Jardine writes an autosegmental representation as a labeled mixed graph ⟨V, E, A, ℓ⟩: nodes,
undirected association edges, directed arcs ordering each string, and a labeling, and the
well-formedness axioms of his dissertation (§4.2) carve the representations out of these graphs.
AR.toGraph reads a representation as such a graph, with the precedence order of each tier as
its arcs. The reading satisfies Axioms 1–3, it is injective up to isomorphism, and every finite
graph satisfying Axioms 1–3 is isomorphic to one; on it, Axioms 5 and 6 are the No-Crossing
Constraint and the OCP of the representation.
Main definitions
Main results
Implementation notes
The axiom numbering follows the dissertation; Jardine and Heinz number the NCC and the OCP as 4
and 5 and have no saturation axiom. Saturation, Goldsmith's original well-formedness condition,
is stated but never imposed. The arcs of AR.toGraph are the precedence order of each tier,
transitively closed, so the OCP reads adjacency as the covering relation of the arcs. Jardine's
2019 graphs carry the successor relation instead (Fig. 4), and their subgraphs (Definition 3)
are read in that signature, which AR.FactorEmbeds implements in position coordinates; such
subgraph-based notions are sensitive to the signature, as Jardine's analysis of the logical
complexity of these representations shows. Axiom 5 is Coleman and Local's definition of
crossing lines (p. 304) as the dissertation adopts it, with strict precedence; Jardine and Heinz
state their Axiom 4 with the reflexive order, which read literally forbids a node from linking to
two others. Coleman and Local also argue (pp. 297–298) that the NCC amounts to graph planarity
and so restricts nothing in multiplanar representations, the weaker reading that the 2019 paper
sets aside (fn. 11).
TODO
- The successor reading of a representation, with
AR.FactorEmbeds as the connected subgraph
embedding of the 2019 paper's Definition 3 in it.
References