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Phonological events

Bird, S. & Klein, E. (1990). Journal of Linguistics, 26(1), 33–56

Bird and Klein read an autosegmental diagram in time: the left-to-right order of a tier is precedence, and an association line is overlap. Their axioms (13) govern the two relations on events: overlap is reflexive and symmetric, precedence is asymmetric and disjoint from overlap, and an event preceding one member of an overlapping pair precedes every event following the other. Intervals of points satisfy the axioms under the definitions (11), and the No-Crossing Constraint is not stipulated but derived (Section 1.1): two crossing lines would make an event precede an event it overlaps. A representation is consistent when an event structure on its positions extends the order of its tiers and makes its lines overlaps (26).

Main definitions

IsEventStructure prec ov: the axioms (13).
IsEventStructureOf X prec ov: an event structure on the positions of a representation, extending its tiers and associations.

Implementation notes

The paper's event structures are the structures of precedence and overlap of the logic of time; they are unrelated to the decompositions of ArgumentStructure.EventStructure. In modern terms, a precedence satisfying (13) with overlap its incomparability is an interval order. The endpoint definitions (12), which the paper considers and sets aside, state overlap strictly, so under them a one-point interval does not overlap itself; the point-set definitions (11) read off closed intervals are those of NonemptyInterval. The cycle of three tiers is a consequence of the axioms that the paper does not discuss: they agree with the classical pairwise No-Crossing Constraint when lines join only tiers adjacent in a forest, and are strictly stronger once lines join three tiers in a cycle.

References

Bird & Klein 1990, Phonological events
Sagey 1988, On the ill-formedness of crossing association lines
Fishburn 1970, Intransitive indifference with unequal indifference intervals

The axioms for events

L61–73

L77–79

The axioms of an event structure entail the No-Crossing Constraint (Section 1.1).

theorem IsEventStructure.not_crossing L81–84

{E : Type u_1} {prec ov : E → E → Prop} (h : IsEventStructure prec ov) {w x y z : E} (hwx : prec w x) (hyz : prec y z) (hwz : ov w z) (hxy : ov x y) :

False

Built from

With overlap the incomparability, the axioms say exactly that precedence is an interval order.

theorem isEventStructure_iff_isIntervalOrder L86–104

{E : Type u_1} {prec : E → E → Prop} :

(IsEventStructure prec fun (x y : E) => ¬prec x y ∧ ¬prec y x) ↔ IsIntervalOrder E prec

Built from

Intervals

Intervals form an event structure under the precedence and overlap of (11).

theorem isEventStructure_intervals L114–133

{T : Type u_1} [LinearOrder T] :

IsEventStructure (fun (x y : NonemptyInterval T) => ∀ p ∈ x, ∀ q ∈ y, p < q) fun (x y : NonemptyInterval T) => ∃ p ∈ x, p ∈ y

Built from

Phonological event structures

L143–150

A representation with an event structure obeys the No-Crossing Constraint.

theorem IsEventStructureOf.noCrossing L154–160

{ι : Type u_1} {τ : ι → Type u_2} {X : Autosegmental.AR τ} {prec ov : X.Pos → X.Pos → Prop} (h : IsEventStructureOf X prec ov) :

X.NoCrossing

Built from

L162–167

Examples

L175–180

L182–185

(27) is consistent: it has an event structure.

theorem clements_consistent L187–190

∃ (prec : clements.Pos → clements.Pos → Prop) (ov : clements.Pos → clements.Pos → Prop), IsEventStructureOf clements prec ov

Built from ,

L192–195

L197

Yet the cycle has no event structure: the precedences chain through the overlaps from the last tier's first position to the first tier's second, which it overlaps.

theorem not_isEventStructureOf_cycle L199–209

(prec ov : cycle.Pos → cycle.Pos → Prop) :

¬IsEventStructureOf cycle prec ov

Built from ,

L211–214