Comparing Positional Licensing Patterns in HG and OT

Positional licensing refers to the observation that elements (e.g. particular feature values or feature value combinations) can be limited to specific positions (e.g. syllable onsets, initial syllables, stressed syllables, etc.). Positional licensing patterns have been analyzed using either positional markedness or positional faithfulness constraints in OT and HG. In this paper we demonstrate that the predictions of OT and HG diverge in deep but structured ways once there are more than two licensing positions. We propose an account for this structured divergence based on 3-position systems, and confirm the validity of that account with an analysis of 4-position systems. We also describe how conjoined constraints impact positional licensing patterns, and in doing so provide a counter-example to a claim made in our previous work (Mai & Baković 2020).


Introduction
Positional licensing refers to the observation that elements (e.g. particular feature values or feature value combinations) can be limited to specific positions (e.g. syllable onsets, initial syllables, stressed syllables, etc.). These elements are said to be licensed in those positions. A simplified schematic example is shown in Figure 1, with just one licensing position P among the set of all positions E. The element x is licensed in P, but not elsewhere in E; we will conventionally refer to the complement of P asP. Positional licensing patterns have been analyzed using either positional markedness or positional faithfulness constraints (Beckman, 1997; Lombardi, 2001; Zoll, 2004. The effect of a positional markedness constraint is schematically represented on the left in Figure 2: the constraint *x/P is violated when the element x occurs inP -that is, outside of the licensing position P. The effect of a positional faithfulness constraint is schematically represented on the right: the constraint *x → ∅/P is violated when x is mapped unfaithfully in the licensing position P. (Unfaithful mappings are represented by deletion throughout.)  Jesney (2016) investigates the typologies of different positional licensing systems in OT (Prince & Smolensky, 2004) and HG (Legendre et al., 1990). 1 She finds that when there are two (or more) licensing positions, P 1 and P 2 , two additional types of licensing pattern are possible depending on (a) whether licensing is analyzed with markedness or faithfulness and (b) on the form of constraint interaction, OT ranking or HG weighting. One of these additional licensing pattern types is disjunctive: x is licensed when it is either in P 1 or in P 2 (or in both; disjunction is inclusive). The other type of licensing pattern is conjunctive: x is licensed only when it is both in P 1 and in P 2 . These two patterns are schematically depicted in Figure 3.

Figure 3: Disjunctive licensing (left) and conjunctive licensing (right)
All of Jesney's systems consist of candidate sets with two structures each, one faithfully realizing the input element x and the other unfaithfully deleting x. A general markedness constraint *x is violated by the faithful structure and a general faithfulness constraint *x → ∅ is violated by the unfaithful structure in every case. (Given the lack of ambiguity, we henceforth abbreviate *x → ∅ to *∅.) The systems differ in privileging up to three fully intersectable licensing positions. 2 Additional candidate sets correspond to the various possible additional fates for x: faithfully realized or deleted in each licensing position or intersection of licensing positions. The systems also differ in whether every licensing position P n is associated with a positional markedness constraint *x/P n , a positional faithfulness constraint *∅/P n , or both.
For the OT systems, Jesney (2016) finds that positional markedness predicts conjunctive licensing patterns while positional faithfulness predicts disjunctive licensing patterns; with both types of constraints, both conjunctive and disjunctive licensing patterns are predicted. For the HG systems, on the other hand, Jesney (2016) finds that both conjunctive and disjunctive licensing patterns are predicted regardless of whether there are only one or both of positional markedness and positional faithfulness constraints in the system. The claims resulting from Jesney's (2016)  To explore how Jesney's (2016) generalizations scale with larger numbers of licensing positions and the concomitant increase in possible conjunctive and disjunctive environments, we created sets of positional licensing systems with 2, 3, and 4 licensing positions, as described in Section 2. Despite the suggestion in Figure 4 that the set of patterns predicted by the S OT system is equivalent to the set of patterns predicted by the M/F/S HG systems, we demonstrate in Section 3 that these predictions diverge in deep but structured ways once there are more than two licensing positions. In Section 4, we propose an account for this structured divergence based on 3position systems, and confirm the validity of that account with an analysis of 4position systems. In Section 5, we describe how conjoined constraints impact positional licensing patterns, and in doing so provide a counterexample to a claim made in our previous work (Mai & Baković, 2020): whereas the typological predictions of an HG system that includes conjoined constraints and the typological predictions of the corresponding OT system with conjoined constraints are equivalent, predictive equivalence cannot be guaranteed between an HG system without conjoined constraint and the corresponding OT system with conjoined constraints. In Section 6 we conclude.

