Logic and computational complexity

这项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。 典型的此类问题是那些必须满足一定数量的简单约束的问题，例如在构建进度表时。 这类问题的特征（以NP完全问题的技术名称而闻名）是，如果找到了解决方案，则很容易检查，但在最坏的情况下，决定这样的解决方案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类型的证明的长度的下限来证明在最坏的情况下实际上需要指数级的长时间。 由于一个计算表明一个解不存在可以被认为是一种证明，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的算法来解决这类问题）在最坏的情况下是无效的。

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.