1 Introduction
An area of theoretical interest in Optimality Theory (OT; Prince & Smolensky Reference Prince and Smolensky1993) is the make-up of the constraint set, Con, and particularly whether various constraints exist as single general constraints, or are split into multiple versions, specific to feature values, position, etc. For instance, is there a single Ident constraint that assesses violations for all features, as opposed to a family of distinct Ident[F] constraints that each assess just one feature (Itô et al. Reference Itô, Mester and Padgett1995: 586, McCarthy & Prince Reference McCarthy, Prince, Beckman, Dickey and Urbanczyk1995: 16)? A candidate that violates three single-feature faithfulness constraints, Ident[F], Ident[G] and Ident[H], will also violate an all-feature Ident constraint three times (once each for [F], [G] and [H]). This kind of relationship is termed summing: the violation profiles of several constraints (‘summands’), when pooled together, add up to the violation profile of the combined (‘summed’) constraint.
Comparing and evaluating analyses that differ in this way requires knowing the effect of such summing on the full typology. This in turn depends on the relationship of the summands in the full system, which can be difficult to ascertain from inspecting violation profiles in a violation tableau, where numerical quirks and other interactions can obscure the interactions. This paper uses Property Theory (Alber et al. Reference Alber, DelBusso and Prince2016, DelBusso Reference DelBusso2018, Alber & Prince Reference Alber and Princein preparation) to show the effects of summing constraints in two distinct kinds of relationships. The results show the predictable changes, and allow for the systematic comparison of related typologies.
Property Analysis provides a key to identifying the constraint relationships – and so too to delineating the effects of summing constraints. We show this by analysing two cases where the summands stand in specific relations, which we call Summing Across (SA; §4) and Summing In (SI; §5). These names refer to how the summand constraints are related in a property analysis: either distributed across different properties with parallel structures or found in the same property. The properties isolate the core constraint interactions that define the grammars of the typology. Both SA and SI collapse the typology systematically, by eliminating or equalising properties and grammars; see (1).
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SA equalises properties and eliminates grammars: when properties become the same, some grammars are no longer possible (i.e. they are harmonically bounded; Samek-Lodovici & Prince Reference Samek-Lodovici and Prince2005). SI eliminates properties and equalises grammars: when properties are lost, grammars differing only in the values of these become the same, and their languages are co-optimal.
While the two cases are developed using abstract OT systems, both are manifested in concrete OT systems as well, including the typologies analysed in Bennett & DelBusso (Reference Bennett and DelBusso2018) and DelBusso & Bennett (Reference DelBusso and Bennett2019), which are based on the literature on Agreement by Correspondence (Rose & Walker Reference Rose and Walker2004, Hansson Reference Hansson2010, Bennett Reference Bennett2015). The SA case connects to theories that posit families of parallel constraints, such as constraint schemata that make the same kind of choice independently available for different features, arrangements, directions, categories, etc. The SI case connects to the relationship between the two main Agreement by Correspondence constraint types, Corr and CC.Id, which act together as a class to enforce harmony (or dissimilation).
2 Summing in concrete OT systems: Agreement by Correspondence
Agreement by Correspondence (ABC) – and related theories examined in Bennett & DelBusso (Reference Bennett and DelBusso2018) and DelBusso & Bennett (Reference DelBusso and Bennett2019) – provides examples of both types of summing analysed here.
In ABC theories, there are two main constraint types: Corr, which assigns violations when some designated segments (e.g. those that share a given feature) are not in surface correspondence with one another, and CC.Id, which evaluates feature agreement between segments that stand in correspondence with each other. Both types of constraints are satisfied when output segments either (i) are in correspondence and in agreement for the feature values specified by CC.Id (producing assimilation), or (ii) are not in correspondence and disagree in feature values specified by Corr (producing dissimilation). Most ABC theories assume families of Corr and CC.Id constraints, with members of each family specified for different features or domains, etc., resulting in parallelisms among the properties of the typologies of such systems. (For example, the interaction between Corr[F] and IO.Id[F] occurs in a predictable way, regardless of which particular feature [F] represents.) There have also been proposals to combine constraints both within and across the two families, similar to the summing operations here. The violation tableau in (2) shows the violations of both the unsummed and summed constraints (the latter outlined in bold) for two candidate sets; only non-harmonically bounded candidates are shown.Footnote 1
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The ABC systems and sums thereon exemplify the typological effects of the two types, collapsing the typologies in different but predictable ways. Corr[vce] + Corr[cont] is an example of Summing Across (SA); Corr[vce] + CC.Id[cont] is an example of Summing In (SI). These are discussed in further detail in the respective sections.
