Optimality Theory (frequently abbreviated OT; second word normally capitalized by convention) is a linguistic model proposing that the observed forms of language arise from the interaction between conflicting constraints. OT differs from other approaches to phonological analysis, such as autosegmental phonology and linear phonology (SPE), which typically use rules rather than constraints. OT models grammars as systems that provide mappings from inputs to outputs; typically, the inputs are conceived of as underlying representations, and the outputs as their surface realizations.
There are three basic components of the theory:
- GEN takes an input, and generates the list of possible outputs, or candidates,
- CON provides the criteria, in the form of strictly ordered violable constraints, used to decide between candidates, and
- EVAL chooses the optimal candidate based on the constraints, and this candidate is the output.
Optimality theory assumes that these components are universal. Differences in grammars reflect different rankings of the universal constraint set, CON. Part of language acquisition can then be described as the process of adjusting the ranking of these constraints.
Optimality theory was originally proposed by the linguists Alan Prince and Paul Smolensky in 1993, and later expanded by Prince and John J. McCarthy. Although much of the interest in optimality theory has been associated with its use in phonology, the area to which optimality theory was first applied, the theory is also applicable to other subfields of linguistics (e.g. syntax and semantics).
Optimality theory also has roots in neural network research. It arose in part as an alternative to the connectionist theory of Harmonic Grammar, developed in 1990 by Géraldine Legendre, Yoshiro Miyata and Paul Smolensky. Variants of Optimality theory with connectionist-like weighted constraints continue to be pursued in more recent work (Pater 2009).
Input and GEN: the candidate set
Optimality theory supposes that there are no language-specific restrictions on the input. This is called richness of the base. Every grammar can handle every possible input. For example, a language without complex clusters must be able to deal with an input such as /flask/. Languages without complex clusters differ on how they will resolve this problem; some will epenthesize (e.g. [falasak], or [falasaka] if all codas are banned) and some will delete (e.g. [fas], [fak], [las], [lak]). Given any input, GEN generates an infinite number of candidates, or possible realizations of that input. A language's grammar (its ranking of constraints) determines which of the infinite candidates will be assessed as optimal by EVAL.
CON: the constraint set
In optimality theory, every constraint is universal. CON is the same in every language. There are two basic types of constraints. Faithfulness constraints require that the observed surface form (the output) match the underlying or lexical form (the input) in some particular way; that is, these constraints require identity between input and output forms. Markedness constraints impose requirements on the structural well-formedness of the output. Each plays a crucial role in the theory. Faithfulness constraints prevent every input from being realized as some unmarked form, and markedness constraints motivate changes from the underlying form.
The universal nature of CON makes some immediate predictions about language typology. If grammars differ only by having different rankings of CON, then the set of possible human languages is determined by the constraints that exist. Optimality theory predicts that there cannot be more grammars than there are permutations of the ranking of CON. The number of possible rankings is equal to the factorial of the total number of constraints, thus giving rise to the term Factorial Typology. However, it may not be possible to distinguish all of these potential grammars, since not every constraint is guaranteed to have an observable effect in every language. Two languages could generate the same range of input-output mappings, but differ in the relative ranking of two constraints which do not conflict with each other. If rankings with ties are allowed, then the number of possibilities is an ordered Bell number rather than a factorial, allowing a significantly larger number of possibilities.
EVAL: definition of optimality
Given two candidates, A and B, A is better than B on a constraint if A incurs fewer violations than B. Candidate A is better than B on an entire constraint hierarchy if A incurs fewer violations of the highest-ranked constraint distinguishing A and B. A is optimal in its candidate set if it is better on the constraint hierarchy than all other candidates. For example, given constraints C1, C2, and C3, where C1 dominates C2, which dominates C3 (C1 >> C2 >> C3), A is optimal if it does better than B on the highest ranking constraint which assigns them a different number of violations. If A and B tie on C1, but A does better than B on C2, A is optimal, even if A has 100 more violations of C3 than B. This comparison is often illustrated with a tableau. The pointing finger marks the optimal candidate, and each cell displays an asterisk for each violation for a given candidate and constraint. Once a candidate does worse than another candidate on the highest ranking constraint distinguishing them, it incurs a crucial violation (marked in the tableau by an exclamation mark). Once a candidate incurs a crucial violation, there is no way for it to be optimal, even if it outperforms the other candidates on the rest of CON.
