On the Link between Partial Meet, Kernel, and Infra Contraction and its Application to Horn Logic

Standard belief change assumes an underlying logic containing full classical propositional logic. However, there are good reasons for considering belief change in less expressive logics as well. In this paper we build on recent investigations by Delgrande on contraction for Horn logic. We show that the standard basic form of contraction, partial meet, is too strong in the Horn case. This result stands in contrast to Delgrande’s conjecture that orderly maxichoice is the appropriate form of contraction for Horn logic. We then define a more appropriate notion of basic contraction for the Horn case, influenced by the convexity property holding for full propositional logic and which we refer to as infra contraction. The main contribution of this work is a result which shows that the construction method for Horn contraction for belief sets based on our infra remainder sets corresponds exactly to Hansson’s classical kernel contraction for belief sets, when restricted to Horn logic. This result is obtained via a detour through contraction for belief bases. We prove that kernel contraction for belief bases produces precisely the same results as the belief base version of infra contraction. The use of belief bases to obtain this result provides evidence for the conjecture that Horn belief change is best viewed as a hybrid version of belief set change and belief base change. One of the consequences of the link with base contraction is the provision of a representation result for Horn contraction for belief sets in which a version of the Core-retainment postulate features.

💡 Research Summary

The paper investigates belief contraction in Horn logic, a fragment of propositional logic where every clause is a Horn clause (at most one positive literal). While classical belief change theory assumes a full classical propositional language, many practical applications—such as rule‑based systems, databases, and logic programming—operate within Horn fragments. Building on Delgrande’s recent work on Horn contraction, the authors first demonstrate that the standard partial‑meet contraction, which selects the intersection of all maximal remainder sets, is overly strong for Horn belief sets. Because Horn clauses impose a strong syntactic restriction, the family of maximal Horn‑consistent remainder sets is often very small, and the partial‑meet operator collapses many plausible contraction outcomes into a single, sometimes unintuitive, result. This finding contradicts Delgrande’s conjecture that “orderly maxichoice” (a refined maxichoice operator) would be the appropriate contraction method for Horn logic.

To address the inadequacy of partial‑meet, the authors introduce infra contraction, a new basic contraction operator inspired by the convexity property that characterises partial‑meet in full propositional logic. Convexity states that if two remainder sets A and B are ordered by inclusion (A ⊆ C ⊆ B), then any intermediate set C is also a remainder set. By extending this idea to Horn logic, the authors define infra remainder sets: these are not required to be maximal Horn‑consistent subsets, but any set that lies between two maximal remainder sets is admissible. This relaxation yields a richer family of possible contractions while still preserving the essential rationality postulates (success, inclusion, and vacuity).

The central technical contribution is a two‑step equivalence proof:

1. Horn‑set level: The authors show that infra contraction on Horn belief sets coincides exactly with Hansson’s classical kernel contraction when the latter is restricted to Horn languages. Kernel contraction removes a kernel—a minimal subset whose removal makes the target belief no longer entailed. By constructing kernels from infra remainder sets, the paper proves that every infra contraction result can be obtained by a kernel contraction and vice‑versa.

2. Belief‑base level: To obtain the above result, the authors first establish that infra contraction and kernel contraction are identical on belief bases (sets of formulas considered without closure under logical consequence). Because bases are more fine‑grained than closed belief sets, the proof can exploit the combinatorial structure of Horn clauses more directly. The equivalence on bases then lifts to the set level via the standard closure operation.

These equivalences provide strong evidence for the authors’ conjecture that Horn belief change should be viewed as a hybrid of belief‑set change (which emphasizes closure) and belief‑base change (which emphasizes syntactic granularity). In Horn logic, the two perspectives converge: the syntactic flexibility of bases compensates for the limited expressive power of Horn clauses, while the closure‑based kernel view guarantees rationality.

A further contribution is a representation theorem for Horn contraction on belief sets. The theorem adapts the classic Core‑retainment postulate (which requires that any removed belief be part of some minimal “core” that must be given up) to the Horn setting, using infra remainder sets as the underlying structure. The resulting postulate—call it Horn‑Core‑retainment—captures exactly the behavior of infra (and thus kernel) contraction in Horn logic.

The paper concludes with several implications and future directions. It suggests that the infra‑kernel correspondence may extend to other restricted logics (e.g., Krom or 2‑CNF), and that practical systems employing Horn knowledge bases could adopt infra contraction to achieve more intuitive belief revision while preserving computational tractability. Moreover, the hybrid viewpoint encourages further research into unified frameworks that seamlessly integrate belief‑set and belief‑base operations across a spectrum of logical languages.