Algebraic methods in computational complexity and decidability

这是一个资助我正在进行的泛代数研究的建议，泛代数是纯数学的一个分支。广义地说，我所建议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）在适当的一般性水平上研究这些模式;（3）通过研究局部发生的现象，提供对全球结构的洞察力;（4）通过拓宽各自领域的知识，我的研究福尔斯主要集中在方程逻辑及其算法方面。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在多大程度上可以被描述。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;确定全局结构在多大程度上会受到这种局部行为的影响;以及这些性质是否可以在算法上被识别;以及，如果是这样的话，这样的算法是否是计算上易处理的。该研究提案的动机，除其他问题外，在理论计算机科学领域的基本问题之一，自然发生在人工智能领域的计算查询的复杂性。

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 one of the fundamental questions in the area of theoretical computer science about the complexity of computational queries occurring naturally in the area of artificial intelligence.