A Declarative Foundation of λProlog with Equality

We build general model-theoretic semantics for higher-order logic programming languages. Usual semantics for first-order logic is two-level: i.e., at a lower level we define a domain of individuals, and then, we define satisfaction of formulas with respect to this domain. In a higher-order logic which includes the propositional type in its primitive set of types, the definition of satisfaction of formulas is mutually recursive with the process of evaluation of terms. As result of this in higher-order logic it is extremely difficult to define an effective semantics. For example to define T p operator for logic program P, we need a fixed domain without regard to interpretations. In usual semantics for higher-order logic, domain is dependent on interpretations. We overcome this problem and argue that our semantics provides a more suitable declarative basis for higher-order logic programming than the usual general model semantics. We develop a fix point semantics based on our model. We also show that a quotient of the domain of our model can be the domain of a model for higher-order logic programs with equality