Constraints are ranked in a hierarchy of strict domination. The strictness of strict domination means that a candidate who violates only a high-ranked constraint does worse on the hierarchy than one that doesn't, even if the second candidate fared worse on every other lower-ranked constraint. This also means that constraints are violable; the winning candidate need not satisfy all constraints. Within a language, a constraint may be ranked high enough that it is always obeyed; it may be ranked low enough that it has no observable effects; or, it may have some intermediate ranking. The term the emergence of the unmarked describes situations in which a markedness constraint has an intermediate ranking, so that it is violated in some forms, but nonetheless has observable effects when higher-ranked constraints are irrelevant.
An early example proposed by McCarthy & Prince (1994) is the constraint NoCoda, which prohibits syllables from ending in consonants. In Balangao, NoCoda is not ranked high enough to be always obeyed, as witnessed in roots like
taynan (faithfulness to the input prevents deletion of the final /n/). But, in the reduplicated form
ma-tayna-taynan 'repeatedly be left behind', the final /n/ is not copied. Under McCarthy & Prince's analysis, this is because faithfulness to the input does not apply to reduplicated material, and NoCoda is thus free to prefer
ma-tayna-taynan over hypothetical
ma-taynan-taynan (which has an additional violation of NoCoda). Constraints are also violable; the winning candidate need not satisfy all constraints, as long as for any rival candidate that does better than the winner on some constraint, there is a higher ranked constraint on which the winner does better than that rival.
Some optimality theorists prefer the use of comparative tableaux, as described in Prince (2002). Comparative tableaux display the same information as the classic or "flyspeck" tableaux, but the information is presented in such a way that it highlights the most crucial information. For instance, the tableau above would be rendered in the following way.
|A ~ B||e||W||L|
Each row in a comparative tableau represents a winner-loser pair, rather than an individual candidate. In the cells where the constraints assess the winner-loser pairs, there is a W if the constraint in that column prefers the winner, an L if the constraint prefers the loser, and an e if the constraint does not differentiate between the pair. Presenting the data in this way makes it easier to make generalizations. For instance, in order to have a consistent ranking some W must dominate all L's. Brasoveanu and Prince (2005) describe a process known as fusion and the various ways of presenting data in a comparative tableau in order to achieve the necessary and sufficient conditions for a given argument.
As a simplified example, consider the manifestation of the English plural:
/ˈkæt/ + /z/ → [ˈkæts] (cats)(also smirks, hits, crepes)
/ˈdɒɡ/ + /z/ → [ˈdɒɡz] (dogs)(also wugs, clubs, moms)
/ˈdɪʃ/ + /z/ → [ˈdɪʃɨz] (dishes)(also classes, glasses, bushes)
Also consider the following constraint set, in descending order of domination (M: markedness, F: faithfulness):
M: *SS - Sibilant-Sibilant clusters are ungrammatical: one violation for every pair of adjacent sibilants in the output.
M: Agree(Voi) - Agree in specification of [voi]: one violation for every pair of adjacent obstruents in the output which disagree in voicing.
F: Max - Maximize all input segments in the output: one violation for each segment in the input that doesn't appear in the output (This constraint prevents deletion).
F: Dep - Output segments are dependent on having an input correspondent: one violation for each segment in the output that doesn't appear in the input (This constraint prevents insertion).
F: Ident(Voi) - Maintain the identity of the [voi] specification: one violation for each segment that differs in voicing between the input and output.
|dish + z||*SS||Agree||Max||Dep||Ident|
|dog + z||*SS||Agree||Max||Dep||Ident|
|cat + z||*SS||Agree||Max||Dep||Ident|
No matter how the constraints are re-ordered, the 'is' allomorph will always lose to 'iz'. This is called harmonic bounding. The violations incurred by the candidate 'dogiz' are a subset of the violations incurred by 'dogis'; specifically, if you epenthesize a vowel, changing the voicing of the morpheme is gratuitous violation of constraints. In the 'dog + z' tableau, there is a candidate 'dogz' which incurs no violations whatsoever. Within the constraint set of the problem, 'dogz' harmonically bounds all other possible candidates. This shows that a candidate does not need to be a winner in order to harmonically bound another candidate.
