该谓词。我们还定义了一类特殊的等层谓词——孤立谓词。除非NP=RP，否则此类谓词也不存在高效算法。先前研究表明，具有高效推进能力的孤立谓词可在多项式时间内检测。我们引入两个性质——反单调推进和高效拒绝，使得孤立谓词可在NC类中被检测。最后，我们识别出最小生成树、最短路径和合取谓词检测等满足这些性质的问题，从而给出它们属于NC类的新验证。