Systems
We began by analyzing 10 systems, differing along two dimensions: the number (0-3) of licensing positions A, B, C referenced by constraints, and whether those are markedness (M), faithfulness (F), or both (S). Again, the three positions A, B, C are fully intersectable; e.g. if A = 'initial syllable', B = 'syllable onset', and C = 'stressed syllable', then A∩B∩C = 'initial stressed syllable onset' (recall footnote 2). The constraint sets for these 10 systems are summarized in Figure 5. All candidate sets contain two structures, one with input x realized faithfully and the other with x deleted unfaithfully. Having no positional constraints, system S 0 effectively has only one candidate set. Systems M 1 , F 1 , and S 1 have two candidate sets each: x ∈ A and x ∈Â. Systems M 2 , F 2 , and S 2 have four candidate sets each: x ∈ A, x ∈ B, x ∈ A∩B, and x ∈Â∩B. Finally, systems M 3 , F 3 , and S 3 each have eight candidate sets: x ∈ A, x ∈ B, x ∈ C, x ∈ A∩B, x ∈ A∩C, x ∈ B∩C, x ∈ A∩B∩C, and x ∈Â∩B∩Ĉ.

Patterns
Factorial typologies under OT and HG were computed and analyzed using OTWorkplace (Prince et al., 2007(Prince et al., -2020. HG was simulated with the equalizer algorithm of Mai & Baković (2020) and confirmed with OTHelp (Staubs et al., 2010); but see the Appendix for a qualification. Nine sets of patterns are predicted, summarized graphically in Figure 6 (for 0-2 licensing positions) and Figure 7 (for 3 licensing positions).   Jesney (2016:194) also observes that the 3way unionofintersections pattern noted at the end of the previous section is a prediction unique to the 3position HG systems, the one pattern in twenty that is not predicted by the most inclusive 3position OT system S OT 3 , and concludes from this 5% discrepancy that "[t]ypological expansion due to cumulative constraint interaction is quite limited" Jesney (2016: 195).

Analysis
Our goal in this section is to explain why this pattern is missing from S OT 3 , which otherwise predicts 2way unionofintersections patterns ( §4.1), and how it is that all of the 3position HG systems predict it ( §4.2). We confirm our explanation by widening our view to consider 4position systems ( §4.3), and conclude that the typological expansion due to cumulative constraint interaction is better described as structured, not limited.

3
An arbitrary member of the set of three 2way union ofintersection patterns, [A∩B]∪[B∩C], is schematically illustrated in Figure 8, along with the OT constraint ranking responsible for it. The first thing to note is that, given that there are only three positions and two intersections in the pattern, there is guaranteed to be a unique position that is common to both intersections (in this case, B). We call this unique shared position the anchor, and the remaining positions the complement. Moving from the top of the constraint ranking down, we can see that the presence of a unique anchor is critical to predicting a unionofintersections pattern in S OT 3 -and thus why the 3way unionofintersections pattern is not predicted, given that it does not have a unique anchor. Starting from the top of the ranking on the right in Figure 8: the positional markedness constraint that references the anchor, *x/B, excludes x from the subset of the complement A and C that does not intersect with the anchor. This relationship between the ranking and the illustration is indicated by the solid lines around both in Figure 8. The positional faithfulness constraints that reference the complement, *∅/A and *∅/C, then serve to protect x in the intersections between the complement and the anchor, A∩B (abbreviated AB in the illustration) and B∩C (abbreviated BC). This relationship is indicated by dotted lines in Figure 8. Finally, the remaining markedness constraints *x/Â, *x/Ĉ, and *x exclude x from the nonintersecting subset of the anchor, B. This relationship is indicated by dashed lines in Figure 8.

