Algebraic Methods for Quantified Constraints

约束满足问题(CSP)是一种能够以自然的方式表达计算机科学许多领域及更多领域中出现的广泛问题的范例，如人工智能、计算语言学、计算生物学、组合学和数据库。CSP实例包括一个有限的变量集、一个值集(域)和一个有限的约束集。任务是将变量赋给值，以便满足所有约束。CSP可以看作是对一阶逻辑片段的模型检验问题，该一阶逻辑片段只有存在量化、合取和相等。当加入泛量化时，新的范式就是量化约束满足问题(QCSP)。当问题被允许的约束语言参数化时，对于某些类型的约束，找到解的任务在计算上很容易(即，需要诸如运行时间和内存的可行的计算资源量)，但对于许多类型，它在计算上是困难的。CSP的跨有限约束语言的复杂性分类于2017年由Bulatov和朱独立完成，现在已知是多项式时间和NP-完全之间的二分法。QCSP的类似分类问题代表了一阶逻辑中唯一一个基于连接的句法片段，其中结果未知。最近，朱和马丁(2019)驳斥了陈猜想，即只有多项式时间、NP-完全和P空间-完全的复杂性才会在这种分类中。目前已知的奇异复杂类，如DP-完全、Theta^P_2-完全和Pi^P_2-完全可以在QCSP中实现。朱和马丁(2019)完成了基于多项式时间、NP-完全、co-NP-完成和Pspace-完成的三元组域分类。这项提议旨在将朱的新方法超越三要素的情况，扩展到更大的领域。该建议将考虑所有有限域，以及一些无限域，在这些域中，约束语言建模来自时间推理的概念。该提案的中心目标是绘制出这些约束语言之间计算复杂性的图景。

The constraint satisfaction problem (CSP) is a paradigm in which it is possible to express, in a natural way, a wide range of problems arising in many areas of Computer Science and beyond, e.g. Artificial Intelligence, Computational Linguistics, Computational Biology, Combinatorics and Databases. A CSP instance involves a finite set of variables, a set of values (the domain) and a finite set of constraints. The task is to assign the variables to the values so as to satisfy all of the constraints. The CSP can be seen as a model-checking problem for the fragment of first-order logic that has just existential quantification, conjunction and equality. When universal quantification is added, the new paradigm is the quantified constraint satisfaction problem (QCSP). When the problem is parameterised by the language of constraints permitted, one finds for some types of constraint, the task of finding a solution is computationally easy (i.e. requires a feasible amount of computational resources such as running time and memory), but for many types, it is computationally hard. The complexity classification across finite constraint languages for the CSP was completed in 2017, independently by Bulatov and Zhuk, and is now known to be a dichotomy between polynomial time and NP-complete. The similar classification problem for the QCSP represents the only connective-based syntactic fragment of first-order logic where the outcome is not known. Recently, Zhuk and Martin (2019) refuted the Chen Conjecture that only complexities of polynomial time, NP-complete and Pspace-complete would in this classification. It is now known that exotic complexity classes such as DP-complete, Theta^P_2-complete and Pi^P_2-complete can be realised in QCSPs. Zhuk and Martin (2019) completed the three-element domain classification as a tetrachotomy between polynomial time, NP-complete, co-NP-complete and Pspace-complete. This proposal aims to take Zhuk's new methods beyond the three-element case to larger domains. The proposal will consider all finite domains, as well as some infinite domains where the constraint languages model notions from temporal reasoning. The central objective of the proposal is to map out landscapes of computational complexity across these constraint languages.