USC Graduate Phonology ✳︎ Fall 2019 ✳︎ Smith

I say I listen to all voices but mine’s the final decision. […] I hear the voices and I read the front page and I know the speculation but I’m the decider and I decide what is best."

George W. Bush

Optimality Theory

• We’re going to be discussing what’s sometimes called Classic Optimality Theory, consisting of:
  • Optimality Theory (Prince & Smolensky 1993)
  • Correspondence Theory (McCarthy & Prince 1995), which revised faithfulness constraints.
  • Generalized Alignment (McCarthy & Prince 1993), which introduced alignment constraints.
• Keep in mind that nearly every imagineable revision to OT assumptions has been pursued at some point in the past 20+ years, and much of the research in OT has been devoted to answering the question “What are the constraints?”

An /input/→[output] derivation in OT

• Start with an input.
• Generate a set of candidates (possible outputs).
• Pick the optimal candidate, the candidate that is best according to the constraint ranking for the language.
• The optimal candidate is the output of the grammar.

Core proposals of OT

• The grammar generates mappings via a strict ranking of violable constraints.
  • Constraints penalize marked structures in SRs, e.g. “Don’t end a word with a voiced obstruent,” or deviating from URs, e.g. “Don’t devoice.”
  • Constraints are in a strict ranking (strict total order).
    • For every pair of contraints, A and B, there is an ordering between them. Either A ranks above B, or B ranks above A.
    • Ranking is transitive. If A ranks above B, and B ranks above C, then A ranks above C.
  • When two constraints have conflicting requirements, satisfaction of the higher ranked constraint takes priority (no exceptions!).
    • What happens if the constraint “Don’t end a word with a voiced obstruent” takes priority over “Don’t devoice”?
    • What happens if the constraint “Don’t devoice” takes priority over “Don’t end a word with a voiced obstruent”?
  • If constraint A is ranked above constraint B, we say that A dominates B, and can write this with a much-greater-than symbol: A ≫ B.
• Constraint interaction aims to capture all aspects of the phonological system in a language.
  • The ranking of constraints accounts for the segmental inventory of a language, its phonotactics, and its alternations.
  • A particular language has the same ranking of constraints for all parts of the language.
  • Important to note: although each language has the same ranking of constraints for every part of the language, we usually cannot discover a single strict total order of constraints.
    • It’s nearly always the case that a set of mappings can be generated by many different strict total orders of constraints.
    • When we construct a ranking for a particular language, we describe only the rankings that are necessary to generate the attested mappings, narrowing down the space of possibilities.
• The ranking of constraints is the only way languages differ from one another (except for vocabulary).
  • Constraints are universal.
  • The mechanism of candidate generation is universal.
  • The mechanism for choosing the optimal candidate is universal.
  • There are no language-particular restrictions on URs.
• Constraint re-ranking captures cross-linguistic variation and typology.
  • Every ranking of constraints is predicted to be a possible language.
  • Factorial typology: There are n! orderings of constraints (n = number of constraints), and thus n! possible rankings.
    • n! (n factorial) is the product of all positive integers less than or equal to n.
    • 5! = 5 * 4 * 3 * 2 * 1 = 120
    • Although there are 120 rankings of 5 constraints, chances are that many of these rankings will generate the same input-output mappings, so there are likely fewer than 120 different sets of input-output mappings.

The architecture of an OT grammar

• The lexicon: contains a list of URs, and other unpredictable information
• GEN: the function that generates candidates
• CON: the set of universal constraints
• A language-particular ranking of constraints from CON: sometimes written as ℋ for hierarchy
• EVAL: the function that selects the optimal candidate(s) from the candidate set, based on the ranking of constraints
• A full derivation looks like this:
  • 1. Start with an input: /ab/
  • 2. Generate the candidate set: GEN(/ab/)
  • 3. Pick the output: EVAL(input, candidates, ℋ)
  • This can be written as a single line: EVAL(/ab/, GEN(/ab/), ℋ)
• Since ranking is the only language-particular part of the grammar, GEN, CON, and EVAL are the same for every language.