3
The illustration of the 3way unionof intersections pattern in Figure 9  There are many and varied constraint weighting conditions in the S HG 3 system that will describe the pattern in Figure 9, so we focus here on the more unified conditions in the M HG 3 and F HG 3 systems. 3 First, M HG 3 . Recall that in this system there is only one, general faithfulness constraint (*∅) and the full array of markedness constraints, general (*x) and positional (*x/Â, *x/B, and *x/Ĉ). The weight of the faithfulness constraint must be greater than the weight of each of the markedness constraints in order to allow x to surface at all (1a), but the cumulative weights of each possible pairing of positional markedness constraints must in turn be greater than the weight of the faithfulness constraint in order to penalize instances of x surfacing in any single position that does not intersect with at least one other position (1b).
( Note that the cumulative weight of each pair of positional markedness constraints in (1b) is crucial. Each of the positional markedness constraints on its own is too specific to penalize x when it surfaces in any non intersecting position; for example, *x/Â on its own can only penalize x when it surfaces anywhere outside of licensing position A. The general markedness constraint, on the other hand, is too general, penalizing x everywhere. Acting together, though, two positional markedness constraints give just the right result. *x/Â and *x/B penalize x when it surfaces anywhere outside the union of the licensing positions A and B -that is, in licensing position C or elsewhere in the set of all positions E. The other two pairings of positional markedness constraints complete the picture, leaving only the positional intersections as possible licensors.

1) Weighting conditions for [A∩B]∪[A∩C]∪[B∩C] in M HG
Recall now that in the F HG 3 system there is only one, general markedness constraint (*x) and the full array of faithfulness constraints, general (*∅) and positional (*∅/A, *∅/B, and *∅/C). The weight of the markedness constraint must be greater than the weight of each of the faithfulness constraints in order to prevent x from surfacing in any single position (2a), but the cumulative weights of each possible pairing of positional faithfulness constraints must in turn be greater than the weight of the markedness constraint in order to allow instances of x to surface in any of the three positional intersections (2b).
(2) Weighting conditions for The cumulative interactions in (2b) are again crucial, but for a slightly different reason in this case even though they look very similar to the previous case in (1b). Each of the faithfulness constraints on their own, 3 Recall that the constraint set of S HG 3 is the union of the constraint sets of M HG 3 and F HG 3 , so any patterns describable by the latter two systems is at least describable by the former, more inclusive system with a similar set of weighting conditions. both general and positional, are too general to allow x to surface only in positional intersections; for example, *∅/A on its own protects x when it surfaces anywhere in the licensing position A. But acting together, two positional faithfulness constraints do the trick: *∅/A and *∅/B protect x from deletion anywhere in the intersection of the licensing positions A and B. The other two pairings of positional faithfulness constraints complete the picture, again rendering only the positional intersections as possible licensors.

4position systems
The unique anchor position analysis discussed in §4.1 is confirmed by further analysis of 4position systems. A unique anchor position is shared in each of the unionofintersections patterns predicted by S OT 4 (46 of 96 total patterns predicted), and no unique anchor position is shared in any of the additional unionofintersections patterns predicted by M/F/S HG 4 (54 of 150 total patterns predicted). One example of each type of pattern is shown in Figure 10 As already noted above, less than twothirds of the patterns predicted by the 4position HG systems are predicted by the most inclusive 4position OT system, S OT 4 (96 of 150). Broadening our scope even wider to 5position systems, about one in five patterns predicted by the HG systems are predicted by S OT 5 (669 of 3287). It's clearly not the case that the typological consequences of cumulative constraint interaction are limited, at least not numerically or proportionally -instead they are structurally delimited, with uniquely shared anchor positions in unionofintersections patterns forming the basis of that delimiting structure.

