Mathematical Sciences: Recursion Theory

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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.