Algebraic methods in computational complexity and decidability

计算复杂性和可判定性的代数方法

• 批准号: 249684-2012
• 负责人: Delic, Dejan
• 金额: $ 0.87万
• 依托单位: Ryerson University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635410
• 关键词: Algebraic; methods; computational; complexity; decidability

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.

这是一项资助我正在进行的泛代数研究的提案，泛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）发现数学知识各个分支的统一模式，例如如逻辑、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) 通过研究局部发生的现象来洞察全球结构； （4）通过拓宽各自领域的知识，或者通过寻找多学科应用，向一般数学界返回一些有价值的东西。我的研究主要落在方程逻辑及其算法方面的范围内。方程逻辑的主要研究对象是代数（方程理论）及其抽象模型的非标准版本。一般问题是确定这些非标准代数版本的模型可以在多大程度上被描述。更详细地说，我对使用组合几何性质的工具来研究代数的局部结构和关系结构特别感兴趣；确定全球结构将在多大程度上受到这种本地行为的影响；以及这些属性是否可以通过算法识别；如果是这样，这样的算法在计算上是否易于处理。 除其他问题外，该研究提案的动机是理论计算机科学领域的基本问题之一，即人工智能领域自然发生的计算查询的复杂性。