The tableaux from above are repeated below using the comparative tableaux format.
|dog + z||*SS||Agree||Max||Dep||Ident|
|dogz ~ dogiz||e||e||e||W||e|
|dogz ~ dogis||e||e||e||W||W|
|dogz ~ dog||e||e||W||e||e|
|dogz ~ dogs||e||W||e||e||W|
From the above tableau for dog + z, it can be observed that any ranking of these constraints will produce the observed output dogz. Because there are no loser-preferring comparisons, dogz wins under any ranking of these constraints; this means that no ranking can be established on the basis of this input.
|cat + z||*SS||Agree||Max||Dep||Ident|
|cats ~ catiz||e||e||e||W||L|
|cats ~ catis||e||e||e||W||e|
|cats ~ catz||e||W||e||e||L|
|cats ~ cat||e||e||W||e||L|
The tableau for cat + z contains rows with a single W and a single L. This shows that Agree, Max, and Dep must all dominate Ident; however, no ranking can be established between those constraints on the basis of this input. Based on this tableau, the following ranking has been established:
- Agree, Max, Dep >> Ident
|dish + z||*SS||Agree||Max||Dep||Ident|
|dishiz ~ dishis||e||e||e||e||W|
|dishiz ~ dishz||W||W||e||L||e|
|dishiz ~ dish||e||e||W||L||e|
|dishiz ~ dishs||W||e||e||L||W|
This tableau shows that several more rankings are necessary in order to predict the desired outcome. The first row says nothing; there is no loser-preferring comparison in the first row. The second row reveals that either *SS or Agree must dominate Dep, based on the comparison between dishiz and dishz. The third row shows that Max must dominate Dep. The final row shows that either *SS or Ident must dominate Dep. From the cat + z tableau, it was established that Dep dominates Ident; this means that *SS must dominate Dep.
So far, the following rankings have been shown to be necessary:
- *SS, Max >> Dep >> Ident
While it is possible that Agree can dominate Dep, it is not necessary; the ranking given above is sufficient for the observed for fishiz to emerge.
When the rankings from the tableaux are combined, the following ranking summary can be given:
- *SS, Max >> Agree, Dep >> Ident
- *SS, Max, Agree >> Dep >> Ident
There are two possible places to put Agree when writing out rankings linearly; neither is truly accurate. The first implies that *SS and Max must dominate Agree, and the second implies that Agree must dominate Dep. Neither of these are truthful, which is a failing of writing out rankings in a linear fashion like this. These sorts of problems are the reason why most linguists utilize a lattice graph to represent necessary and sufficient rankings, as shown below.
A diagram that represents necessary rankings of constraints in this style is a Hasse diagram.
It is claimed that optimality theory cannot account for phonological opacity (see Idsardi 2000, e.g.). In derivational phonology effects may be seen that are inexplicable at the surface level but which are explainable through "opaque" rule ordering; but in optimality theory, which has no intermediate levels for rules to operate on, these effects are difficult to explain.
For example, in Québécois French high front vowels triggered affrication of /t/, (e.g. /tipik/ → [tˢpɪk]) but the loss of high vowels (visible at the surface level) leaves the affrication with no apparent source. Derivational phonology can explain this by saying that vowel syncope (the loss of the vowel) "counterbled" affrication - that is, instead of vowel syncope occurring and "bleeding" (i.e. preventing) affrication, we say that affrication applies before vowel syncope, so that the high vowel is removed and the environment destroyed which had triggered affrication. Such counterbleeding rule orderings are therefore termed opaque (instead of transparent), because their effects are not visible at the surface level.
