Distance-Based Belief Merging Operators

• Distance-based belief merging operators are procedures that resolve inconsistencies across multiple knowledge bases by selecting interpretations that minimize a defined distance under given integrity constraints.
• They employ metrics like Hamming and drastic distances combined with aggregation functions such as sum and lexicographic orderings to accurately fuse diverse belief sets.
• Their applicability spans symbolic AI, disagreement-aware summarisation, and logic program merging, balancing expressivity with computational complexity and fragment-specific requirements.

Distance-based belief merging operators are a family of aggregation procedures for resolving inconsistencies across multiple knowledge bases or belief sets, relying on metrics over propositional interpretations. These operators select merged beliefs that minimize a defined notion of "distance" from the input profile, subject to integrity constraints. Their theoretical underpinnings, postulate compliance, and fragment-dependent expressivity have made them central both to symbolic AI research and recent applications in disagreement-aware information fusion.

1. Formal Definition and Construction

Let $U$ denote a finite set of propositional atoms. An interpretation $\omega \subseteq U$ is a subset of atoms set to $\text{true}$, and a knowledge base $K$ is a set of propositional formulas, with $\operatorname{Mod}(K)$ as its model set. For a profile $E = (K_1, \ldots, K_n)$ of knowledge bases and an integrity constraint $\mu$, a distance-based merging operator $\Delta^{d, \otimes}_\mu$ is defined as follows. Fix a pseudo-distance $d: 2^U \times 2^U \rightarrow \mathbb{R}_{\geq 0}$ and a monotone aggregation function $\otimes$ over $n$ arguments.

• The distance between an interpretation $\omega$ and a knowledge base $K_i$ is $$d(\omega, K_i) = \min\{d(\omega, \omega'): \omega' \in \operatorname{Mod}(K_i)\}$$.
• The aggregated distance to profile $E$ is $$d^\otimes(\omega, E) = \otimes(d(\omega, K_1), \ldots, d(\omega, K_n))$$.
• The merged models are:

$$\operatorname{Mod}(\Delta^{d, \otimes}_\mu(E)) = \operatorname{Argmin}_{\omega \in \operatorname{Mod}(\mu)} d^\otimes(\omega, E).$$

Common choices for $d$ include the Hamming distance $$d_H(\omega, \omega') = |\{p \in U : \omega(p) \ne \omega'(p)\}|$$ and the drastic distance $d_D(\omega, \omega') = 0$ if $\omega = \omega'$, $1$ otherwise. Standard aggregators are sum ($\Sigma$), leximax (GMax), and leximin (GMin) (Haret et al., 2016, Creignou et al., 2014, Aghaebe et al., 8 Jan 2026).

2. Rationality Postulates and Representation

Distance-based merging operators were designed to satisfy the rationality postulates (IC0–IC8) introduced by Konieczny and Pino Pérez, which include consistency, syntax irrelevance, and majority/reinforcement principles. In particular:

• Any $\Delta^{d, \otimes}$ with $d$ a metric and $\otimes$ a monotone aggregator (such as sum or GMax) satisfies all IC0–IC8 postulates in full classical logic.
• The outcome depends only on the semantics $\operatorname{Mod}(K_i)$, not formula syntax.
• For drastic distance, all monotone aggregators coincide: counting bases violated is invariant to aggregation function.
• Every operator satisfying IC0–IC8 and natural continuity conditions is representable as some $\Delta^{d, \otimes}$ (Haret et al., 2016).
• In fragments, refined operators may lose satisfaction of IC5–IC8 but always preserve IC0–IC3; IC4 (unbiasedness) can be regained via “fairness” in the refinement scheme (Creignou et al., 2014).

3. Expressivity and Behaviour in Logical Fragments

Fragments of propositional logic, such as the Horn, Krom/2CNF, or 1CNF classes, are closed under specific operations (e.g., conjunction or $\mathrm{MAJ}_3$). The expressibility and distributability of knowledge bases with respect to merging operators is highly fragment- and distance-dependent.

• Drastic Distance: Any knowledge base $K$ (regardless of complexity) can be $\mathcal{F}$-distributed with respect to $\Delta^{D, \otimes}$. For each model $\omega \in \operatorname{Mod}(K)$, a singleton base with model $\{\omega\}$ is formed, and their merging with an integrity constraint yields $K$. However, $\mathcal{F}$-simplifiability (profile size $1$) is possible only if $K$ is $\mathcal{F}$-expressible (Haret et al., 2016).
• Hamming Distance:
  • For $1$CNF, only $1$CNF-expressible $K$ can be distributed—the method cannot recover non-$1$CNF $K$.
  • For $2$CNF, every $K$ is $2$CNF-simplifiable under Hamming+sum (construction with fresh atoms).
  • For Horn, some non-Horn $K$ with at most two models can be simplified, but the general case remains open (Haret et al., 2016, Creignou et al., 2014).
• In compositional fragments, the merged result may not lie in the fragment even if inputs do, making refinements necessary to guarantee closure (Creignou et al., 2014).

