Many computational problems can be modelled as the class of all finite relational structures $\mathbb A$ that satisfy a fixed first-order sentence $\phi$ hereditarily, i.e., we require that every substructure of $\mathbb A$ satisfies $\phi$. In this case, we say that the class is in HerFO. The problems in HerFO are always in coNP, and sometimes coNP-complete. HerFO also contains many interesting computational problems in P, including many constraint satisfaction problems (CSPs). We show that HerFO captures the class of complements of CSPs for reducts of finitely bounded structures, i.e., every such CSP is polynomial-time equivalent to the complement of a problem in HerFO. However, we also prove that HerFO does not have the full computational power of coNP: there are problems in coNP that are not polynomial-time equivalent to a problem in HerFO, unless E=NE. Another main result is a description of the quantifier-prefixes for $\phi$ such that hereditarily checking $\phi$ is in P; we show that for every other quantifier-prefix there exists a formula $\phi$ with this prefix such that hereditarily checking $\phi$ is coNP-complete.

翻译：许多计算问题可以建模为所有有限关系结构$\mathbb A$的类，这些结构在遗传意义上满足一个固定的一阶语句$\phi$，即我们要求$\mathbb A$的每个子结构都满足$\phi$。在这种情况下，我们说该类属于HerFO。HerFO中的问题总是属于coNP，有时是coNP完全的。HerFO还包含许多P类中的有趣计算问题，包括许多约束满足问题（CSPs）。我们证明HerFO刻画了有限有界结构归约的CSPs补类，即每个这样的CSP在多项式时间等价于HerFO中某个问题的补问题。然而，我们也证明了HerFO并不具有coNP的全部计算能力：存在coNP中的问题在多项式时间下不等价于HerFO中的任何问题，除非E=NE。另一个主要结果是描述了使得遗传检验$\phi$属于P的$\phi$量词前缀；我们证明对于其他所有量词前缀，都存在具有该前缀的公式$\phi$，使得遗传检验$\phi$是coNP完全的。