RIMRULE: A Cross-Domain Rule Framework

• RIMRULE is a class of interpretable, modular rule-based frameworks applied across domains such as statistical process control, antenna interference mitigation, anomaly detection, LLM adaptation, and belief updating.
• It enhances system performance by detecting small process shifts, optimizing spatial nulls in antennas, condensing anomaly rules, and dynamically injecting rules to improve LLM accuracy.
• The framework’s modular design facilitates integration with legacy systems, reduces computational overhead, and enables expert-driven oversight across diverse technical fields.

RIMRULE denotes a class of rule-based frameworks, algorithms, and update mechanisms that have been introduced independently in multiple technical domains, including statistical process control, anomaly detection, antenna interference mitigation, language-model tool adaptation, and belief updating. Despite their differing contexts, RIMRULE methods are unified by the central concept of explicitly codified rule structures, typically interpretable and modular, that can be optimized, consolidated, or adaptively injected to improve system-level decision-making or adaptation.

1. RIMRULE in Statistical Process Control

In the field of statistical process monitoring, RIMRULE refers to the modified $r/m$ (or M-$r/m$) runs-rule chart, which extends the classical Shewhart $\bar X$ chart by incorporating more sensitive supplementary runs rules for detecting small process shifts (Antzoulakos et al., 2010). The standard “$r$-of-$m$” rule signals an out-of-control condition if, in any $m$ successive subgroup averages, at least $r$ exceed (or fall below) the control limits. RIMRULE generalizes this: an out-of-control signal is triggered if $r$ “extreme” values (out-of-limits) occur with up to $(m-r)$ “inside” (but in-control) values interspersed, rather than requiring immediacy.

Key characteristics:

• Control limits: $UCL = \mu_0 + k\sigma/\sqrt{n}$, $LCL = \mu_0 - k\sigma/\sqrt{n}$.
• ARL (Average Run Length) computations incorporate combinatorics for $r/m$ rules, and Markov-chain methods for exactness.
• Modified M-$r/m$ schemes (RIMRULE) show superior detection of small to moderate mean shifts ($\delta\in[0.4,1.6]$), reducing ARL by up to 50% compared to standard Shewhart or $r/m$ methods.
• Implementation requires only a rule-change—no extra data collection or processing overhead.

This modification preserves the operational simplicity of Shewhart charts while offering performance comparable to complex multi-rule schemes such as the Western Electric Rules (Antzoulakos et al., 2010).

2. RIMRULE for Antenna Interference Mitigation

In satellite ground-station communications, RIMRULE refers to a rim-based interference-mitigation approach employing reconfigurable regions on the edge (rim) of a parabolic reflector to synthesize deep spatial nulls against Low Earth Orbit (LEO) satellite interference (Santana et al., 18 Dec 2025). The method is implemented in the SatTrack simulation framework to evaluate gain, interference suppression, and throughput trade-offs.

Core aspects include:

• Physical optics (PO) model: The reflector is discretized into “fixed” and “reconfigurable rim” regions, with independent control of phase (or binary polarity) in the rim alone.
• Optimization objective: Minimize the magnitude of the far-field co-polar field in the interference direction by adjusting rim weights $w_i$:

$S(\mathbf{w}) = q_{\rm fix} + \sum_{i=1}^N w_i q_i$

• Algorithmic methods: Serial greedy, greedy bit-flip, simulated annealing (with support for $M$-ary complex phase weights), and majorization-minimization optimizations, enabling scalable real-time control and flexible trade-offs between suppression depth and runtime.
• Performance: Binary rim control achieves 40–55 dB average suppression; advanced $M=16$ phase optimization exceeds 65–70 dB (best-case $P_r < -220$ dBm). Nearly full link throughput is preserved except during main-lobe crossings.
• Scalability: Computations scale linearly with the number of rim elements, not total aperture area.
• Limitation: Rim-only nulling cannot suppress main-lobe crossings without disturbance to boresight gain; such events are rare due to narrow main-beamwidths (Santana et al., 18 Dec 2025).

RIMRULE thus enables highly efficient, low-overhead spatial filtering against transient LEO interference.

3. RIMRULE for Multivariate Anomaly Explanation in Networks

In the domain of radio access network (RAN) anomaly detection, RIMRULE identifies a semi-autonomous rule-mining system that translates statistical anomaly detections from “black-box” multivariate detectors into interpretable, actionable rule sets (Isaac et al., 11 Jan 2025). The paradigm blends automated candidate generation with domain-expert feedback to ensure operational relevance and human interpretability.

Methodological flow:

• Anomaly scores are generated from an underlying detector (e.g., isolation forest, deep autoencoder).
• Detected anomalies are collated into contiguous episodes and features are summarized (deltas, medians, etc.).
• Rules are candidate conjunctions over features: each rule encodes atomic conditions (above/below/near reference), enabling a Boolean representation as ternary vectors.
• A scoring and ranking procedure highlights candidate rules for Subject Matter Expert (SME) appraisal: accept, whitelist (dismiss), split, or combine.
• Final rule sets are concise (e.g., $K\approx42$, $3$–$6$ conditions per rule), maintain high recall ($\approx 0.90$), and reduce false alarms by $\approx 50\%$ over raw anomaly detection (Isaac et al., 11 Jan 2025).