The lexicon

• The lexicon contains all contrastive properties of morphemes, including phonological, morphological, syntactic, and semantic properties.
• There are no language-particular restrictions on the phonological forms of URs.
  • Constraints evaluate SRs or similarity between URs and SRs.
• The lack of restrictions on URs is a property called Richness of the Base, and means the set of logically possible URs is the same for every language.

GEN: the generator function

• The function that generates a set of candidate outputs from the input.
• One way to think of it: apply all possible operations to the input, any number of times (deletion, insertion, feature changing, maybe changing order)
  • The resulting set of candidates is infinite (even if we assume a finite alphabet of symbols).
  • The candidate set always contains a faithful candidate, an output candidate which is identical to the input.
• Candidates have an input portion (UR) and an output portion (SR), and nothing else. There are no intermediate forms as there are for SPE-style rules.
• GEN(/ab/) = {[ab], [a], [b], [ba], [ ], [ta], [at], [ae], [diii], …}
• The property that GEN can generate any conceivable output candidate for some input is called Freedom of Analysis.
  • That’s not to say GEN can generate anything. Candidates must still be composed of elements from the universal vocabularies of linguistic representation, e.g. segments, features, syllables.
  • Whatever limits there are on GEN, those limits are universal.

CON: the constraint set

• There are two kinds of constraints:
  • Markedness constraints impose requirements on output candidates.
    • *[+voice] = don’t have a voiced segment
    • Agree(voice) = adjacent segments should agree in voicing
    • Onset = Syllables should have onsets
    • NoCoda = Syllables should not have codas
  • Faithfulness constraints militate against deviating from the input.
    • Dep = don’t epenthesize
    • Max = don’t delete
    • Ident([feature]) = don’t change the value of [feature]
• A markedness constraint can be treated as a function from a candidate output to a positive integer (the number of violations).
  • Violations are a penalty score: fewer violations is better.
  • Constraints only assign violations. They never directly reward candidates.
  • *CC([bak]) = 0
  • *CC([tikpad]) = 1
• A faithfulness constraint can be treated as a function from an input-output pair to a positive integer (the number of violations):
  • Max(/bak/, [ba]) = 1
  • Max(/bak/, [bak]) = 0
• The best way to write constraints is to describe the function explicitly, starting with “Assign one violation for every…”, and give it a descriptive name (often written in small caps and/or with a *).
• Here are examples of constraints that have been proposed in the OT literature:
  • *[+voice] = Assign one violation for every segment in the output that has the feature [+voice].
  • *Onset = Assign one violation for every syllable in the output without an onset.
  • *NoCoda = Assign one violation for every syllable in the output with at least one coda consonant.
  • *Struc = Assign one violation for every segment in the output.
  • Agree(voice) = Assign one violation for every sequence of two segments in the output which do not have identical value for the [voice] feature.
  • *CC = assign one violation for every sequence of two consonants in the output..
  • Max = Assign one violation for every segment in the input that isn’t in the output.
  • Dep-V = Assign one violation for every vowel in the output that isn’t in the input.
• Let’s assess the violations of *CC:
  • *CC([bakta]) = ?
  • *CC([abtako]) = ?
  • *CC([abitaki]) = ?
  • *CC([brabtakpi]) = ?

EVAL: the evaluator function

• The function that selects the winner from the set of candidates.
  • Function arguments: input, set of output candidates, ranking of constraints 

  • Output of function: subset of the candidates that are optimal
  • EVAL(/input/, GEN(/input/), ranking) = {[output]} 

