The semantic transformation calculus

The semantic transformation calculus constitutes the generative core of the A-LLM. It is built around two primary classes of generative operations: vertical transformations and horizontal transformations. Vertical transformations are abstractions of languages' syntactic and semantic properties; horizontal transformations express conceptual relations on semantic geometry [Gardenfors2000], [Lehrer1990]. We define the set of vertical transformations V and the set of horizontal transformations H explicitly as V = {\mathit{noun}, \mathit{verb}, \mathit{adjective}} ∪ {\mathit{hypernym}} ∪ {\mathit{instrument}, \mathit{cause}, \mathit{result}} and H = {\mathit{opposite}, \mathit{orthogonal}, \mathit{opposite-orthogonal}} ∪ {\mathit{before}, \mathit{after}}. For example, with the root concept largeness: \mathit{smallness} = \mathit{opposite}(\mathit{largeness}), \mathit{reduce_(to)} = \mathit{verb}(\mathit{smallness}), and \mathit{longness} = \mathit{orthogonal}(\mathit{largeness}). These conceptual nodes represent A-LLM's lexical concepts connecting the conceptual space to lexical items such as largeness, smallness, reduce (to), and longness. The semantic relation hypernymy is of specific interest. It is represented as vertical transformations (in V) and will be discussed in more detail in Section 6.

Both vertical and horizontal transformations are compositional and may be applied recursively on conceptual representations in order to generate higher-order structures. To maintain semantic coherence, A-LLM imposes algebraic constraints on their interaction: vertical transformations preserve the conceptual dimension and identity of the base concept, while horizontal transformations preserve the relational topology of the conceptual field. We assume that vertical and horizontal operators commute for FoC root concepts; that is, for a FoC root concept c, v ∈ V, and h ∈ H and the following holds: v(h(c)) = h(v(c)). This is a design choice that holds for the core transformation pairs (e.g. adjective/verb with opposite/orthogonal); it may not hold for all operator combinations in natural language, and the consequences of this assumption are discussed where relevant. This parallels the rules of operator interaction observed in formal grammar systems [ChierchiaMcConnellGinet2000], but is applied to concepts rather than lexical items.

The next section extends the conceptual architecture by examining the semantic graph structure and explains how conceptual distance metrics support both inference and semantic search.