On the Cryptographic Uses of TLAs—Dash Ŋ. Ooba-Nuhd SpecGram Vol CLXXIII, No 3 Contents The C-Rhyme and Pun-ish-ment of St. Alvin—Jerry Fyodor & Josef Dobrovskýevsky

The Topology of Syntax

Iain A. Plicable
Lecturer in Mathematical Linguistics
University of Ledworth

A key theorem of Universal Grammar is that lines do not cross in tree diagrams. However, critics of Universal Grammar challenge even such a basic result as this on the basis of sentences such as the following, taken from Virgil’s Eclogues:


“Now the last age of Cumaean song comes.”

Figure 1: An invalid tree diagram

Leaving aside the fact that such forays into historical linguistics tell us nothing about the necessarily synchronic nature of Universal Grammar, we must address the fact that a naive attempt to parse this sentence leads to profoundly unsatisfactory results, as seen in Figure 1.

Sceptics claim that this shows that Latin phrases are bound together by dependency relationships rather than constituency relationships. This hypothesis, however, assumes that morphology is governing syntax, in a clear violation of the natural hierarchy of information structures. A more rigorous, mathematical approach provides a better solution.

While the surface structure of language is linear, the existence of tree diagrams shows that this structure is embedded in a 2-dimensional surface. Were it not so, we should be incapable of any grammar other than clause chaining. For convenience, this surface is usually depicted as a simple plane. However, just as the linear structure is embedded in a 2-dimensional surface, the surface may itself be embedded in 3-dimensional space. This allows us to envisage it as a curved surface, such as a torus, familiar as the shape of a doughnut, or alternatively a coffee cup.

Figure 2: A torus and its representation as a topological diagram

The topological diagram in Figure 2 represents how a torus may be constructed from a flat sheet by joining opposite edges. That corresponding arrows point in the same direction indicates that the torus is an orientable surface, non-orientable grammar being an issue for future research. Using the curvature of the torus, we can now produce the sentence diagram in Figure 3.

Figure 3: Using a toroidal surface produces a valid tree diagram

Nothing could be simpler.

On the Cryptographic Uses of TLAsDash Ŋ. Ooba-Nuhd
The C-Rhyme and Pun-ish-ment of St. AlvinJerry Fyodor & Josef Dobrovskýevsky
SpecGram Vol CLXXIII, No 3 Contents