• We can think of many ways that EVAL could work…
  • Strict domination is the mechanism used in standard OT for adjudicating harmony disagreements among constraints.
  • Satisfaction of a higher-ranked constraint takes absolute priority over the satisfaction of all lower-ranked constraints.
  • A possibly helpful analogy: you can think of the constraint ranking as a hierarchy of authoritarian autocrats, each of whom defers to his superior and doesn’t care about the opinions of his underlings.
• To find just the winners, if you have n constraints…
  • Find the candidates that tie for being “best” (fewest violations) on the top-ranked constraint C₁; discard the rest.
    • Because of strict domination, none of the lower-ranked constraints matter – just the constraint currently under consideration.
  • Of the remaining candidates, find those the next constraint, C₂, deems best; discard the rest.
  • Repeat for C₃, …, C_n, stopping if you have only one candidate left.
  • As a slogan: “Take the best, ignore the rest.”¹
  • Whatever candidates are still left at the end are tied for being the winner.
  • If you have enough constraints, there is normally just one winner.
• Given how EVAL works, what a constraint must be able to do is identify the best candidate(s) out of a set of candidates.
  • Treating constraints as functions that return positive integers simplifies this calculation.
  • But we could in theory have a constraint like AlphabeticalOrder, which favors candidates that are earlier in alphabetical order. Since this constraint is able to pick the best candidates out of a set, it can get the job done.

An example

• What does output does the grammar generate for:
  • Input: /ab/
  • Ranking: *ab ≫ Max ≫ *a ≫ *b
• First, generate a candidate set: GEN(/ab/) = {[ab], [a], [b], []}
• Pick the best candidate with EVAL:
  • EVAL(/ab/, {[ab], [a], [b], []}, ranking)
  • For the remaining candidates, {[ab], [a], [b], []}
    check the first constraint: *ab ≫ Max ≫ *a ≫ *b
    • *ab: assign one violation for every [a] followed by [b]
    • *ab([ab]) = 1
    • *ab([a]) = 0
    • *ab([b]) = 0
    • *ab([]) = 0
    • Take the candidates that have the fewest violations and discard the rest.
    • Remaining candidates: {[a], [b], []}
    • Remaining constraints to check: Max ≫ *a ≫ *b
    • Stop if either:
      • There is a single remaining candidate. (Nope)
      • There are no more constraints to check. (Nope)
  • For the remaining candidates, {[a], [b], []},
    check the next constraint: *ab ≫ Max ≫ *a ≫ *b
    • Max: assign one violation for every segment in the input that isn’t in the output.
    • Max(/ab/, [a]) = 1
    • Max(/ab/, [b]) = 1
    • Max(/ab/, []) = 2
    • Take the best, ignore the rest.
    • Remaining candidates: {[a], [b]}
    • Remaining constraints to check: *a ≫ *b
    • Stop if either:
      • There is a single remaining candidate. (Nope)
      • There are no more constraints to check. (Nope)
  • For the remaining candidates, {[a], [b]},
    check the next constraint: *ab ≫ Max ≫ *a ≫ *b
    • *a: assign one violation for every [a].
    • *a([a]) = 1
    • *a([b]) = 0
    • Take the best, ignore the rest.
    • Remaining candidates: {[b]}
    • Remaining constraints to check: *b
    • Stop if either:
      • There is a single remaining candidate. (Yes! Stop!)
      • There are no more constraints to check. (Nope)
• Output: [b]
• If we repeat this for different inputs, we get a set of mappings generated by the ranking.
  • Mappings generated by *ab ≫ Max ≫ *a ≫ *b:
    • /ab/ → [b]
    • /b/ → [b]
    • /a/ → [a]
    • /ba/ → [ba]
    • etc.
  • As mentioned a few times above, different rankings will often produce the same mappings.
• We considered only four candidates from the candidate set of /ab/. Is there a candidate for /ab/ that we didn’t consider, which could’ve beaten [b]?

The OT tableau

• The tableau (from French, pronounced [ˌtʰæˈblow] in English, plural tableaux [ˌtʰæˈblow] or [ˌtʰæˈblowz]) is a way to illustrate a fragment of this calculation.
• Not all candidates can be shown; not all constraints are shown, either.
• The tableau is not a full calculation, but rather an illustrative device, like a derivation in rule-based analyses.
• This tableau shows a ranking argument for *CC ≫ Dep-V.

