Omniscience Index: Metrics and Implications

• Omniscience Index is a rigorously defined metric that quantifies an agent’s or system’s capability to achieve full, calibrated knowledge under specific operational constraints across diverse domains.
• It employs specialized formulations—ranging from scalar evaluations in language models and submodular optimization in networked data exchange to iterative reasoning in games—to assess performance accurately.
• Practical applications include selecting reliable AI models, optimizing secure network communication protocols, and refining equilibrium concepts in epistemic game theory through iterative logical deletion.

The Omniscience Index is a rigorously defined metric that quantifies reliability, completeness, or epistemic depth in various domains—ranging from algorithmic knowledge testing in AI to optimal data exchange rates in networked information theory to formal epistemic reasoning in game theory. Depending on the context, its formulation, computational methodology, and theoretical significance differ substantially, but each instantiation addresses a fundamental question: How reliably can an agent, system, or model achieve "omniscience"—i.e., full or maximally calibrated knowledge—under precise operational constraints?

1. Definitions and Formalizations Across Domains

LLM Evaluation: In AA-Omniscience (Jackson et al., 17 Nov 2025), the Omniscience Index (OI) is a scalar in $[-100,100]$ that measures an LLM’s factual recall while penalizing hallucinations and rewarding abstention. Formally,

$$\mathrm{OI} = 100 \cdot \frac{c - i}{c + p + i + a}$$

where $c$ is the count of correct answers, $i$ incorrect, $p$ partially correct, and $a$ abstentions across a fixed evaluation set. OI is $+100$ for perfect correctness, $-100$ for perfect hallucination, and $0$ for a model that answers as many questions correctly as incorrectly or that abstains on all.

Networked Data Exchange and Secret Key Rates: In communication for omniscience (CO) (Ding et al., 2019, Ding et al., 2016, Nitinawarat et al., 2010), the Omniscience Index denotes the minimum sum-rate required for a set of agents to collectively reconstruct a distributed discrete source by exchanging information. Given entropy structure $H(\cdot)$ over subsets $V$, the CO Omniscience Index $\alpha^*$ satisfies

$$\alpha^* = \max_{P \in \Pi(V),\ |P|>1} \sum_{C \in P}\frac{H(V) - H(C)}{|P| - 1}$$

where $\Pi(V)$ is the set of partitions of $V$. In the PIN secrecy model, it equals the minimal public communication needed for perfect omniscience, with optimal secret key capacity given by $C(V) = H(V) - \alpha^*$.

Epistemic Game Theory: Within Kripke semantics for transparent equilibrium (Fourny, 2018), the Omniscience Index $\Omega(G)$ of a game $G$ quantifies the minimal depth $k$ of iterated logical omniscience (counterfactual reasoning) required such that the only strategy profiles surviving $k$ rounds of rationality and transparency deletion form the unique Perfectly Transparent Equilibrium:

$\Omega(G) := \min\{k \geq 1 : S_k \neq \emptyset\}$

with $S_k$ the set of profiles surviving $k$ levels of iterated deletion.

2. Algorithmic Computation and Optimization

AA-Omniscience Benchmark: All candidate LLMs are subjected to 6,000 expert-level factual questions spanning 42 topics in six major domains. Responses are graded by a hybrid human and automated protocol into CORRECT, PARTIAL_CORRECT, INCORRECT, or NOT_ATTEMPTED; tallies feed directly into the OI formula. No external tools or retrieval are permitted, isolating parametric recall.

Submodular Optimization in CO: The minimum sum-rate $R_{\rm aco}(V)$ is computed through submodular function minimization (SFM), Dilworth truncation, and principal sequence of partitions (PSP) (Ding et al., 2016, Ding et al., 2019, Ding et al., 2016). Efficient algorithms (PAR, MDA, CoordSatCapFus) reduce complexity from $O(|V|^2 \cdot \mathrm{SFM}(|V|))$ to as low as $O(|V| \cdot \mathrm{SFM}(|V|))$. PSP analysis yields hierarchical decompositions, secret capacities, and clustering bases.

Successive Omniscience in Cooperative Data Exchange: Multi-round, nested-group generalizations (SLO, SGO) resolve the tuple of minimum sum-rates via lexicographic multi-objective linear programming or systems of linear equations under random packet distributions; explicit closed-form solutions can be derived in two-group special cases (Heidarzadeh et al., 2017).

