Inference mechanisms

The inferential architecture of the Axiomatic Large Language Model (A-LLM) emerges from the interaction of definitional primitives, transformation operators, and functional compounds within the graph that form the conceptual space. Because every concept in the system is reducible to the Longman Defining Vocabulary primitives introduced in Section 3, semantic interpretation proceeds by tracing definitional chains that reveal the internal structure of complex concepts.

Horizontal transformations yield relational inferences such as opposition, orthogonality, complementarity, and scalar adjacency, in alignment with conceptual space theory [Gardenfors2000]. For instance, from tallness the system may infer shortness through an opposition operator, or map it orthogonally to width or breadth. In A-LLM, such inferences are not heuristic or corpus-derived but formal consequences of its algebraic structure.

The compound mechanism defines the major class of inferential processes: functional composition. Because compounds encode conceptual functions like \mathit{mother}(x), \mathit{pressure}(x), \mathit{instrument}(x), inference follows the structure of functional application. Kinship terms provide a canonical example. \mathit{grandmother}(x) expands to \mathit{mother}(\mathit{parent}(x)), capturing classical relational analyses in formal semantics [KampReyle1993]. In general, A-LLM treats functional expressions as concept-constructing operations, creating a system in which inferential chains emerge directly from conceptual structure.

Graph-theoretic inference constitutes a third inferential mode. Because the semantic graph encodes definitional, vertical, and horizontal relations as directed and typed edges, path traversal corresponds directly to meaningful semantic transformations. The shortest path between two nodes approximates semantic similarity while weighted paths allow for differential treatment of conceptual transformations.

Last but not least, inference arises from semantic reduction. Because A-LLM preserves definitional lineage across all operations, any complex concept can be reduced to its primitive components. Reduction provides a mechanism for semantic explanation, enabling users or algorithms to understand how a concept is constructed and how deeply it depends on specific primitives. This aligns with goals in explainable AI where models must reveal the internal basis of their outputs rather than provide opaque statistical approximations [Steels2008]. Through reduction, A-LLM contributes a uniquely transparent mode of reasoning within computational semantics.

A-LLM and its inference mechanism can be used in a variety of ways. Some of them, in particular how to use it in cooperation with a modern language processing AI system, will be presented in the concluding section.

The inferential architecture of 3A-LLM emerges from the interaction of definitional primitives, transformation operators, and functional compounds within the graph. Because every concept in the system is reducible to the LDV primitives, semantic interpretation proceeds by tracing definitional chains that reveal the internal structure of complex concepts.

Horizontal transformations yield relational inferences such as opposition, orthogonality, complementarity, and scalar adjacency, in alignment with conceptual space theory [Gardenfors2000]. For instance, from tallness the system may infer shortness through an opposition operator, or map it orthogonally to width or breadth. In 3A-LLM, such inferences are not heuristic or corpus-derived but formal consequences of its algebraic structure.

The compound mechanism defines the major class of inferential processes: functional composition. Because compounds encode conceptual functions like \mathit{mother}(x), \mathit{pressure}(x), \mathit{instrument}(x), inference follows the structure of functional application. Kinship terms provide a canonical example: \mathit{grandmother}(x) expands to \mathit{mother}(\mathit{parent}(x)), capturing classical relational analyses in formal semantics [KampReyle1993]. The functional view aligns with type-theoretic approaches in lexical semantics. Löbner [Lobner2011] introduces functional nouns as concepts whose meaning takes the possessor or bearer as argument. These correspond exactly to 3A-LLM's handling of compounds: role terms such as \mathit{father}(\mathit{Rosi}), \mathit{president}(\mathit{France}); unique parts such as \mathit{cover}(\mathit{computer}); abstract aspects such as \mathit{name}(\mathit{Pope}), \mathit{age}(\mathit{Picasso}), or \mathit{meaning}(\mathit{meaning}). Possessive and property chains become nested applications, e.g. \mathit{car}(\mathit{mother}(\mathit{wife}(\mathit{father}(\mathit{my})))). Inferences arise by substitution: with \mathit{Rosi} = \mathit{mother}(\mathit{Mary}) and \mathit{Tom} = \mathit{father}(\mathit{Rosi}) it follows that \mathit{Tom} = \mathit{father}(\mathit{mother}(\mathit{Mary})); with \mathit{grandfather_maternal} = \mathit{father}(\mathit{mother}) we can reduce the chain and get \mathit{Tom} = \mathit{grandfather_maternal}(\mathit{Mary}).

Graph-theoretic inference constitutes a third inferential mode: path traversal corresponds directly to meaningful semantic transformations; the shortest path between two nodes approximates semantic similarity, and weighted paths allow differential treatment of conceptual transformations. Last, inference arises from semantic reduction: any complex concept can be reduced to its primitive components, providing a mechanism for semantic explanation and alignment with explainable AI goals [Steels2008].