Roberts/HT 2013 Week 6


Factorial Typology

Recall that the following are axioms of OT:

The constraint set (Con) is universal.
Specifying the ranking of the constraints in Con is necessary and sufficient to model the behaviour of a phonology.

One consequence of these axioms is that OT makes typological claims; if we know the composition of Con, we can answer the question “What is/is not a possible phonology?”.

The formal answer to this question is given by factorial typology (see Kager 1999: 34ff.).

Factorial, notated n! ≝ product of positive integers ≤ n: 5! = 5 × 4 × 3 × 2 × 1 = 120

The number of permutations of a set with cardinality n is equal to n!, therefore there are n! possible rankings of Con.

(|Con|)! is a hard maximum on the number of possible grammars; the actual number will be lower, since not every pair of constraints is crucially ranked.

Factorial typology is the subject of this week’s practical exercise.

Extensions of OT:

A problem: opacity

There are four possible interactions between rules that affect the same structures (terminology due to Kiparsky 1968):


Feeding: the first rule creates the structure the second rule targets.

E → D / A _CAD
A → B / C _ DCBD

Bleeding: the first rule destroys the structure the second rule targets.

D → ∅ / _#CA
A → B / C _ D

Counterfeeding: the inverse of feeding. The second rule creates the target “too late” for the first rule to affect it. The first rule underapplies.

A → B / C _ D
E → D / A _CAD

Counterbleeding: the inverse of bleeding. The second rule destroys the target “too late” for the first rule to ignore it. The first rule overapplies.

A → B / C _ DCBD
D → ∅ / _#CB

Classic OT (OT as originally formulated by Prince and Smolensky 1993), can only model the transparent interactions: feeding and bleeding.

Nevertheless, counterbleeding and counterfeeding are attested, e.g. in American English, where Pre-Fortis Clipping and (in Canada) Diphthong Raising are counterbled by [t]-Flapping.

Pre-Fortis Clipping: a vowel/diphthong preceding a voiceless (fortis) consonant is realized as shorter than one preceding a voiced (lenis) consonant. ride → [ɹɑɪd]; write → [ɹɑ̆ɪt].
Canadian Diphthong Raising: a historical development of Pre-Fortis Clipping. Diphthongs in the fortis context have higher on-glides: write → [ɹəɪt] (see Chomsky and Halle 1968: 342).
[t]-Flapping: the coronal stops /t/ and /d/ are realized as [ɾ] when between vowels in a syllable that is not foot-initial (roughly).

The interaction between these rules is opaque in that rider [ɹɑɪɾɚɹ] and writer [ɹɑ̆ɪɾɚɹ]/[ɹəɪɾɚɹ] are not homophones. Pre-Fortis Clipping or Canadian Raising applies, even though the [t] that conditions them is eliminated by Flapping.

If we try to model this interaction in Classic OT, we arrive at a harmonic bounding condition: [ɹəɪɾɚɹ] cannot possibly win.


ClipDiph: Requires the distance between elements of diphthongs occuring before voiceless obstruents to be minimal. (Bermúdez-Otero, 2003, p. 8)
ClearDiph: Requires the distance between elements of diphthongs to be maximal. (Bermúdez-Otero, 2003, p. 8)
*ˈVtV: Penalises occurrence of tV where it is preceded by a stressed vowel
Input: /ɹɑɪtɚɹ/ Clip­Diph *ˈVtV Clear­Diph Ident
a. ɹɑɪtɚɹ *! *
b. ɹəɪtɚɹ *! *
c. 💣 ɹɑɪɾɚɹ *
d. ɹəɪɾɚɹ *! **

Note that candidate c.’s violations are a proper subset of those of candidate d. Candidate d, which is the attested form, cannot win under any ranking of these constraints.

Proposed solutions

Sympathy Theory (McCarthy 1999; summary in Kager 1999:387‒392)

Sympathy theory handles opacity by enforcing similarity between pairs of output candidates.

One faithfulness constraint is used as the selector constraint (marked ✯). This constraint selects the sympathetic candidate (marked ❀), which is the most harmonic candidate that has no violations of the selector constraint.

