Optimality Theory (OT) is an approach to
linguistics that derives
output forms from
inputs on the assumption that there are many competing
constraints on possible output forms. Depending on how the constraints are ranked, outputs will be
faithful to some features of the input, or change them to more natural values.
There is no reason in principle why this approach shouldn't be applied to fields other than linguistics. Within the discipline it is now the dominant approach in phonology, and some linguists have worked on optimality-theoretic accounts of syntax.
The approach contrasts with more traditional rule-based approaches. Given an input form, you would successively apply rules to it. Any number of rules might apply, and it might make a big difference which order they were applied in. Later, some constraints were introduced into rule-based accounts, but they acted as filters: a rule-generated output was compared to the filter criterion and rejected if it failed it.
The fundamental ideas of Optimality Theory are firstly that there are no rules, everything being done by constraints; and secondly that all of the constraints are allowed to be violated. It is not violation as such that fails a candidate output, but ranking. If one constraint is more important, more highly ranked, than another, then anything that violates that one is rejected, and the remaining candidate may be acceptable even if it goes on to violate a lower-ranked constraint. A third idea is that the set of constraints is part of Universal Grammar: all languages have the same ones, they just order them differently.
Phonology is the study of the relationship between the sounds actually made (phonetics) and the structure of the words using them. For example, the German verb binden means 'to bind', and the past tense is pronounced [bant] in the singular and [bandn] in the plural. The alternation between [t] and [d] needs to be explained. The simplest explanation is that the underlying form has a [d] and something causes it to become [t] at the end of a word. A rule can do this easily. Optimality Theory has to propose what seems like a lot of set-up just to explain the same thing.
The constraints in Optimality Theory are of two kinds, faithfulness constraints and markedness constraints. There are three fundamental faithfulness constraints, visible in all languages: don't delete anything, don't add anything, and don't change anything. Numerous others need to be introduced for special purposes in some languages: don't swap the order of anything, don't change how syllables are aligned with the edges of words, etc. Faithfulness constraints keep the output close to the input.
The other kind is markedness constraints. A feature (such as a sound or a pattern) is more or less marked: a marked one is in some way less natural. Languages generally prefer some sounds more than others, some syllable shapes more than others; and some sounds work in some positions better than others. These preferences are believed to be universal. Unmarked features are more common in languages across the world than marked ones.
In the German example the marked feature is the voiced coda. The coda is the consonant at the end of a syllable: languages generally prefer voiceless [t] in this position. So we postulate a constraint called *VOICEDCODA. The * means it's bad: "no voiced coda". Working against this is the fact that we want the underlying form to contain [d], because it's a [d] in all the other forms of the verb, so it's simplest to say it's a [d] everywhere. There is a constraint NOCHANGE which says don't change the [d] to anything else.
If German had the ranking *VOICEDCODA >> NOCHANGE then anything that violated *VOICEDCODA, such as [band], is ruled out. The surviving possibility [bant] passes the higher constraint, so it has won, even though it violates the lower constraint NOCHANGE. The lower violation doesn't matter so much. On the other hand if German had the opposite ranking order then the changed form [bant] would lose to the higher constraint, and the form [band] would win even though it had a voiced coda. This is not what happens in German, so this is the wrong ranking.
Optimality Theory doesn't specify any particular set of candidates. The generator allows for anything. It is the task of the evaluation by the constraints to weed out all the unacceptable ones. Other possibilities generated would be deleting the voiced coda completely, giving [ban], and adding another vowel so that the offending [d] is now at the beginning of a syllable, no longer a coda. These two candidates violate NODELETION and NOINSERTION respectively. As German doesn't do these, the constraints against them must be high-ranked.
The results of numerous candidates evaluated against various possible rankings of constraints are depicted in two-dimensional tableaux, which I'm not going to try to reproduce here. Some of the terminology is rather painfully unobvious: the three faithfulness constraints, for example, are actually called MAXIO (meaning don't delete), DEPIO (don't insert), and IDENTIO (don't change the input).
Optimality Theory originated at Rutgers University, and the world on-line resource centre for OT papers is their archive at http://roa.rutgers.edu