Applications demonstrate an 80% reduction in operator validation workload, as only exemplars per rule are inspected, rather than every anomaly.

4. RIMRULE for LLM Tool Adaptation via Neuro-Symbolic Rule Injection

RIMRULE also designates a neuro-symbolic method for adapting LLMs to domain-specific tool use, addressing the challenge of unreliable API or tool calls in zero- and few-shot settings (Gao et al., 31 Dec 2025). The method induces, consolidates, and dynamically injects rules derived from LLM failure traces, using a minimum description length (MDL) objective for rule compactness and generality.

Key elements:

• Failure capture and rule proposal: For each incorrect or tool-error trace, the LLM proposes a candidate rule in natural language.
• Symbolic compilation: Rules are mapped to a fixed schema (Domain, Qualifier, Action, Strength, ToolCategory).
• MDL-guided consolidation: Rule subsets $H$ are selected to minimize $\mathrm{MDL}(H) = L(H) + L(D|H)$, balancing symbolic complexity and empirical correction rate.
• Inference-time injection: On new queries, top-$k$ rules—filtered and ranked via symbolic or embedding-based similarity—are injected into the prompt, guiding action selection without model retraining.
• Empirical gains: Rule injection yields 2–4 point accuracy improvements on both in-distribution and out-of-distribution (unseen tool) splits, is complementary to standard fine-tuning, and is portable across LLM families (rules distilled from Llama3.2 boost GPT-4o, O1, and vice versa).

RIMRULE thus establishes a new paradigm beyond prompt tuning or finetuning: dynamic, interpretable, plug-and-play rule-based LLM adaptation (Gao et al., 31 Dec 2025).

5. RIMRULE in Ambiguous-Belief Updating—Relative Maximum Likelihood

In decision theory under ambiguity, the term “Relative Maximum Likelihood” (RML) describes an updating rule for beliefs represented as a convex set of priors (Cheng, 2019). The RIMRULE in this context contracts the original prior set $C\subseteq\Delta(\Omega)$ linearly towards those priors in $C$ that maximize the observed event's likelihood.

Formal construction:

• Prior contraction: For observed event $E$,

$C^*(E) := \operatorname{argmax}_{p\in C} p(E)$

$$C^{RML}_\alpha(E) = \alpha \cdot C^*(E) + (1-\alpha)\cdot C = \{ \alpha p + (1-\alpha)q: p\in C^*(E), q\in C \}$$

where $\alpha\in[0,1]$ parametrizes the contraction.

• Posterior updating: Posterior set after event $E$,

$$C^{RML}_\alpha|E = \{ r(\cdot|E) : r \in C^{RML}_\alpha(E) \}$$

• Special cases: $\alpha=0$ yields Full Bayesian (update all priors), $\alpha=1$ gives Maximum Likelihood (update only maximal-likelihood priors), $0<\alpha<1$ interpolates between these extremes.
• Axiomatization: Characterization is given in terms of Maxmin Expected Utility (MEU) preference axioms; RIMRULE is pinned down by weakened dynamic consistency and contingent reasoning, and, with appropriate event consistency, admits a common $\alpha$ across all events (Cheng, 2019).

RIMRULE thus formalizes belief updating rules that systematically interpolate between conservative and aggressive updating in ambiguous settings.

6. Domain-Specific Summary Table

Context/Domain	RIMRULE Interpretation	Core Mechanism / Optimization
Process Control	Modified runs-rule control chart	Flexible “$r$ of $m$ with inserted in-control”
Antenna Engineering	Rim-based reflector interference nulling	Rim-phase optimization for spatial nulls
RAN Anomaly Detection	Human-in-the-loop anomaly rule mining	Greedy conjunction search + SME curation
LLM Adaptation	Neuro-symbolic tool-adaptation via rule injection	MDL-pruned rule sets, dynamic prompt injection
Ambiguous Belief	Relative Maximum Likelihood updating	Linear contraction of prior set toward ML

This overview highlights the cross-disciplinary role of RIMRULE as a design pattern for interpretable, modular, and performance-improving rule-based governance, irrespective of the underlying substrate.

7. Interpretability, Modularity, and Cross-Domain Impact

Across domains, RIMRULE frameworks share critical properties:

• Interpretability: Rules are simple, often human-readable conjunctions or schema-derived predicates, facilitating domain expert involvement and trust.
• Modularity: Rules can be pruned, generalized, recombined, or retargeted with minimal architectural disturbance.
• Integrability: RIMRULE can plug atop legacy detectors (e.g., for anomaly explanation), existing control charts, or LLMs, without model retraining or hardware change.
• Portability: Symbolic rules have demonstrated transferability across model architectures (LLM context) and system settings.

A plausible implication is that the modular, interpretable ethos of RIMRULE is a robust design pattern for bridging the gap between statistical/algorithmic performance optimization and practical, expert-driven deployment in diverse technical fields.