3 Formal background
3.1 Elementary Ranking Conditions and fusion
Elementary Ranking Conditions (ERCs; Prince Reference Prince2002) encode the rankings necessary for one candidate x to be more optimal than y. ERCs are vectors with a column for each constraint with one of three values: W (prefers x), L (prefers y) or e (does not distinguish the candidates). For example, the ERC WeL indicates that the first constraint must dominate the last for the chosen candidate x to be optimal. An ERC is satisfied when all Ls are preceded by a W. The ranking encoded in the ERC makes x optimal over y. A non-trivial ERC contains at least one W and one L, and is satisfied when at least one constraint with a W dominates all constraints with Ls in the grammar. A trivial ERC with only Ws and e's is satisfied by any ranking; one with no Ws (only Ls and e's) cannot be satisfied (Prince Reference Prince2002). The latter is referred to as ℒ+. An OT grammar is defined by a set of ERCs (Merchant & Prince Reference Merchant and Princeto appear), as in (3).
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A fundamental operation on ERCs is fusion (∘; Prince Reference Prince2002). Fusion takes a set of ERCs and produces the ERC entailed by the set. Column values are combined across the rows as follows: L∘X = L (i.e. L is dominant); W∘e = W (i.e. e acts as identity); X∘X = X (where X is any value). In (4), the C1 column contains a single W and 2 e's, fusing to W; remaining columns contain at least one L, fusing to L. The fused ERC encodes the ranking information that C1⪢C3, entailed by the fused ERC set but not explicit in any individual ERC.
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The kind of constraint summing introduced in this paper reduces two or more constraints to one by fusing across the summand columns in an ERC set. The value in the summed column is determined just as in fusion. In (5), C1 and C2 in ERC3 are fused into C1+2 in ERC3′. If a candidate violates either of C1 or C2, then it violates the summed C1+2.Footnote 2 Such fusion can also produce trivial ℒ+ ERCs, as is the case if C1 and C2 in ERC1 are fused, yielding LeL.
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3.2 Property Theory
The set of ERCs delineating a grammar need not define a total – or partial – order over the entirety of Con: not all constraints are crucially ranked relative to one another in a given typology (see Prince Reference Prince2017 on representing grammars). Identifying which rankings are crucial is the aim of Property Theory (see Alber et al. Reference Alber, DelBusso and Prince2016, DelBusso Reference DelBusso2018 and Alber & Prince Reference Alber and Princein preparation for introductions to Property Theory).
A Property Analysis of a typology is a set of properties that, collectively, define all of the grammars in the typology, i.e. the set of choices sufficient to distinguish all languages in a typology. In a valid Property Analysis of a typology, the possible value combinations of the property set correspond one-to-one to the grammars.
A property is a binary choice between two mutually exclusive ranking conditions. A property has two constraints (or groups of constraints) as antagonists, X⪡⪢Y, defining two mutually exclusive values, α. X⪢Y and β. Y⪢X. Any grammar in which X and Y are crucially ranked has one value or the other. The values define an ERC (set) (α) and its negation (β); these are the value ERCs. A property grammar is defined as a set of values that generates the ERC set of the grammar.
