Model construction for general constrained Horn clauses

Develop a method for constructing models from backward-execution saturation for arbitrary constrained Horn clause sets, including sets whose saturated goals contain non-linear clauses.

Background

The paper develops backward execution (BE) using hyper-SLD resolution and shows how, when BE terminates with satisfiability and the saturated goals are linear, a model can be extracted by interpreting each predicate through the conjunction of the negated constraints associated with that predicate’s goals. The authors explain that this construction does not directly extend to non-linear goals because such goals express dependencies between predicate interpretations, and a greatest fixed point need not exist.

The unresolved problem is therefore to provide a general model-construction procedure for backward execution beyond the linear-goal fragment. Such a procedure would make BE capable not only of refuting unsatisfiable clause sets but also of explicitly producing models for satisfiable sets with arbitrary non-linear goals.