Logic and computational complexity

逻辑和计算复杂性

• 批准号: 105666-2006
• 负责人: Urquhart, Alasdair
• 金额: $ 2.26万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=365048
• 关键词: Logic; computational; complexity

The main aim of the research is to understand why certain problems are difficult to solve using computers.  Typical of such problems are those in which a certain number of simple constraints must be satisfied, as in constructing schedules.  The characteristic feature of such problems (known by the technical name of NP-complete problems), is that a solution, if found, is easy to check, but deciding whether or not such a solution exists in the worst case requires an exponentially long time relative to the size of the input.   The principal aim of the theory of complexity theory is to decide whether or not such problems, that seem to require exponentially long computation times in the worst case, really do require such times. My own research aims to work towards showing that exponentially long times are in fact required in the worst case by proving lower bounds on the length of certain kinds of proofs.  Since a computation showing that a solution doesn't exist can be considered as a proof of a kind, lower bounds on the length of proofs show that certain kinds of algorithms (including the algorithms most often used in practice to solve such problems) cannot be efficient in the worst case.

研究的主要目的是了解为什么某些问题很难用计算机解决。典型的这类问题是那些必须满足一定数量的简单约束的问题，如在制定进度计划时。这类问题的特征是：（被称为NP完全问题的技术名称），是一个解决方案，如果找到，很容易检查，但是在最坏的情况下决定这种解是否存在需要相对于输入的大小呈指数增长的时间。 复杂性理论的主要目的是确定在最坏情况下似乎需要指数级长计算时间的问题是否真的需要这样的时间。我自己的研究旨在通过证明某些证明长度的下限来证明在最坏的情况下实际上需要指数级长的时间。由于计算显示不存在解决方案可以被视为一种证明，证明长度的下限表明，某些类型的算法（包括在实践中最常用于解决这些问题的算法）在最坏的情况下不能是有效的。