Dr. Terry Halpin's PhD thesis aimed to provide a homomorphic or isomorphic mapping between sentences of a graphical language called NIAM (Natural language Information Analysis Method) and sentences of a theory of logic under finite model theory.

NIAM is a graphical language used for conceptual modelling of information systems, where graphical models are used to represent the concepts, relationships, and constraints of the system being modelled. However, in the absence of any formally proved homomorphic or isomorphic mapping to sentences of a formal theory of logic, NIAM would be just a colllection of drawings without a formal, mathematical foundation and lacking the rigor of a logical language.

To address this issue, Dr Halpin proposed a mapping between NIAM models and a logical language based on finite model theory, which is a branch of model theory that deals with the study of interpretations of sentences in finite structures.

The aim of the mapping was to provide a rigorous and formal foundation for NIAM modelling and to enable NIAM models to be verified, validated, and analysed using formal logical methods. The mapping was based on the idea of translating NIAM models into logical sentences, and vice versa, so that the meaning of the models could be expressed formally in logical language.

The mapping involved defining a set of rules for translating NIAM models into logical sentences and for interpreting the logical sentences as NIAM models. The mapping, within the thesis, provided evidence of a homomorphism between sentences of NIAM and KL (Knowledge Language), a formal theory under finite model theory, meaning that it preserved the structure of the models and the meaning of the concepts and relationships represented in the models to the extent possible under a homorphism.

The mapping enabled NIAM models to be analysed using logical techniques, such as proof theory and model checking, and also allowed for the automatic generation of database schemas and programming code from the models.