The research proposed centers on investigations of the structures of sets and functions ordered by relative complexity of computation. Particular emphasis will be placed on issues of definability and automorphisms. Also included in this area is the analysis of the relations between the difficulty of computing functions and other issues such as rates of growth, complexity of their definitions in arithmetic and the strength of axiom systems needed to prove their existence (reverse mathematics). The emphasis in reverse mathematics will be on analyzing basic combinatorial principles that seem to lie outside the scope of the standard theories studied. Applications of the methods of pure computability theory will be made in the areas of model theory and notions used in differential geometry. The emphasis in computable model theory will be on the ways in which choosing different representations of a given structure affect the computational complexity of various relations or procedures on the elements of the structure. Issues in automata theory and especially automatic structures that deal with computable complexity will also be addressed.
The proposed project includes research into a broad range of topics in computability theory (recursion theory) and logic both theoretical and applied to other areas of mathematics and computer science. At the foundational level, this work illuminates the nature of relative complexity of computation, the strength of axioms needed to prove standard mathematical theorems and the relations between these areas. In practical terms, results in this area (computable mathematics and model theory as well as reverse mathematics) at times indicate that there are no algorithms for certain important tasks or that more information than might have been expected is needed to write programs calculating the desired results. The work related to automata theory and automatic structures is based on a very limited model of computation that is often relevant to practical computing problems. The theoretical and foundational analysis of structures whose basic relations and functions are computable by such automata should also eventually be of practical significance.