Algebraic Methods for Quantified Constraints

约束满足问题（CSP）是一种范式，其中可以以自然的方式表达计算机科学及其他许多领域中出现的各种问题，例如人工智能，计算语言学，计算生物学，组合学和数据库。CSP实例涉及有限的变量集、值集（域）和有限的约束集。任务是将变量分配给值，以满足所有约束。CSP可以被看作是一个模型检查问题的片段的一阶逻辑，只有存在量化，合取和平等。当加入泛量化时，新的范例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，人们发现对于某些类型的约束，找到解决方案的任务在计算上是容易的（即需要可行数量的计算资源，例如运行时间和内存），但对于许多类型，它在计算上是困难的。CSP的有限约束语言复杂度分类于2017年由Bulatov和Zhuk独立完成，现在已知是多项式时间和NP完全之间的二分法。类似的分类问题的QCSP代表唯一的连接为基础的句法片段的一阶逻辑的结果是未知的。最近，Zhuk和Martin（2019）反驳了Chen猜想，即只有多项式时间，NP-完全和P空间-完全的复杂性才能在这种分类中。目前已知奇异复杂性类如DP-完全、Theta^P_2-完全和Pi^P_2-完全都可以在QCSP中实现。Zhuk和Martin（2019）完成了三元素域分类，作为多项式时间，NP完全，co-NP完全和Pspace完全之间的四分法。该提案旨在将Zhuk的新方法超越三要素的情况，以更大的域。该建议将考虑所有有限域，以及一些无限域的约束语言模型的概念从时间推理。该提案的中心目标是绘制出这些约束语言的计算复杂性的景观。

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.