4. Refinement and Fragment-Closure

Direct application of $\Delta^{d,\otimes}_\mu$ in fragments typically does not preserve fragment closure. To address this, $\Delta$-refinements are introduced:

• A $\Delta$-refinement operator $\Delta^*$ guarantees outcomes remain in the fragment by applying a closure operator based on a designated Boolean function (e.g., conjunction for Horn).
• Refined merging ensures fragment-located results by mapping the model set of the full merging to the closest $\beta$-closed superset, where $\beta$ is the closure operator of the fragment.
• Closure-based refinements always preserve (IC0–IC3); lexicographic refinements can sometimes preserve additional postulates, notably (IC5) and (IC7), but combination and reinforcement postulates generally fail (Creignou et al., 2014).

Table: Refinement Properties in Horn/Krom Fragments

Refinement type	IC0–IC3	IC4	IC5+IC7	IC6+IC8
Closure-based	Yes	If fair	No	No
Lexicographic	Yes	Yes	Yes	No
Mixed	Yes	Sometimes	No	No

5. Computational Complexity

Complexity of distance-based merger computation is dictated by fragment, profile size, and distance function:

• Model-checking a single distance-based merge in full propositional logic is coNP-complete.
• Reasoning in Horn or Krom fragments is intractable for revision and merging (PSPACE-hard in general).
• ASP encodings exist for distance-based merge in logic programs under answer set semantics, with complexity $\Pi^P_2$-complete for essential entailment checks. Cardinality-based revision becomes $P^{NP[\log n]}$ (0912.5511).
• In practical aggregation for small $m$ ($m=7$ aspects), brute-force enumeration is polynomial-time for all intended use-cases (e.g., $2^7 = 128$ worlds, $n \sim 100$ critics yields negligible cost) (Aghaebe et al., 8 Jan 2026).

6. Applications and Instantiations

Distance-based merging operators are instantiated across diverse reasoning tasks:

• Fragmented KB Distribution: Systematically reconstructing a complex $K$ from simple or fragment-restricted bases via selected merging operators (Haret et al., 2016).
• Disagreement-aware Summarisation: Aggregating conficting beliefs at the opinion-aspect level using $\Delta^{d,\otimes}$ (specifically sum-of-Hamming) delivers summaries faithful to underlying disagreement, outperforming direct generation-level fusion in LLMs for smaller models (Aghaebe et al., 8 Jan 2026).
• Logic Program Merging: In answer set programming, arbitration and IC-merging versions of distance-based operators are encoded in ASP, maintaining the original complexity class and enabling monotonic representations under the SE-model framework (0912.5511).
• Strict Fragment-Adherence: Use in Horn and Krom, via appropriate closure or lexicographic refinement, supports robust “AGM-style” change within tractably-representable fragments; a rich space of tradeoffs among postulate satisfaction, closure, and aggregation remains (Creignou et al., 2014).

7. Limitations, Trade-offs, and Open Problems

• The expressive power and distributability of distance-based operators vary markedly with the logic fragment and the chosen distance: drastic merging is universally distributable; Hamming merging exhibits fragment-sensitive expressivity (Haret et al., 2016).
• No refinement simultaneously retains all merging postulates (IC0–IC8) and ensures fragment closure; practical design requires trade-off decisions between postulate strength and syntactic tractability (Creignou et al., 2014).
• For certain fragments—notably Horn—full characterisation of which beliefs are distributable (or simplifiable) under Hamming-based merging remains unresolved.
• Efficient computation in strict fragments and large-scale settings is still poorly understood, especially for refined merging in nontrivial fragments (Creignou et al., 2014).
• Empirical findings indicate that belief-level distance-based aggregation offers robust, model-agnostic performance in summarisation tasks, especially where capturing persistent disagreement is crucial (Aghaebe et al., 8 Jan 2026). A plausible implication is that belief-level aggregation augments or stabilizes LLM outputs across architectures, outperforming direct early-fusion methods in data regimes with high opinion heterogeneity.

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Key references: “Distributing Knowledge into Simple Bases” (Haret et al., 2016), “Belief merging within fragments of propositional logic” (Creignou et al., 2014), “A general approach to belief change in answer set programming” (0912.5511), and “Faithful Summarisation under Disagreement via Belief-Level Aggregation” (Aghaebe et al., 8 Jan 2026).