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The antagonists in a property (X and Y) may be single constraints, or sets of (sets of) constraints. The latter case is a situation where crucial rankings are not strictly pairwise relationships between individual constraints, but between groups of constraints that act together. Such groups form a constraint class, abbreviated κ, which is appended with an operator, dom or sub, designating the highest or lowest member of the class, respectively, in their linear ordering (Alber et al. Reference Alber, DelBusso and Prince2016, DelBusso Reference DelBusso2018 and Alber & Prince Reference Alber and Princein preparation). For example, {X,Y}.dom means that the dominant member of {X,Y} interacts with the antagonist. A κ.dom in a property generates an ERC with multiple Ws or Ls; when class κ dominates the antagonist, the dominant constraint in the class ranks above the antagonist; when κ is subordinate, the antagonist ranks above the dominant constraint in κ, and thus transitively above all κ.dom members. A κ.sub, however, generates a set of ERCs. This set is conjunctive when the class is dominant; i.e. if the lowest-ranked member of κ.sub dominates the antagonist, then all other constraints in the class also dominate the antagonist (by dominating the lowest). When the class is subordinate, the set is disjunctive: the antagonist does not need to dominate all members of the class in order to dominate the lowest-ranked member. If a Property Analysis contains a property ranking a κ.sub relative to an antagonist, it must also have some other property that antagonises the members of κ.sub among each other, and so defines which member is subordinate. The ERCs for a class with two constraints are shown in (7).
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In a Property Analysis, properties can be related in complex ways, indicated by their scope (Alber et al. Reference Alber, DelBusso and Prince2016, Alber & Prince Reference Alber and Princein preparation), defined in (8).
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For a wide-scope property, every grammar in the typology has a value. For a narrow-scope property, only some grammars in the typology have a value. Narrow scope is defined by the values of other properties: only if another specific ranking condition occurs is the ranking in the narrow-scope property needed. Thus scope reflects dependency between different properties. For a grammar outside the scope of a property, the choice made by that property is meaningless and/or inconsistent with the ranking conditions defining it.
4 Summing Across: constraints in parallel properties
The SA case sums distinct constraints that play parallel roles in a typology, interacting with the same antagonists in the same ways, as, for example, the Corr constraints with the IO.Id[F] constraints. This relationship is manifested in Property Analyses by the presence of parallel properties, defined in (9), which have the same structure, but differ minimally in a constraint or a κ.op. Summing the differing constraints merges the parallel properties, resulting in loss of those grammars differing in their values.Footnote 3
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The SA case is first presentedFootnote in (10) with a simple system called ‘SA Base’, consisting of three constraints. C1 and C2 are the prospective summands, and X (either a constraint or a κ.op) is the antagonist in parallel properties P1 and P2. These have the same form, differing in the lefthand side (C1 or C2) and sharing the antagonist X. Additionally, their scopes are equivalent (both are wide-scope). The property values are freely combinable, as C1 and C2 can be ranked relative to X independently of one another, generating the four grammars in the value table in (10b), which shows the value combinations that define them. The values generate ERCs, which combine to give the resulting grammars. ERCs are given in the order C1-C2-X.
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When C1 and C2 are summed, their ERC columns fuse, as in (11), and the P1 and P2 values thus become identical. When C1 and C2 are combined in a single constraint, there are only two possible rankings: C1+2⪢X and X⪢C1+2. As a result, any combination of α and β values for the parallel Ps are contradictions, fusing to ℒ+. Grammars with mixed values of this sort rank the summands on either side of the shared antagonist X, an impossibility when these are a single constraint, so pre-sum grammars Lg2 and Lg3 are harmonically bounded (shaded in (11)), leaving two grammars, defined by the values of P1 and P2.
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These results generalise to the context of additional properties and constraints. Even when more than two constraints are summed across more than two properties, the consequence is that the parallel properties are merged into a single property. This eliminates from the typology grammars that have mixed values on the parallel properties in the pre-sum system. The choice after summing is limited to just α or β: a mixture of α and β is impossible. Consequently, the grammars that were defined by mixtures of values on the parallel properties in the pre-sum system become impossible after summing. The SA formal result is spelled out in (12), along with its justification. To generalise across all property values, the following notations are used: ‘v’ for a property value ERC, ‘v̄’ for its reverse and ‘…’ for other constraints with the same values (W/L/e) in all ERCs.
