A Fixpoint Semantics for Ordered Logic

TLDR

Classical logic programs with negation by failure are special cases of OL theories. The authors develop semantics for ordered logic (OL), a logic that captures key aspects of object‑oriented programming such as object identity, multiple inheritance, and defaults. OL is built on a partially ordered structure of logical theories that act as objects, and a non‑deterministic fixpoint procedure generates all models, which for a well‑behaved subclass yields exactly the preferred models based on lack of assumptions. OL is non‑monotonic under its natural semantics, and the fixpoint procedure can generate the preferred models for a well‑behaved subclass, enabling a syntactic characterization of logic programs with stable models.

Abstract

We develop semantics for a logic, called ordered logic (OL), which models the most important aspects of object-oriented programming languages, such as object identity, multiple inheritance and defaults. The logic is based on a partially ordered structure of logical theories, which play the role of objects. OL is non-monotonic under the natural modeltheoretic semantics. A non-deterministic procedure is defined that has all models as fixpoints. It is shown that, for a well-behaved subclass of theories, this procedure can be used to generate exactly the set of ‘preferred’ models, where preference is based on the lack of ‘assumptions’. Classical logic programs with negation by failure are special cases of OL theories. From the above we can then derive a syntactic characterization of logic programs with stable models.