The opacity of such phenomena finds no straightforward explanation in optimality theory, since intermediate forms are not accessible (constraints refer only to the surface form and/or the underlying form). There have however been a number of proposals designed to account for it; but most of these proposals significantly alter optimality theory's basic architecture, and therefore tend to be highly controversial. Frequently, such alterations add new types of constraints (which are not universal faithfulness or markedness constraints), or change the properties of GEN (such as allowing for serial derivations) or EVAL. Some well-known examples of these include John J. McCarthy's sympathy theory and candidate chains theory, and there are many others.
A relevant issue is the existence of circular chain shifts, i.e. cases where input /X/ maps to output [Y], but input /Y/ maps to output [X]. Many versions of optimality theory predict this to be impossible (see Moreton 2004, Prince 2007).
Optimality theory is also criticized as being an impossible model of speech production/perception: computing and comparing an infinite number of possible candidates would take an infinitely long time to process. Idsardi (2006) argues this position, though other linguists dispute this claim on the grounds that Idsardi makes unreasonable assumptions about the constraint set and candidates, and that more moderate instantiations of optimality theory do not present such big computational problems (see Kornai (2006) and Heinz, Kobele, and Riggle (2009)). Another common rebuttal to this criticism of optimality theory is that the framework is purely representational. In this view, optimality theory is taken to be a model of linguistic competence and is therefore not intended to explain the specifics of linguistic performance.
Another objection to optimality theory is the claim that it is not technically a theory, in that it does not make falsifiable predictions. The source of this issue is terminology: the term "theory" is used differently here than in physics, chemistry, and other sciences. Specific instantiations of optimality theory may make falsifiable predictions, in the same way that specific proposals within other linguistic frameworks can. What predictions are made, and whether they are testable, depends on the specifics of individual proposals (most commonly, this is a matter of the definitions of the constraints used in an analysis). Thus, optimality theory as a framework is best described as a scientific paradigm.
Some criticisms of optimality theory appear to be based on a misunderstanding of how it works. A well-known example of this is Noam Chomsky's widely-repeated assertion that optimality theory would predict every lexical input to be reduced to a single optimal syllable (e.g. every word is realized as [ba]). In fact, under the premises of optimality theory universal neutralization of this type would only be predicted if there were no faithfulness constraints (see McCarthy 1997). In a sense, the diametrically opposite kind of criticism comes from Morris Halle: "... the existence of phonology in every language shows that Faithfulness is at best an ineffective principle that might well be done without." By "phonology," Halle clearly means disparity between inputs and outputs of a phonological system. Optimality theory would fail to predict such deviations from an underlying form only if there were no markedness constraints. In OT, input-output disparity is normally understood as the result of markedness constraints being ranked over faithfulness constraints (M >> F).
Theories within optimality theory
In practice, implementations of optimality theory often assume other related theories, such as Syllable theory, Moraic theory, or Feature Geometry. Completely distinct from these, there are sub-theories which have been proposed entirely within optimality theory, such as positional faithfulness theory, correspondence theory (McCarthy & Prince 1995), sympathy theory, and a number of theories of learnability, most notably by Bruce Tesar. There are also a range of theories specifically about optimality theory. These are concerned with issues like the possible formulations of constraints, and constraint interactions other than strict domination.
Use outside of phonology
Optimality Theory is most commonly associated with the field of phonology, but has also been applied to other areas of linguistics. Jane Grimshaw, Geraldine Legendre and Joan Bresnan are well known for developing instantiations of OT within syntax. Optimality theoretic approaches are also relatively prominent in morphology (and the morphology-phonology interface in particular).
- Ellison, T. Mark; Klein, Ewan (2001), Review: The Best of All Possible Words (review of Optimality Theory: An Overview, Archangeli, Diana & Langendoen, D. Terence, eds., Blackwell, 1997), Journal of Linguistics 37 (1): 127–143, JSTOR 4176645.
- Chomsky (1995)
- Dresher (1996)
- Hale & Reiss (2008)
- Halle (1995)
- Idsardi (2000)
- Idsardi (2006)
- Kager, René (1999). Optimality theory. Section 1.4.4: Fear of infinity, pp. 25–27.
