We have already characterized the relations partial rankings bear to equivalence classes of tableaux and sets of tableaux in which they are true. But partial rankings (and their dual objects, of course) also have relations between themselves. One of those we have looked at already in some detail: entailment. In this section, we will look more closely into the internal structure of the domain Φ of partial rankings created by different relations that hold between them. First, that will allow us to understand better what sort of creatures
partial rankings are. Second, when we take on sets of partial rankings in Section 5, the knowledge we will acquire by the end of this section will become handy.
Here is how the relation of entailment between rankings organizes their domain, for a constraint set with just 3 constraints C1, C2, C3 (recall that Λ is the minimal ranking which only has Ci≫ ∅ atomic rankings):
(38) Λ
yyrrrrrrrrrrrrrrrrrrrrrrrrr
:
This picture shows only the small part of the structure, namely the part which is dominated by the partial ranking (C1 ≫C2). Every other ranking in the picture is its refinement. Those refinements belong to one of the two levels: (C1≫C2) ∧ (C1≫C3) and (C1 ≫ C2) ∧ (C3 ≫ C2) are still underspecified, they do not decide, respectively, the ranking between C2 and C3, and between C1 and C3. The three rankings at the bottom of the picture are total rankings: they are fully specified in our small Con. If there is a direct arrow fromφ toψ in the picture, it means that we can getψ from φ by adding one atomic ranking, and transitively closing the result. So each arrow connects two rankings which are at minimal distance from each other in the structure. The reason why a partial ranking can have immediate descendants at different levels is transitive closure.
For instance, when we add to C1≫ C2 an atomic ranking C2≫C3, we effectively add also the third atomic rankingC1≫C3 as well, by transitivity. Had partial rankings not been necessarily transitively closed, the structure would have looked differently. Note that there is no direct arrow from(C1≫C2) to (C1≫C3) ∧ (C3≫C2) ∧ (C1≫C2): there is no single atomic ranking that we can add to the former to immediately get the latter.
The entailment relation obeys reflexivity (every ranking entails itself), antisymmetry (if ranking φentails ψ and they are not equal, then ψ does not entail φ) and transitivity (ifφentailsψ, andψ entailsξ, thenφ also entailsξ), and thus imposes a partial order on the rankings. The structure⟨Φ,Λ,⊧⟩ (the set Φ of partial rankings, with a special element Λ, plus a relation of entailment between rankings) is a poset (partially ordered set) with a bound on one side.We can thus say that more underspecified rankings are greater than their refinements, treating⊧ as≥. The minimal ranking Λ is the maximal element, or 1.
We can call the refinements of a ranking its daughters ordescendants, bringing in the tree terminology, as our poset structure can be viewed as a tree with the empty ranking as the root. Then we have a natural name for the inverse of the refinement relation: a rankingais aparent for a rankingbiff bis a refinement of a.
Recall that an M-maximal ranking is such a ranking that it is true inM, but none of its immediate predecessors is. All descendants of an M-maximal ranking in the structure
⟨Φ,Λ,⊧⟩ have to be true in M, so the maximal ranking defines a “triangle” of rankings compatible with M, and is the greatest element in this “triangle”.
If φ⊧ψ, then all atomic rankings of φare also in ψ. But it is not the only relation of interest possible between rankings. Take some arbitrary φand ψ. First, they may have a common set of atomic rankings — a (ranking-)intersection. Second, when all the atomic rankings from both are combined together, they define a bigger ranking which contains bothφ andψ — a (ranking-)union.
Why should we be interested in those? There are two reasons. First and foremost, we will heavily use the operations on individual rankings we will now define in our analysis of sets of OT rankings. In that sense, we now simply build the foundation which will allow us to do much less trivial work further down the road. And the second reason is less grounded in the concerns relevant for the current paper, but instead can be appreciated right away, rather than after we see how we put those operations to use in Section 5: the proposed operations have nice potential practical applications which we illustrate using hypothetical scenarios involving grammar learners.
Imagine we have a learner comparing two partial ranking hypotheses φ and ψ about what the grammar is, knowing that each hypothesis may be wrong about any atomic ranking it contains. Then the intersection ofφandψcontains the atomic rankings present in both hypotheses. Other things being equal, in our hypothetical scenario those atomic rankings are less likely to be wrong than the atomic rankings present in only one ofφand ψ.
For union, imagine a learner which acquired two partial ranking grammar hypotheses φand ψfrom different sources (e.g., from two sets of data processed in parallel threads of computation), and now wants to merge them into another partial ranking which is able to account for all the dataφorψ can account for in the most conservative manner possible.
Such a learner will have to find the minimal partial ranking which is entailed by both φ and ψ— their union.
