Feasibly categorical models

Abstract

We define a notion of a Scott family of formulas for a feasible model and give various conditions on a Scott family which imply that two models with the same family are feasibly isomorphic. For example, if A and B possess a common strongly p-time Scott family and both have universe {1}*, then they are p-time isomorphic. These results are applied to the study of permutation structures, linear orderings, equivalence relations, and Abelian groups. For example, conditions on two permutation structures (A, f) and (B, g) are given which imply that (A, f) and (B, g) are p-time isomorphic.