- Prince, Alan and Paul Smolensky. (2004): Optimality Theory: Constraint Interaction in Generative Grammar. Section 10.1.1: Fear of Optimization, pp. 215–217.
- de Lacy (editor). (2007). The Cambridge Handbook of Phonology, p. 1.
- Chomsky (1995)
- Halle (1995)
- Prince (2007), pp. 22-24
- McCarthy, John (2001). A Thematic Guide to Optimality Theory, Chapter 4: Connections of Optimality Theory.
- Legendre, Grimshaw & Vikner (2001)
- Trommer (2001)
- Wolf (2008)
- Brasoveanu, Adrian, and Alan Prince (2005). Ranking & Necessity. ROA-794.
- Chomsky (1995). The Minimalist Program. Cambridge, MA: The MIT Press.
- Dresher, Bezalel Elan (1996): The Rise of Optimality Theory in First Century Palestine. GLOT International 2, 1/2, January/February 1996, page 8 (a humorous introduction for novices)
- Hale, Mark, and Charles Reiss (2008). The Phonological Enterprise. Oxford University Press.
- Halle, Morris (1995). Feature Geometry and Feature Spreading. Linguistic Inquiry 26, 1-46.
- Heinz, Jeffrey, Greg Kobele, and Jason Riggle (2009). Evaluating the Complexity of Optimality Theory. Linguistic Inquiry 40, 277-288.
- Idsardi, William J. (2006). A Simple Proof that Optimality Theory is Computationally Intractable. Linguistic Inquiry 37:271-275.
- Idsardi, William J. (2000). Clarifying opacity. The Linguistic Review 17:377-50.
- Kager, René (1999). Optimality Theory. Cambridge: Cambridge University Press.
- Kornai, Andras (2006). Is OT NP-hard?. ROA-838.
- Legendre, Géraldine, Jane Grimshaw and Sten Vikner. (2001). Optimality-theoretic syntax. MIT Press.
- McCarthy, John (2007). Hidden Generalizations: Phonological Opacity in Optimality Theory. London: Equinox.
- McCarthy, John (2001). A Thematic Guide to Optimality Theory. Cambridge: Cambridge University Press.
- McCarthy, John and Alan Prince (1993): Prosodic Morphology: Constraint Interaction and Satisfaction. Rutgers University Center for Cognitive Science Technical Report 3.
- McCarthy, John and Alan Prince (1994): The Emergence of the Unmarked: Optimality in Prosodic Morphology. Proceedings of NELS.
- McCarthy, John J. & Alan Prince. (1995). Faithfulness and reduplicative identity. In J. Beckman, L. W. Dickey, & S. Urbanczyk (Eds.), University of Massachusetts occasional papers in linguistics (Vol. 18, pp. 249–384). Amherst, MA: GLSA Publications.
- Moreton, Elliott (2004): Non-computable Functions in Optimality Theory. Ms. from 1999, published 2004 in John J. McCarthy (ed.), Optimality Theory in Phonology.
- Pater, Joe. (2009). Weighted Constraints in Generative Linguistics. "Cognitive Science" 33, 999-1035.
- Prince, Alan (2007). The Pursuit of Theory. In Paul de Lacy, ed., Cambridge Handbook of Phonology.
- Prince, Alan (2002). Entailed Ranking Arguments. ROA-500.
- Prince, Alan (2002). Arguing Optimality. In Coetzee, Andries, Angela Carpenter and Paul de Lacy (eds). Papers in Optimality Theory II. GLSA, UMass. Amherst. ROA-536.
- Prince, Alan and Paul Smolensky. (1993/2002/2004): Optimality Theory: Constraint Interaction in Generative Grammar. Blackwell Publishers (2004) (2002). Technical Report, Rutgers University Center for Cognitive Science and Computer Science Department, University of Colorado at Boulder (1993).
- Trommer, Jochen. (2001). Distributed Optimality. PhD dissertation, Universität Potsdam.
- Wolf, Matthew. (2008). Optimal Interleaving: Serial Phonology-Morphology Interaction in a Constraint-Based Model. PhD dissertation, University of Massachusetts. ROA-996.