Categoricity of computable infinitary theories

Computable structures of Scott rank are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of , this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank whose computable infinitary theories are each -categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank , which guarantee that the resulting structure is a model of an -categorical computable infinitary theory. Work on this paper began at the Workshop on Model Theory and Computable Structure Theory at University of Florida Gainesville, in February, 2007. The authors are grateful to the organizers of this workshop. They are also grateful for financial support from National Science Foundation grants DMS DMS 05-32644, DMS 05-5484. The second author is also grateful for the support of grants RFBR 08-01-00336 and NSc-335.2008.1.