In reaction systems, preimages and nth ancestors are sets of reactants leading to the production of a target set of products in either 1 or n steps, respectively. Many computational problems on preimages and ancestors, such as finding all minimum-cardinality nth ancestors, computing their size or counting them, are intractable. In this paper, we characterize all nth ancestors using a Boolean formula that can be computed in polynomial time. Once simplified, this formula can be exploited to easily solve all preimage and ancestor problems. This allows us to directly relate the difficulty of ancestor problems to the cost of the simplification so that new insights into computational complexity investigations can be achieved. In particular, we focus on two problems: (i) deciding whether a preimage/nth ancestor exists and (ii) finding a preimage/nth ancestor of minimal size. Our approach is constructive, it aims at finding classes of reactions systems for which the ancestor problems can be solved in polynomial time, in exact or approximate way.

在反应系统中，原像和n个祖先是分别以1步或n步生成目标产物集的反应物组。在原像和祖先的许多计算问题，比如找出所有最小基数ñ个祖先，计算它们的大小或计数他们，是很棘手的。在本文中，我们描述了所有n祖先使用可以在多项式时间内计算的布尔公式。一旦简化，就可以利用该公式轻松解决所有原像和祖先问题。这使我们可以将祖先问题的难度与简化成本直接联系起来，从而可以获得对计算复杂性调查的新见解。特别地，我们关注两个问题：（i）确定是否存在原像/第n个祖先，以及（ii）寻找最小像元/第n个祖先。我们的方法是建设性的，旨在寻找可以在多项式时间内以精确或近似方式解决祖先问题的反应系统类别。