Similarity between the other candidates and the sympathetic candidate is enforced by sympathy constraints, which follow the same Max, Dep and Ident schemas as IO-faithfulness constraints, and are also marked with ❀.

Input: /ɹɑɪtɚɹ/ Clip­Diph *ˈVtV ❀Ident-V Clear­Diph ✯Ident-C Ident-V
a. ɹɑɪtɚɹ *! * *
b. ɹəɪtɚɹ *! *
c. ɹɑɪɾɚɹ *! *
d. ɹəɪɾɚɹ * * *

Output-output correspondence

Opacity effects are more often (though not exclusively) to be found in morphologically complex environments. It has been suggested that the grammar should enforce phonological similarity between morphologically related forms (see e.g. the notion of uniform exponence in Kenstowicz 1996).

The theory of output-output correspondence in Benua (1997) formalises this notion in OT. For every morphological alternation in which one form is phonologically opaque (it is claimed), there is a transparent base form, and a family of OO-faithfulness constraints enforcing similarity to it.

Input: /ɹɑɪtɚɹ/ Clip­Diph *ˈVtV OO-Ident-V Clear­Diph Ident-C Ident-V
a. ɹəɪt ~ ɹɑɪtɚɹ *! * *
b. ɹəɪt ~ ɹəɪtɚɹ *! *
c. ɹəɪt ~ ɹɑɪɾɚɹ *! *
d. ɹəɪt ~ ɹəɪɾɚɹ * * *

But, note that the opaque interaction between Raising and Flapping is not confined to morphologically complex environments: we also have mitre [məɪɾɚɹ]!

Local Conjunction

The earliest attempt to address opacity in OT was made by Smolensky (1995): Local Conjunction is a constraint schema that builds a new constraint out of the conjunction of two others in a specific domain:


[C1 & C2]δ

A constraint of this form will be violated iff both C1 and C2 are violated by a structure within a domain δ.

Modelling the interaction between Raising and Flapping using Local Conjunction is impossible under the assumptions we’ve been working with so far.

We can model the interaction between Pre-Fortis Clipping and Flapping, however, if we assume that Pre-Fortis Clipping is actually what we might call Pre-Lenis Lengthening: that is, if rider is pronounced [ɹɑˑɪɾɚ], rather than writer being pronounced [ɹɑ̆ɪɾɚ] (where this leaves Canadian Raising is unclear).

Input: /ɹɑɪtɚɹ/ *ˈVtV [Ident-V & Ident-C]σσ LongDiph Ident-V Ident-C
a. ɹɑˑɪtɚɹ *! *
b. ɹɑɪtɚɹ *!
c. ɹɑˑɪɾɚɹ *! * *
d. ɹɑɪɾɚɹ * *
(Lee 2006)

Stratal Optimality Theory…

…is the formalism I work in, and therefore the subject of next week’s lecture.


Benua, Laura (1997) Transderivational identity: phonological relations between words. PhD thesis, University of Massachusetts at Amherst.

Chomsky, Noam and Halle, Morris (1968) The sound pattern of English. New York: Harper and Row.

Kager, René (1999) Optimality Theory. Cambridge University Press.

Kenstowicz, Michael (1996) “Base-Identity and Uniform Exponence: alternatives to cyclicity” in Durand, Jacques and Bernard Laks (eds.) Current trends in phonology: models and methods. University of Salford: ESRI.

Kiparsky, Paul (1968) “Linguistic universals and linguistic change.” in Bach, Emmon W., Robert T. Harms and Charles J. Fillmore (eds.) Universals in Linguistic Theory. London: Holt, Rinehart and Winston. pp. 170‒202.

Lee, Yongsun (2006) “OT-CC and English Opacity in Flapping.” Saehan English Language and Literature 48:313‒334.

McCarthy, John J. (1999) “Sympathy and phonological opacity.” Phonology 16:331-399.

Prince, Alan S. and Smolensky, Paul (1993) [2004] Optimality Theory: constraint interaction in generative grammar. Oxford: Blackwell.

Smolensky, Paul (1995) On the structure of the constraint component Con of UG. Handout of talk at UCLA, 4/7/95. Available from the Rutgers Optimality Archive as ROA-86.