Three Formal Extensions to Primitive Optimality Theory

This paper proposes an expansion of set of primitive constraints available within the Primitive Optimality Theory framework (Eisner, 1997a). This expansion consists of the addition of a new family of constraints--existential implicational constraints, which allow the specification of faithfulness constraints that can be satisfied at a dis tance--and the definition of two ways to combine simple constraints into com: plex constraints, that is, constraint disjunction (Crowhurst and Hewitt, 1995) and local constraint conjunction (Smolensky, 1995). 1 I n t r o d u c t i o n Primitive Optimality Theory (OTP) (Eisner, 1997a), and extensions to it (e.g., Albro (1998)), can be useful as a formal system in which phonological analyses can be implemented and evaluated. However, for certain types of constraints, translation into the primitives of O T P (Eisner (1997b)) can only be accomplished by adding to the grammar a number of ad hoc phonological tiers. Because these tiers serve no phonological purpose other than to allow calculation of the constraints without adding new primitives, and because the addition of phonological tiers to an O T P grammar can have a dramatic negative impact on the efficiency of OTP implementations 1, it is preferable to avoid the addition of ad hoc tiers by adding new primitives to the system. This paper looks at three types of constraints employed throughout the Optimality Theoretic literature that cannot be translated in to the 1The computat ion time for an Optimali ty Theoretic derivation within the implementation of Albro (1998) increases exponentially with the number of tiers. The same is true for the implementation described in Eisner (1997a), although a proposal is given there for a method that might improve the situation. primitives of O T P without reference to ad hoc tiers, and proposes a formalization of these constraints that is compatible with the finite state model described in Eisner (1997a) and Albro (1998). These are constraints of existential implication (that is, of faithfulness without the requirement of alignment), constraint disjunction, and local constraint conjunction. 2 E x i s t e n t i a l I m p l i c a t i o n 2.1 M o t i v a t i o n OWP as described in Eisner (1997a) provides some support for correspondence constraints ( input-output only). These may be defined by means of implication constraints of the form P --4 P or P --+ P, which can be interpreted as requiring, in the first case, that each surface constituent representing property P be aligned with an underlying constituent representing that property, and in the second case that every underlying constituent representing property P be aligned with a surface consti tuent representing that property. Constraints of this type may be employed to require correspondence between the underlying representation and the surface representation where corresponding constituents must be aligned with one another. However, natural languages also seem to follow weaker constraints requiring only that for each underlying constituent there be a corresponding surface constituent, regardless of the position of that constituent relative to its position in the underlying representation. For example, in Sanskrit roots with at least two voiced stops, where the root ends in a voiced aspirated stop, the underlying aspiration of the root-final stop can be realized upon that stop in the surface representation only when the root is followed by a suffix beginning with a vocoid or a nasal (data from Whitney (1889)): 21 /da,9a/ reach to [dagfli~janti] (Fut.) / b u d ~ / know, wake [bod~i] (Aor.) / d a b ~ / harm [dab~ati] (Pres.) Otherwise, the aspirat ion is realized on the preceding stop: / d a g ~ / reach to [d~ak] (root noun) / b u d a / know, wake [baut] (root noun) / d a b ~ / harm [dfiap] (root noun) In these forms it is clear t ha t aspiration is being preserved, but tha t it is surfacing in a position t ha t cannot overlap with the underlying form. Another example is the Bantu language Chizigula (Kenstowicz and Kisseberth, 1988), in which roots with underlying high vowels appear on the surface with a single high tone in the penul t imate syllable of the word, where this syllable could belong to a suffix. Additionally, if a prefix with an underlying high tone is prefixed to a root with no underlying high tone, the high tone of the prefix appears in the penul t imate syllable of the resulting word. The existence of a high tone in the underlying form implies the existence of a high tone in the surface form, but the position where tha t high tone occurs in the underlying form has nothing to do with where the tone appears on the surface. 2.2 F o r m a l i z a t i o n Existential implication constraints can be used to drive correspondence effects such as the above. These constra ints take the form "If X appears within domain D, then Y must appear within D as well. " Using the terms of OTP, this family of constraints can be wri t ten as a'l A . A am ~ /31 V . V/3n/9,1 A . A 9,pHere each a'i or /3j is a cons t i tuent interior or edge, one of tier, ]tier, or [tier, where tier represents a cons t i tuent type, and each 9,k mus t be a cons t i tuent interior (tier). The constraint represented by this nota t ion ou tpu t s a violation for each domain 9,, where 9' represents the intersection of the domains 9,k, in which the t ime slice represented by the oq occurs, but no/3j occurs. Using the FST notat ion of Eisner (1997a), the implementa t ion for this constraint would be the following FST: [/0 N/0