Typed feature logics have been employed as description languages in modern type-oriented grammar theories like HPSG and have laid the theoretical foundations for many implemented systems. However, recursivity pose severe problems and have been addressed through specialized powerdomain constructions which depend on the particular view of the logician.
In this paper, we argue that definite equivalences introduced by Smolka can serve as the formal basis for arbitrarily formalized typed feature structures and typed feature-based grammars/lexicons, as employed in, e.g., TFS or TDL. The idea here is that type definitions in such systems can be transformed into an equivalent definite program, whereas the meaning of the definite program then is identified with the denotation of the type system. Now, models of a definite program P can be characterized by the set of ground atoms which are logical consequences of the definite program. These models are ordered by subset inclusion and, for reasons that will become clear, we propose the greatest model as the intended interpretation of P, or equivalent, as the denotation of the associated type system.
Our transformational approach has also a great impact on nonmonotonically defined types, since under this interpretation, we can view the type hierarchy as a pure transport medium, allowing us to get rid of the transitivity of type information (inheritance), and yielding a perfectly monotonic definite program.