An uninterpreted program (UP) is a program whose semantics is defined over the theory of uninterpreted functions. This is a common abstraction used in equivalence checking, compiler optimization, and program verification. While simple, the model is sufficiently powerful to encode counter automata, and, hence, undecidable. Recently, a class of UP programs, called coherent, has been proposed and shown to be decidable. We provide an alternative, logical characterization, of this result. Specifically, we show that every coherent program is bisimilar to a finite state system. Moreover, an inductive invariant of a coherent program is representable by a formula whose terms are of depth at most 1. We also show that the original proof, via automata, only applies to programs over unary uninterpreted functions. While this work is purely theoretical, it suggests a novel abstraction that is complete for coherent programs but can be soundly used on arbitrary uninterpreted (and partially interpreted) programs.