Complexity and decidability of algebraic and relational structures

这是一份资助我正在进行的普适代数研究的提案，普适代数是纯数学的一个分支。从广义上讲，我提出的研究的主要长期目标是：(1)在数学知识的各个分支中发现统一的模式，如逻辑、代数和离散数学；(2)在适当的普遍性水平上研究这些模式；(3)通过研究局部现象来洞察全球结构；(4)通过拓宽各自领域的知识，或通过寻找多学科的应用，将一些有价值的东西回报给一般的数学界。我的研究主要集中在等式逻辑和它的算法方面。方程逻辑的主要研究对象是代数的非标准版本（方程理论）及其抽象模型。一般的问题是确定这些代数的非标准版本的模型能被描述到什么程度。更详细地说，我特别感兴趣的是使用组合几何性质的工具来研究代数和关系结构的局部结构；确定这种局部行为将在多大程度上影响全球结构；这些属性是否可以被算法识别；如果是这样，那么这种算法在计算上是否可处理。研究计划的动机，除其他问题外，是由一个长期开放的问题，在普遍代数领域，这是表征所有有限可决定的局部有限变量。我的目标是通过使用驯服同余理论和模型理论的方法解决这个问题，在我的研究领域做出重大贡献。

This is a proposal to fund my ongoing research in universal algebra, which is a branch of pure mathematics. Broadly speaking, the main long-term goals of my proposed research are (1) to discover unifying patterns in various branches of mathematical knowledge, such as logic,algebra, and discrete mathematics; (2) to study these patterns at an appropriate level of generality; (3) to provide insight into global structure by studying phenomena that occur locally; and (4) to return something of the value to the general mathematical community, by either broadening the knowledge in the respective areas, or by finding multi-disciplinary applications. My research falls mostly within the boundaries of equational logic and its algorithmic aspects. The main object of study of equational logic are nonstandard versions of algebra (equational theories) and their abstract models. The general problem is to determine to what extent the models of these nonstandard versions of algebra can be described. In more detail, I am particularly interested in using the tools of combinatorial-geometric nature to study the local structure of algebras and relational structures; to determine to what extent the global structure will be influenced by this local behaviour; and whether such properties can be recognized algorithmically; and, if so, whether such an algorithm is computationally tractable. The research proposal is motivated, among other questions, by a long-standing open problem in the field of universal algebra, which is to characterize all finitely decidable locally finite varieties.   I aim to make a major contribution in my field of research by solving this problem using the methods of tame congruence theory and model theory.