• Important parts of the tableau:
  • /input/ – sometimes this is in the upper lefthand corner
  • [output] candidates – which are usually lettered for reference
  • pointing finger – indicates winner
    • An equivalent rightward pointing arrow is sometimes used instead.
    • The name of the pointing finger in typography is the “index” or “manicule.”
  • constraints – the columns are in descending order of rank: higher-ranking constraints are on the left
  • asterisks – one for each constraint violation
    • Numbers are now commonly used instead
• Sometimes there are different types of lines between constraints:
  • solid line between constraints – represents a crucial ranking
  • dashed line between constraints – represents a non-crucial ranking
  • I’m not going to make this distinction in any tableaux, but you should be aware of the difference. (We’ll dicuss why dashed vs. solid lines is an unhelpful and insufficient distinction on the next handout.)
• Below is a tableau with shading and exclamation marks added.
  • exclamation marks – suffixed to the asterisk that “kills” a candidate (the fatal violation)
  • shading – shows that a cell is not relevant, either because the winner has already been chosen, or because that candidate has already been ruled out.
    • All cells to the right of an exclamation mark are shaded.
    • Cells in the winner’s row may also be shaded, if the corresponding constraints were not consulted by EVAL.
    • A constraint whose column is shaded is inactive in the candidate selection process.
  • The pointing finger, exclamation marks, and shading are predictable from the asterisks, but they are helpful to the reader.

• We can show multiple inputs in the same tableau, since all inputs in a language are subject to the same ranking of constraints.

Practice tableau

• Let’s make a tableau for the Dutch example from the last handout.
• Recall the problem:
  • Dutch inserts ʔ to avoid two adjacent vowels.
  • But ʔ is only permitted before a stressed vowel.
• Constraints for the tableau (poorly defined):
  • Max-V: don’t delete a vowel.
  • Ident(stress): don’t change a segment’s value for the feature [stress].
  • *VV: avoid two vowels in a row.
  • Dep-C: don’t insert a consonant.
  • *ʔV: don’t have a glottal stop before an unstressed vowel.
• Mappings:
  • /aɔ́rta/ → [aʔɔ́rta]
  • /xáɔs/ → [xáɔs]
• Redefine these constraints so they aren’t ambiguous, make tableaux for both inputs, and find a ranking that works.
  • Once we have better definitions for constraiants, let’s come up with a preliminary candidate set together.
  • The best approach for ranking constraints is to put the constraints into a tableau in a random order and then reason from there about the ranking.

Harmonic Bounding

• A candidate is harmonically bounded if it cannot win under any ranking of constraints.
  • This means that the grammar simply cannot generate the harmonically bounded mapping.

• You can often determine whether a candidate is harmonically bounded simply by looking at its violations.
• Harmonic bounding is dependent on the constraint set and candidate set under consideration.
  • Let’s add a third constraint (however nonsensical) that makes it so candidate c isn’t harmonically bounded.
• A more subtle type of harmonic bounding is collective harmonic bounding, in which a candidate cannot win because multiple other candidates prevent it from being the best on any constraint.
  • In the tableau below, candidate b. isn’t the worst on any constraint, but it’s never the best either.

• How many harmonically bounded candidates are there for each of the inputs above? (If we considered the entire candidate set.)

Recap of major ideas

• Main ideas of OT
  • a ranking of constraints
    • strict domination
    • the ≫ symbol
    • the same ranking is used for all inputs in a language
  • constraints are violable
  • very important idea: all systematic phonological patterns in a language come from the same constraint ranking
    • alternations
    • phonotactics
    • segmental inventory
  • very important idea: all systematic differences between languages come from differences in constraint ranking
    • factorial typology
    • rankings are language-particular
    • GEN, CON, EVAL are universal
    • Richness of the Base
• GEN
  • Freedom of Analysis
  • the faithful candidate
• CON
  • constraints as violation-counting functions
  • markedness
  • faithfulness
  • no constraints on the structure of URs
• EVAL
  • as a slogan: “take the best, ignore the rest”
  • the strictness of strict domination
  • harmonic bounding
• The OT tableau
  • ranking argument
  • crucial ranking
  • violation marks
  • the pointing finger
  • exclamation marks
  • shading
  • don’t bother with solid vs. dashed lines