Mathematical Sciences: Recursion Theory

莱姆普将继续研究递归理论。本文的主要课题是研究包含跳跃算子信息的一阶r.e.度集和图灵度集理论片段的可判定性。Lempp希望解决深度度问题的策略（与Slaman一起）将有助于将偏阶同态嵌入到这些度集合中。其次，Lempp将研究在由迭代跳跃定义的等价关系的r.e.度模中存在一个连接操作。在一个强制论证中，Lempp已经证明了图灵度模这种等价关系中不一定存在连接。此外，Lempp计划通过证明非密度定理在d.r.e.度上的扩展来结束与Cooper和Watson开始的工作。最后，Lempp打算继续他在索引集方面的工作。在他的论文中，Lempp展示了一大族这样的指标集，这些指标集的r.e.超集构成了布尔代数；他现在计划研究非布尔代数的情况和r.e.度的索引集。递归理论是对可计算性的理论极限的研究，这样的研究完成后，很可能会重新接触到它们在电子计算机领域的原始灵感。

Lempp will pursue research in recursion theory. The main project is to study the decidability of fragments of the first- order theories of the sets of r.e. degrees and of Turing degrees where information about the jump operator is included in the language. Lempp hopes that strategies from the solution to the deep degree problem (with Slaman) will help with the necessary embeddings of homomorphisms of partial orders into these sets of degrees. Secondly, Lempp will investigate the existence of a join operation in the r.e. degrees modulo some equivalence relation defined by iterated jumps. In a forcing argument, Lempp has already shown that the join does not necessarily exist in the Turing degrees modulo this equivalence relation. Furthermore, Lempp plans to conclude work begun with Cooper and Watson by proving an extension of the nondensity theorem in the d.r.e. degrees. Finally, Lempp intends to continue his work in index sets. In his thesis, Lempp has exhibited a large family of such index sets for r.e. sets the r.e. supersets of which form a Boolean algebra; he now plans to look at the non-Boolean algebra case and at index sets of r.e. degrees. Recursion theory being the study of the theoretical limits of computability, such investigations when completed may well renew contact with their original inspiration in the world of electronic computers.