Conclusion

We evaluated 3A-LLM as a semantic backbone for LLM-based search on 330 test cases (165 query expansion, 165 implicit concept prediction) drawn from a concept taxonomy of 4,102 triples. 3A-LLM substantially outperforms WordNet and string-similarity baselines and provides explainable results via derivation paths. We reported 2--3 examples per category, including vehicle, building, person, event, animal (with is-chains), kinship, device/tool, artifact, region/country/universe, religion, feeling, science, destruction, food/beverage, illness/health, bodyparts, clothing/textile, imagination/imaginary person, leadership, protection/shelter, measure, synonyms/similar, and preposition-based activities (snorkeling, diving, hiking). We detailed FoC (e.g. at least 24 concepts from a root like tallness), genus--differentiae and is-chains, the handling of synonyms (equal), similar concepts, homonyms, and polysemy, inference mechanisms (including Löbner-style functional nouns and kinship substitution), canonical Cantor encoding, formal properties, quality dimensions, and worked case studies (piano, air pressure, validation). The combination of deterministic expansion, structural explainability, and cross-lingual concept identity makes 3A-LLM a suitable semantic layer for hybrid KG--LLM search.

A current implementation of 3A-LLM is publicly accessible and can be explored interactively. (Note: [URL anonymized for double-blind review.]) Approximately 160 of the most important semantic primes from the LDV have been identified and are visualized together with their semantic relations. The Cantor IDs assigned to primitive concepts and compounds can be inspected on a dedicated search page. (Note: [URL anonymized for double-blind review.]) The knowledge graph underlying 3A-LLM is visualized elsewhere (Note: [URL anonymized for double-blind review.]) and currently comprises approximately 1000 classes, {>}50 million nodes and {>}400 million edges. All graph queries used in the evaluation execute in {<}30 ms.

Future work will add embedding-based and LLM-based baselines, scale to the full taxonomy, and test 3A-LLM in end-to-end hybrid search pipelines.

\appendix

The Axiomatic Large Language Model (A-LLM) presented in this work offers a comprehensive and formally coherent framework for semantic representation. Its architecture integrates definitional primitives derived from the Longman Defining Vocabulary, a mathematically structured transformation calculus, and a mechanism based on compounds to generate the model's graph-theoretic conceptual space.

The practical significance of the Axiomatic Large Language Model (A-LLM) lies in its inference mechanism discussed above and starting from that a wide range of applications, one of them semantic search. Unlike keyword-based retrieval systems, relying on surface form matching, or vector-based systems which approximate conceptual similarity through distributional patterns, A-LLM computes semantic relatedness through explicit graph-theoretic metrics. The semantic graph enables controlled exploration of conceptual neighborhoods by tracing definitional, vertical, and horizontal relations. Because each path in the graph corresponds to a meaningful conceptual transformation, search results remain interpretable. For example, a query for pressure may retrieve air pressure, fluid pressure, \mathit{instrument}(\mathit{pressure}), or conceptual opposites such as vacuum pressure depending on the semantic radius selected. Users but also algorithms can therefore adjust the semantic scope of their search in transparent ways.

The model further supports semantic query expansion, a technique widely used in information retrieval. Traditional expansion methods rely on lexical synonyms, statistical co-occurrence, or precompiled thesauruses. A-LLM, by contrast, derives expansions through the structure of the semantic graph. If a user submits a query for \mathit{instrument}(\mathit{air_pressure}), the system can expand it by including altimeter, barometer, or pressure sensor through functional transformations applied to the conceptual structure. Because each expanded term arises from a definitional or transformational rule, the expansion remains both transparent and semantically precise. This approach is consistent with the conceptual decomposition strategies found in cognitive semantics [Jackendoff1983] but is formalized in a computationally tractable manner.

The most obvious way to use the A-LLM in cooperation with a modern language processing AI system is to let it operate as a separate module within the system. As such it can answer queries from other modules of the AI system, in particular by drawing inferences in the way discussed above.

A full integration of the A-LLM in a modern language processing AI system, however, means to exploit the A-LLM‘s capabilities for the training of the AI system’s LLM. In order to do that, some training texts must first be broken down to sentences. In a second step, for each of the sentences, the most prominent concept, not explicitly mentioned, has to be determined which is the A-LLM‘s task. The AI’s LLM then has not only to predict word by word of the training sentences but also, reaching the training sentence‘s end, has to predict the words that determine the sentence’s most prominent concepts. For example, training from the sentence “Max is playing piano”, the AI‘s LLM learns to predict music and just shifts the meaning of piano in the given context towards the piano’s function whereas in other contexts, e.g. in “Max and Tom are carrying the piano to the fifth floor”, the piano's weight and not its function is in the focus.

The A-LLM determines the most prominent concepts, not explicitly mentioned, by semantic distance. For each of the concepts denoted in an expression, a list of concepts with high semantic distance is generated. The resulting lists are merged. If a concept is in more than one of the initial lists, the assigned semantic distance is the sum of the initial distances. In the end, concepts that are mentioned in the expression but nevertheless are present in the resulting list being near to another concept mentioned, are deleted from the list which after that step includes the most prominent but not mentioned concepts.

\paragraph{Implementation status.} A current implementation of the A-LLM is publicly accessible and can be explored interactively under the links given in the following. Approximately 160 of the most important semantic primes from the LDV have been identified and are visualized together with their semantic relations. (Note: [URL anonymized for double-blind review.]) The Cantor IDs assigned to primitive concepts and compounds (e.g., for the FoC largeness) can be inspected on a dedicated search page. (Note: [URL anonymized for double-blind review.]) The ontology underlying the A-LLM is visualized elsewhere (Note: [URL anonymized for double-blind review.]) and currently comprises approximately 1000 classes.

\section*{Declaration on Generative AI} During the preparation of this work, the authors used ChatGPT and Cursor for: Grammar and spelling check; Improve writing style; Formatting assistance (restructuring portions of the manuscript in response to workshop-oriented formatting requirements). After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the publication's content and claims.

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