Semantic graph structure and semantic distance

The semantic graph constitutes the global structural framework of the A-LLM. It enables the A-LLM to model semantic relatedness, conceptual neighborhoods, inferential pathways, and hierarchical derivations. The semantic graph represents the conceptual space in which meaning is encoded, manipulated, and interpreted.

The graph takes the form of a directed, typed structure G = (N, E) where the set of nodes N corresponds to concepts, both atomic concepts and concepts derived from them, and where the set of edges E encodes the vertical (syntactic) and the horizontal (semantic) relations generated by the transformation mechanisms. Edges in the semantic graph fall into three principal categories. The first category consists of definitional edges which connect composite concepts to their component structures. For example, a compound such as \mathit{pressure}(\mathit{air}) possesses definitional edges linking it to the concepts pressure and air. These edges ensure the full reducibility of complex concepts to LDV primitives, providing semantic transparency not available in distributional models or neural embeddings [Devlin2019]. If in a(b) both a and b are denoted by nouns then a(b) represents a genus--differentiae pattern, where a is the genus and b is the differentiae. This enables to infer that an important semantic relation holds between a and a(b), namely that a(b) is a hyponym of a. This corresponds to the ontological subClassOf relation expressing that a(b) ⊆ a.

The second category consists of vertical edges encoding relations such as nominalization, adjectival formation, and verbalization. They connect the conceptual space to the lexical items of the language that is used in verbal expressions to be processed.

The third category consists of horizontal edges which represent semantic relations among concepts, including opposition, orthogonality, complementarity, and scalar adjacency. These edges echo the structure of conceptual spaces in which semantic dimensions are explicitly represented [Gardenfors2000], [Gardenfors2014] but they do so through discrete, algebraically defined operators rather than continuous spatial mappings.

One of the central consequences of the graph structure is the emergence of conceptual neighborhoods. Nodes with multiple short paths between them form semantically coherent clusters, a phenomenon parallel to semantic fields identified in lexical semantics [Lehrer1990]. These clusters provide a principled basis for semantic search and conceptual inference by relying on “semantic radius” and “semantic distance”. Both are defined in terms of graph-theoretic metrics. Semantic radius is a natural number n: for any concept c, all concepts reachable from c by passing n or fewer edges are in c's n-radius. Semantic distance, by contrast, is a non-negative real number (not a natural number): it takes the weight of the edges into account. Edge weights are numbers between 0.0 and 1.0. The inverted weight is 1.0 minus the weight. To keep things simple, let us assume all edges have the weight 0.5 which means that the inverted weights also are 0.5. The weight of a path from node c_1 to c_2 is the product of the inverted weights of the edges of the path. For example, if c_1 and c_2 are neighbors, the path weight is 0.5; if one node has to be passed, the path weight is 0.5 × 0.5 = 0.25 and so on. The sum of path weights (paths of length less than 3) between c_1 and c_2 yields a semantic proximity (or relatedness) score: higher values indicate closer conceptual relatedness. One may define a proper semantic distance as 1 - \mathit{proximity} if a metric is required. This path-based notion of relatedness is in line with walk-based or path-based similarity measures used in graph kernels and semantic networks [Lehrer1990]; we refer to it as “semantic distance” when used for ranking and retrieval. As a worked example, take the root concept largeness and the derived concepts \mathit{enlarge_(to)} = \mathit{verb}(\mathit{largeness}), \mathit{longness} = \mathit{orthogonal}(\mathit{largeness}), and \mathit{long} = \mathit{adjective}(\mathit{orthogonal}(\mathit{largeness})) = \mathit{adjective}(\mathit{longness}). From largeness, the path to \mathit{enlarge_(to)} has one edge (verb), so path weight 0.5; the path to \mathit{longness} has one edge (orthogonal), so path weight 0.5; the path to \mathit{long} has two edges (orthogonal, then adjective), so path weight 0.5 × 0.5 = 0.25. The semantic distance between \mathit{enlarge_(to)} and \mathit{longness} is the weight of the unique path of length two between them (via largeness), i.e. 0.25; the shortest path between \mathit{enlarge_(to)} and \mathit{long} has length three (via largeness and \mathit{longness}), so with the “less than 3 edges” bound their semantic distance is the sum over shorter paths only, here 0.

More sophisticated metrics may incorporate different edge weights. For example, edge weights may correspond to transformation types. Vertical edges may have assigned higher weights than horizontal edges. This weighting strategy mirrors the intuition that a root concept and its lexical concepts are conceptually closer than concepts connected by semantic relations. Weighted distances allow the graph to approximate semantic similarity measures observed in cognitive psychology [Rosch1975] while retaining definitional interpretability.

The structure of the graph also supports inferential pathways. Because each edge corresponds to a semantic operation, traversing a path yields an explicit conceptual explanation. For example, an inferential chain such as air → air_pressure → \mathit{instrument}(\mathit{air_pressure}) is not only discoverable but interpretable. Each step corresponds to a definitional or transformational edge, making A-LLM an inherently explainable model of semantic inference.

The presentation of how the A-LLM is built will be finalized in the following section that discusses its higher order concept construction.