Cumulative interaction and conjoined constraints
The previous two sections identified how the typological predictions of S OT n diverge from those of M/F/S HG n due to the differential ability of HG and OT to capture combinations of conjunctive and disjunctive licensing environments. However, as shown in Mai & Baković (2020), when conjoined constraints 4 are introduced into these frameworks, the differences in their typological predictions disappear. We tested this result by augmenting each of the systems described in Section 2 such that all possible conjunctions of the positional markedness constraints (in the M n systems), all possible conjunctions of the positional faithfulness constraints (in the F n systems), or both (in the S n systems) are added to the constraint set. 5 So, for example, the F OT 3 and F HG 3 systems contain the constraints { *x, *∅, *∅/A, *∅/B, *∅/C } ; we augmented this set to +F 3 such that it also contains the conjunctions { *∅/AB, *∅/AC, *∅/BC, *∅/ABC } . Contrary to Mai & Baković's (2020) claim that the addition of conjoined constraints to an HG grammar has no effect on the typology it predicts, we found that conjoined constraints can expand both HG and OT typologies under particular conditions, and that the additional patterns in the expanded typologies elucidate the ways that conjoined constraints behave in constraintbased systems. In particular, the M/F/S HG 4 systems all predict the same 150 patterns, and the corresponding OT systems without conjoined constraints produce subsets of those patterns: M/F OT 4 both predict the same 17 patterns, and S OT 4 predicts those same 17 plus 79 more (= 96). Augmenting these systems with conjoined constraints in the manner described above, all six augmented systems predict the same set of 168 patterns, 18 in addition to the 150 predicted by M/F/S The problem becomes apparent on closer inspection of these weighting conditions. Consider conditions (3a,b,d): together, they state that the weight of *∅/C can be added to the weight of either *∅/A (3a) or *∅/B (3b) to overtake *x, but that the combined weight of *∅/A and *∅/B is insufficient to the task (3d). This means that the weight of *∅/C must be greater than the individual weights of *∅/A and *∅/B, leading to a contradiction between conditions (3c) and (3f): if the combined weight of *∅/B and *∅/D is sufficient to overtake *x (3c), and the weight of *∅/C must be greater than the weight of *∅/B as we surmised from (3a,b,d), then the combined weight of *∅/C and *∅/D must be greater than the weight of *x, contrary to (3f). Conditions (3b,c,f) together lead to the same contradiction between conditions (3a) and (3d). These contradictions are succinctly captured by algebraic rearrangement of the inequalities in (3), as shown in (4): conjoined constraints to capture unachored unionofintersection patterns. Unlike what we see with the interaction of the full set of constraints in S OT 4 , the interaction of conjoined constraints with the base set is not greater than the sum of its parts: the subrankings that made [A∩C] impossible in F OT 4 still feature in the rankings responsible for [A∩C]∪[B∩C]∪ [B∩D] in +F OT 4 . The only difference is that now a conjoined constraint can rank where the other half of a contradiction would have. Conjoined constraints thus mediate conditions that would otherwise require impossible rankings or weightings.

Conclusion
We have demonstrated two things in this paper. The first builds on the insights of Jesney (2016), who noted the typological differences between OT and HG systems with positional constraints, as summarized in Figure 4. Jesney's focus was on how either positional markedness or positional faithfulness constraints are sufficient to generate patterns that require both families of positional constraints in OT. We've shown here that there are deep and structured typological differences between these HG systems and the corresponding, most inclusive OT system (S OT ), differences that are revealed once there are positional licensing constraints in those systems that refer to more than two licensing positions. We've also shown that there is a basis for those differences: if a given unionofintersections pattern involves a common position across the intersectionsan anchor -there is a way to generate that pattern in OT; with no anchor, the added power of cumulative constraint interaction in HG is required.
The second thing that we have shown concerns the claim made in Mai & Baković (2020) that conjoined constraints engender no expansion in the typological expressivity of HG. While it remains the case that the addition of conjoined constraints to HG and OT systems with a common base set of constraints "equalizes" their predictive differences, conjoined constraints do so in a way that is distinct from the impact that cumulative constraint interaction has on HG typologies. Adding conjoined constraints to HG and OT systems expands their expressive capacities by introducing the means to sidestep conflicting weighting and ranking conditions. Our future work will investigate the formal consequences of this expansion in greater depth.