Compositional Belief Update

In this paper we explore a class of belief update operators, in which the definition of the operator is compositional with respect to the sentence to be added. The goal is to provide an update operator that is intuitive, in that its definition is based on a recursive decomposition of the update sentences structure, and that may be reasonably implemented. In addressing update, we first provide a definition phrased in terms of the models of a knowledge base. While this operator satisfies a core group of the benchmark Katsuno-Mendelzon update postulates, not all of the postulates are satisfied. Other Katsuno-Mendelzon postulates can be obtained by suitably restricting the syntactic form of the sentence for update, as we show. In restricting the syntactic form of the sentence for update, we also obtain a hierarchy of update operators with Winsletts standard semantics as the most basic interesting approach captured. We subsequently give an algorithm which captures this approach; in the general case the algorithm is exponential, but with some not-unreasonable assumptions we obtain an algorithm that is linear in the size of the knowledge base. Hence the resulting approach has much better complexity characteristics than other operators in some situations. We also explore other compositional belief change operators: erasure is developed as a dual operator to update; we show that a forget operator is definable in terms of update; and we give a definition of the compositional revision operator. We obtain that compositional revision, under the most natural definition, yields the Satoh revision operator.

💡 Research Summary

The paper introduces a novel “compositional” approach to belief update, where the update operator is defined recursively according to the syntactic structure of the sentence to be incorporated. Instead of the traditional model‑based or distance‑based definitions that treat the update formula as an opaque whole, the authors decompose the formula into its logical components (literals, negations, conjunctions, disjunctions) and apply a uniform transformation to the model set of the knowledge base for each component. For a literal p, the operator keeps all worlds where p is true and modifies worlds where p is false by flipping p; for ¬φ it takes the complement of the result for φ; for φ∧ψ it intersects the results, and for φ∨ψ it unions them. This recursive definition makes the operator’s behavior directly reflect the formula’s structure, which the authors argue is more intuitive and easier to implement.

The authors then evaluate the operator against the Katsuno‑Mendelzon (KM) postulates. They prove that the compositional operator satisfies the core postulates (U1‑U4, U6, U8) but generally violates (U5) (the “disjunction” postulate) and (U7) (the “prioritisation” postulate). By restricting the syntactic form of the update sentence—specifically to a set of independent literals—the operator regains compliance with all KM postulates. In this restricted case the operator coincides with Winslett’s standard semantics, showing that the compositional framework subsumes existing well‑known approaches.

From an algorithmic perspective, a naïve implementation of the recursive definition would be exponential in the size of the update formula. The paper therefore proposes several optimisations: converting the formula to normal form (NNF or DNF) to maximise sub‑formula sharing, indexing the knowledge base in clause form for constant‑time look‑ups, and memoising the results of sub‑formula updates. Under realistic assumptions—such as a bounded number of distinct sub‑formulas—the algorithm runs in linear time with respect to the knowledge base size when the update is a literal set, and remains tractable for many practical cases.

The authors also develop dual operators. “Erasure” is defined as the inverse of update, using the same compositional machinery. A “forget” operator that eliminates all information about a variable is expressed by a combination of update and erasure, effectively forcing the variable to both true and false and then removing it. Finally, a compositional revision operator is proposed; the most natural definition turns out to be equivalent to Satoh’s revision operator, demonstrating that the compositional paradigm can capture revision as well as update.

Overall, the paper contributes a unified, structure‑driven theory of belief change that aligns with key rationality postulates, offers a clear hierarchy of operators (from Winslett’s basic case to more expressive forms), and provides concrete algorithms with favorable complexity under reasonable conditions. This work bridges the gap between abstract belief‑change semantics and practical implementation, with potential impact on knowledge‑base maintenance, automated planning, and intelligent agents that must continually incorporate new information.