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The effects of SA are manifested as summing violations across summand columns in a universal violation tableau (Prince Reference Prince2015). A universal violation tableau is a violation tableau that represents a full typology, where each row is a language: the ERCs that result when each row is chosen as optimal and compared to every other row generate the grammar of the language (Prince Reference Prince2015). As constraints in parallel properties interact with the same antagonists independently of one another, there are four violation-profile patterns in the pre-sum system (relative to each shared antagonist): (i) both constraints have minimum violation value (αα);Footnote 5 (ii) neither do (ββ); (iii) one does, but not the other (αβ or βα). The universal violation tableaux for the SA Base pre- and post-sum systems are shown in (13). Summing violations across the rows (the post-sum system) results in the collective harmonic bounding of grammars that violate one of the summands and the antagonist (L2, L3) by those that violate either both summands or the antagonist (L1, L4).
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This pattern is iterated for multiple antagonists, as shown in the universal violation tableau with antagonist {Xa, Xb}.sub in (14). The seven grammars of the pre-sum universal violation tableau reduce to three post-sum grammars.
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The formal changes from summing constraints in parallel properties are manifested extensionally in the parallel treatment of different extensional traits – traits which were determined by separate properties in the pre-sum system. In ABC typologies, a clear example is differences between features. In a system studied by Bennett & DelBusso (Reference Bennett and DelBusso2018), there are two doubly parallel properties, each of which has a class of one feature-specific Corr constraint and one feature-specific CC.Id constraint as one antagonist, and a class of IO.Id constraints as the other, as in (15).
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These properties highlight a parallelism in the extensional typology: languages may have harmony for neither, either or both of the features [voice] and [continuant], and the harmony/no harmony choice depends on P1[vce] and P1[cont] values respectively. In segmental terms, the choice of whether an input with disharmonic stops like /t d/ surfaces faithfully, or assimilates or dissimilates, is a separate choice from whether inputs like /d z/ are faithful, or assimilate or dissimilate. The two choices are intensionally parallel, as Corr and CC.Id constraints impart the same structure onto each ranking choice, abstracting away from differences in the particular features referred to. The choice about how to handle stricture disharmony works in the same way as the choice about how to handle voicing disharmony.
Summing all of the Corr constraints into one constraint and all of the CC.Id constraints into another, as in (16), collapses the two choices into a single one: harmony for both features or for neither (as the summed constraint is violated).
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Grammars differing in values for the combined properties become harmonically bounded: the candidate languages that only have harmony for one feature violate both the IO faithfulness constraints and the ABC constraints, while the all-or-nothing harmony alternatives have violations of only one or the other. The summed constraints are more general, referring to larger sets of features. They are similar to completely general constraints proposed in the literature that lack any kind of feature reference; see McCarthy (Reference McCarthy2010) on Corr and Gallagher & Coon (Reference Gallagher and Coon2009) on CC.Id. The extensional result of summing across features is to yoke the two featural choices together.
5 Summing In: constraints in a class
In many typologies, constraints act together in properties, forming a class, κ. For example, the pairs of Corr and CC.Id constraints form classes in P1[cont] and P1[vce] in (15). SI sums constraints within such a class, resulting in elimination of other properties and equalisation of grammars. These effects arise due to the structure of Property Analyses with κ.op's: the op indicates that the antagonist is ranked relative to the highest or lowest member of κ, but does not determine which constraint this is. Ordering among the κ.op members must occur in another property or set of properties. For example, in P1: {A,B}.sub⪡⪢X, X is ranked relative to the subordinate of A and B, which are not themselves ordered in P1 values. As such, a second property, P2: A⪡⪢B, ranks the members of the class. For P2α, A⪢B, B is subordinate and thus X is ranked relative to B in P1, and for P2β the reverse holds.Footnote 6 When the members of a κ.op are summed, properties antagonising the summands are eliminated, as their values – attempting to rank a constraint relative to itself – become ℒ+. Consequently, grammars of the pre-sum system differing only in the values of such properties become co-optimal (i.e. both are optimal under the same rankings). There is no distinction between optima which are differentiated only by the ranking of the summed constraints.
