By Fagin's Theorem, NP contains precisely those problems that can be described by formulas starting with an existential second-order quantifier, followed by only first-order quantifiers (ESO formulas). Subsequent research refined this result, culminating in powerful theorems that characterize for each possible sequence of first-order quantifiers how difficult the described problem can be. We transfer this line of inquiry to the parameterized setting, where the size of the set quantified by the second-order quantifier is the parameter. Many natural parameterized problems can be described in this way using simple sequences of first-order quantifiers: For the clique or vertex cover problems, two universal first-order quantifiers suffice ("for all pairs of vertices ... must hold"); for the dominating set problem, a universal followed by an existential quantifier suffice ("for all vertices, there is a vertex such that ..."); and so on. We present a complete characterization that states for each possible sequence of first-order quantifiers how high the parameterized complexity of the described problems can be. The uncovered dividing line between quantifier sequences that lead to tractable versus intractable problems is distinct from that known from the classical setting, and it depends on whether the parameter is a lower bound on, an upper bound on, or equal to the size of the quantified set.

翻译：根据费金定理，NP恰好包含那些可由以存在性二阶量词开头、后接仅一阶量词的公式（ESO公式）描述的问题。后续研究深化了这一结论，最终形成了强有力的定理，该定理刻画了每种可能的一阶量词序列所描述问题的难度上限。我们将这一研究方向转移至参数化设定，其中二阶量词所量化的集合的大小即为参数。许多自然参数化问题可通过此类公式结合简单一阶量词序列进行描述：对于团或顶点覆盖问题，两个全称一阶量词即足矣（"对任意顶点对……必须成立"）；对于支配集问题，一个全称量词后接一个存在量词即够（"对所有顶点，存在某个顶点使得……"），诸如此类。我们给出一个完整刻画，表明对每种可能的一阶量词序列，所描述问题的参数化复杂度的上限。新发现的可解问题与不可解问题之间的分界线不同于经典设定中的已知分界线，并且取决于参数是量化集合大小的下界、上界还是等于该大小。