Thus it makes sense to define the natural operations of ranking-intersection ∩r and ranking-union∪r.
(39) Let φand ψbe well-formed formulas of OTR.
φ∩rψ is the smallest well-formed formula ξ s.t.
∀Ci, Cj∶ [(Ci≫Cj) ∈φ ∧ (Ci≫Cj) ∈ψ] → (Ci≫Cj) ∈ξ.
The ranking-intersection∩r behaves simply as set intersection for sets of atomic
rank-ings contained in the argument partial rankrank-ings. If there are contradictory atomic rankrank-ings in φ and ψ under intersection, they are simply excluded from φ∩rψ because the inter-section only includes common atomic rankings of the two. Atomic rankings obtained by transitivity are also fine: if there were two atomic rankings forcing a third by transitivity, in order to get intoφ∩ψ they had to be present in both φand ψ, so then if there were a contradiction, it would have been to be one in individualφandψ already. Thus the result of∩r is always defined.
But ranking-union isnot a simple set union on the sets of atomic rankings correspond-ing toφand ψunder union. Well-formed partial rankings must not contain contradictory atomic rankings, and must be transitively closed. But if we simply combine all the atomic rankings inφandψ, we may, first, create a contradictory ranking, in caseφand ψ contra-dicted each other, and second, the resulting set of atomic rankings may be not transitively closed. So our definition of ranking-union ∪r, first, leaves the operation undefined on contradictory arguments, and second, ensures the result is transitively closed:
(40) Let φand ψbe well-formed formulas of OTR.
φ∪rψ is defined iff¬∃Ci, Cj∶ (Ci≫Cj) ∈φ ∧ (Cj≫Ci) ∈ψ.
If φ∪rψis defined, then it is the smallest well-formed formula ξ s.t.
∀Ci, Cj∶ [(Ci≫Cj) ∈φ ∨ (Ci≫Cj) ∈ψ] → (Ci≫Cj) ∈ξ.
Both union and intersection are obviously associative, commutative, distributive, and obey absorption, provided that all unions are defined.
We can now view the domain of partial rankings as the structure⟨Φ,Λ,⊧,∩r,∪r⟩. Since our ranking-union ∪r is not set union, the structure ⟨Φ,Λ,⊧,∩r,∪r⟩ is not an algebraic substructure of the power set algebra generated by the set of atomic rankings with set-theoretic operations on it. In the powerset algebra, elements are really bags with atomic rankings in them, and since the requirements to be non-contradictory and to be transitively closed are not imposed, there are more elements (“pseudo-rankings”) in it than in Φ. Because of the conditions of consistency and transitive closure, OT carves a non-trivial part of the powerset algebra, creating the domain Φ.
The intersection for partial rankings is just a restriction of the powerset algebra intersection to Φ, and it makes⟨Φ,Λ,⊧,∩r,∪r⟩a semilattice. But it is not a full lattice, because∪r is not always defined. (The minimal element of a full lattice would have been the maximally contradictory ranking which contains all atomic rankings whatsoever.)
It might be interesting to characterize this structure ⟨Φ,Λ,⊧,∩r,∪r⟩ independently, not using the specific OT notions, as if it is just an abstract algebraic structure about whose members we do not know a thing. This would provide an independent characterization of what kind of structure OT actually creates in the single-ranking realm. I leave that for some other occasion.
∩r and ∪r are straightforwardly connected to entailment:
(41)
φ∩rψ ⊧ φ φ ⊧ φ∪rψ
Thusφ∩rψis an ancestor for bothφand ψ, andφ∪rψ (if it exists) is a descendant for both of them. Moreover, the partial rankingφ∩rψis the most specified common ancestor, and φ∪rψ is the least specified common descendant ofφand ψ.
Entailment among partial rankings is definable in terms of∩r: φ⊧ψiff φ∩rψ=φ.
We finish this section with a remark on how the relation of entailment⊧is interpreted in the domains of representative tableauxM− and representative sets of tableaux Σ−.19
For Σ−, the notion of entailment imported from the domain of partial rankings Φ says that σφ ⊧ σψ when σφ ⊆ σψ: if φ ⊧ ψ, then ψ is compatible with all the tableaux φ is compatible with, and then possibly with some more, which exactly describes the situation whenσφ⊆σψ.
For the domain of representative tableauxM−, entailment between equivalence classes MφandMψ is most easily characterized using the representatives of those classes which are entailment closures of their normal formsMφEnand MψEn (see 32 above for the definition).
φ ⊧ ψ exactly when MφEn ⊆ MψEn. This is, of course, not surprising, given the relation betweenσφ and MφEn, 33.