As with SA, SI is first shown with the simple three-constraint SI Base system in (17). P1 has a two-constraint κ.op, {C1, C2}.sub, and its values are a conjunctive (α) and disjunctive (β) ERC set (ERC order C1-C2-X). P2 antagonises the members of the P1 κ.op and is within the scope of P1β: only in grammars with this value is the ranking of C1 and C2 crucial.
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Summing fuses the first two ERC columns, reducing the multi-ERC sets of P1 values to single ERCs (the conjuncts and disjunct become equivalent), as in (18). Additionally, P2 values become ℒ+, as they cannot rank a single constraint relative to itself. Eliminating the illogical P2 makes Lg2 and Lg3 equivalent/co-optimal, as they are no longer distinguished by any property.
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These typological effects of SI shown in the simple system generalise to the more complex cases with larger summed κ.op's, and consequently to the more complex property structures in (19).
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Property Analyses with SI-eligible κ.op's also have characteristic universal violation tableaux that show the summing effects. There are three violation patterns for which a κ.sub and a κ.dom differ in whether a candidate violates neither or both summands respectively, as in (20). Summing results in identical violation profiles for Lg2 and Lg3.
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The loss of properties and subsequent co-optimisation of grammars are manifested extensionally in the loss of a distinguishing trait in the pre-sum system, for example, the distinction between correspondence and non-correspondence in faithful grammars in ABC systems.
The SI case sums the classes (κ.op's) of Corrs and CC.Ids in the properties above. Such a combination is highly similar to Hansson's (Reference Hansson2014) Agreement by Projection (ABP) theory, which combines Corr and CC.Id constraints into single constraints that simultaneously condition agreement on one feature (as in Corr) and enforce agreement on another (as in CC.Id), effectively transmuting a pair of Corr and CC.Id constraints into a single constraint that demands agreement (Agr). Summing the κ.op's into Agr constraints results in the Ps in (21): Agr[cont]/[vce] = Corr[vce] + CC.Id[cont] and Agr[vce]/[cont] = Corr[cont] + CC.Id[vce]. This summing also eliminates from the Property Analysis additional properties antagonising Corr and CC.Id constraints, and in so doing results in languages becoming co-optimal, distinguished by only those values (Bennett & DelBusso Reference Bennett and DelBusso2020). Extensionally, the co-optimised languages have the same segmental surface forms (i.e. [t1d1] or [t1d2]), and differ only in the presence or absence of correspondence indices (which are eliminated in Hansson's proposal). The summed system thus leave only a choice between faithful and unfaithful mappings.
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6 Summary
A question in linguistic analysis concerns the effects of combining certain conceptual pieces (here, constraints) into a single mechanism – which could be argued to be preferable by Occam's Razor, all other things being equal. The converse is also of interest: what are the effects of splitting a single mechanism in an insufficiently rich theory in order to produce more degrees of freedom in a typology? Understanding the answers to these questions is fairly straightforward in any concrete case if we define both theories, produce their typologies and compare the results. But the possibility of a generalised solution holds the promise of obviating that extra legwork, and yielding greater insight into the structure of OT typologies.
The findings in this paper report a means of delineating systematic differences between typologies with related Cons. Using Property Theory, the typological effects of summing constraints in particular relations is predictable. SA applies to Property Analyses with parallel properties, equalising properties and eliminating grammars, while SI applies to Property Analyses with constraint classes, equalising grammars and eliminating properties. In addition to addressing the first question above on the effects of combining constraints, these results are a step towards an answer to the converse: the effects of splitting a constraint are hypothesised to be the reverse of summing. Together, these provide a means of understanding a lattice of typologies projected from any one typology that is fully analysed and understood, related to the others through summing and unsumming (splitting).
