Motivated by applications to open program reasoning such as maximal specification inference, this paper studiesoptimal CHC solving, a problem to compute maximal and/or minimal solutions of constrained Horn clauses (CHCs). This problem and its subproblems have been studied in the literature, and a major approach is to iteratively improve a solution of CHCs until it becomes optimal. So a key ingredient of optimization methods is the optimality checking of a given solution. We propose a novel optimality checking method, as well as an optimization method using the proposed optimality checker, based on a computational theoretical analysis of the optimality checking problem. The key observation is that the optimality checking problem is closely related to the termination analysis of programs, and this observation is useful both theoretically and practically. From a theoretical perspective, it clarifies a limitation of an existing method and incorrectness of another method in the literature. From a practical perspective, it allows us to apply techniques of termination analysis to the optimality checking of a solution of CHCs. We present an optimality checking method based on constraint-based synthesis of termination arguments, implemented our method, evaluated it on CHCs that encode maximal specification synthesis problems, and obtained promising results.

We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of the given first-order-fixpoint logic formula (formally, an instance in a language called µCLP) to a constraint satisfaction problem for a recently introduced predicate constraint language. Coupled with an existing sound-and-relatively-complete solver for the constraint language, this novel reduction alone already gives a sound and relatively complete method for deciding µCLP validity, but we further improve it to a novelmodular primal-dualmethod. The key observations are (1) µCLP is closed under complement such that each (co-)inductive predicate in the originalprimalinstance has a corresponding (co-)inductive predicate representing its complement in thedualinstance obtained by taking the standard De Morgan’s dual of the primal instance, and (2)partial solutionsfor (co-)inductive predicates synthesized during the constraint solving process of the primal side can be used as sound upper-bounds of the corresponding (co-)inductive predicates in the dual side, and vice versa. By solving the primal and dual problems in parallel and exchanging each others’ partial solutions as sound bounds, the two processes mutually reduce each others’ solution spaces, thus enabling rapid convergence. The approach is alsomodularin that the bounds are synthesized and exchanged at granularity of individual (co-)inductive predicates. We demonstrate the utility of our novel fixpoint logic solving by encoding a wide variety of temporal verification problems in µCLP, including termination/non-termination, LTL, CTL, and even the full modal µ-calculus model checking of infinite state programs. The encodings exploit the modularity in both the program and the property by expressing each loops and (recursive) functions in the program and sub-formulas of the property as individual (possibly nested) (co-)inductive predicates. Together with our novel modular primal-dual µCLP solving, we obtain a novel approach to efficiently solving a wide range of temporal verification problems.