Recursive definitions of predicates are usually interpreted either inductively or coinductively. Recently, a more powerful approach has been proposed, called flexible coinduction, to express a variety of intermediate interpretations, necessary in some cases to get the correct meaning. We provide a detailed formal account of an extension of logic programming supporting flexible coinduction. Syntactically, programs are enriched by coclauses, clauses with a special meaning used to tune the interpretation of predicates. As usual, the declarative semantics can be expressed as a fixed point which, however, is not necessarily the least, nor the greatest one, but is determined by the coclauses. Correspondingly, the operational semantics is a combination of standard SLD resolution and coSLD resolution. We prove that the operational semantics is sound and complete with respect to declarative semantics restricted to finite comodels.

A sufficient and necessary condition is given under which least Herbrand models exactly characterize the answers of definite clause programs.

In this paper, we address the problem of defining a fixpoint semantics for Constraint Handling Rules (CHR) that captures the behavior of both simplification and propagation rules in a sound and complete way with respect to their declarative semantics. Firstly, we show that the logical reading of states with respect to a set of simplification rules can be characterized by a least fixpoint over the transition system generated by the abstract operational semantics of CHR. Similarly, we demonstrate that the logical reading of states with respect to a set of propagation rules can be characterized by the greatest fixpoint. Then, in order to take advantage of both types of rules without losing fixpoint characterization, we present a new operational semantics with persistent constraints.

We finally establish that this semantics can be characterized by two nested fixpoints, and we show that the resulting language is an elegant framework to program using coinductive reasoning.

The paper presents a simple and concise proof of correctness of the magic transformation. We believe that it may provide a useful example of formal reasoning about logic programs. The correctness property concerns the declarative semantics. The proof, however, refers to the operational semantics (LD-resolution) of the source programs. Its conciseness is due to applying a suitable proof method.