We introduce model-theoretic semantics  for Higher-Order Horn logic programming language. One advantage of logic programs over conventional non-logic programs has been that the least fixpoint is equal to the least model, therefore it is associated to logical consequence and has a meaningful declarative interpretation. In simple theory of types  on which Higher-Order Horn logic programming language is based, domain is dependent on interpretation . To define T p operator for a logic program P, we need a fixed domain without regard to interpretation which is usually taken to be a set of atomic propositions. We build a semantics where we can fix a domain while changing interpretations. We also develop a fixpoint semantics based on our model, and show that we can get the least fixpoint which is the least model. Using this fixpoint we prove the completeness of the interpreter of our language in .
Bai, Mino and Blair, Howard A., "General Model Theoretic Semantics for Higher-Order Horn Logic Programming" (1992). Electrical Engineering and Computer Science Technical Reports. Paper 173.