Epistemic Reasoning in Games: The iterated deletion structure implemented in the Kripke framework simulates logical omniscience levels. At each step, logical accessibility and closest-state functions quantify which counterfactual worlds and strategies remain epistemically accessible, eventually converging to a profile (or set) that satisfies required rationality and knowledge constraints.

3. Theoretical Properties and Interpretation

Range and Critical Points of OI: The normalization to $[-100,100]$ provides a symmetric, direct measure where positive values indicate net factual reliability, negatives net unreliability, and zero is either neutral or reflects universal abstention. Table interpretations clarify the operational significance.

Scenario	Correct	Incorrect	OI Value
All correct	6000	0	+100
Half correct, half wrong	3000	3000	0
All abstentions	0	0	0
All incorrect	0	6000	-100

Partition-Max Formulations and Structural Duality: In CO and PIN settings, the Omniscience Index is closely tied to entropy-maximizing partitions, with the fundamental partition being the first critical minimizer in the PSP sequence. Duality with clustering, secret key rates, and combinatorial optimization structure (e.g., Steiner tree packing in PIN) is profound.

Logical Omniscience in Games: The Omniscience Index $\Omega(G)$ connects epistemic modal depth directly to solution concepts. As $k$ increases, the set $S_k$ shrinks by eliminating non-rationalizable profiles; stabilization at $k^*$ signals the emergence and uniqueness of the perfectly transparent equilibrium.

4. Empirical Results and Comparative Insights

Frontier LLM Factuality: In AA-Omniscience, Claude 4.1 Opus achieves the best OI (+4.8), with only three models surpassing neutrality. Some models (e.g., Grok 4, GPT-5) have high raw accuracy but suffer negative OI due to high hallucination rates—underscoring the value of calibration. No single model is universally dominant across all domains.

Efficiency of Omniscience Communication: The most efficient algorithms for CO not only improve asymptotic complexity but also readily yield explicit transmission schedules, weighted optimal allocations, and clustering decompositions, ensuring practical deployability for multicasting and secrecy problems (Ding et al., 2019, Ding et al., 2016, Ding et al., 2016).

Game-Theoretic Depth: Not all games admit finite $\Omega(G)$; when they do, the Omniscience Index equals the minimal epistemic depth at which the PTE is forced by iterated deletion. This illustrates the impossibility triangle conjecture relating epistemic omniscience, logical omniscience, and necessary rationality.

5. Practical Significance and Applications

LLM Selection for Knowledge-Intensive Tasks: OI provides practitioners a unified score balancing factual accuracy, hallucination rate, and abstention, directly guiding model selection for critical tasks (e.g., legal drafting, scientific summarization) (Jackson et al., 17 Nov 2025).

Design of Network Coding and Multi-terminal Secrecy Protocols: The Omniscience Index delivers actionable sum-rate minima for distributed file reconstruction, key generation, and secret extraction in networks with diverse side-information structures, facilitating polynomial-time construction of optimal codes and secure keys (Milosavljevic et al., 2011, Nitinawarat et al., 2010).

Epistemic Reasoning and Solution Concepts in Game Theory: $\Omega(G)$ offers a precise epistemic measure for how much logical reasoning is needed before only perfectly transparent (fully rational and mutually known) strategic outcomes remain, informing analyses of robust equilibrium concepts under varied informational assumptions (Fourny, 2018).

6. Generalizations, Limitations, and Future Directions

Dynamic and Domainwise Benchmarking: AA-Omniscience is designed for continual updating. Its public slice enables rapid, domain-specific mini-benchmarks. Cost-performance curves indicate mid-tier models may offer superior OI-per-dollar.

Scalability and Extensibility in CO Algorithms: Recent submodular optimization advances allow omniscience indices to be computed for arbitrary correlation structures (e.g., linear, discrete, non-asymptotic). Fusion and partition methods extend readily to weighted objectives and clustering interpretations.

Controversies in Epistemic Modeling: The impossibility triangle highlights foundational tensions between logical omniscience, rationality, and factual knowledge, motivating further work on multimodal, non-normal, and counterfactual Kripke structures for epistemic game theory.

The Omniscience Index—across instantiations in AI reliability, network coding, and epistemic logic—constitutes a core metric for rigorously quantifying knowledge completeness, reliability, and optimality in systems under stringent operational or rationality constraints.