mathematics

Optimality theory

Optimality theory is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Optimality theory rather than just read about it. In short: Optimality theory (frequently abbreviated OT) is a linguistic model proposing that the observed forms of language arise from the optimal satisfaction of conflicting constraints. OT differs from other approaches to phonological analysis, which typically use rules rather than constraints.

Key takeaways

Optimality theory belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Optimality theory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Optimality theory from memory before moving on to harder problems.

Reference excerpt

Optimality theory (frequently abbreviated OT) is a linguistic model proposing that the observed forms of language arise from the optimal satisfaction of conflicting constraints. OT differs from other approaches to phonological analysis, which typically use rules rather than constraints. However, phonological models of representation, such as autosegmental phonology, prosodic phonology, and linear phonology (SPE), are equally compatible with rule-based and constraint-based models. OT views 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. It is an approach within the larger framework of generative grammar. Optimality theory has its origin in a talk given by Alan Prince and Paul Smolensky in 1991 which was later developed in a book manuscript by the same authors in 1993.

Overview There are three basic components of the theory:

Generator (Gen) takes an input, and generates the list of possible outputs, or candidates, Constraint component (Con) provides the criteria, in the form of strictly ranked violable constraints, used to decide between candidates, and Evaluator (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 as applied to language was originally proposed by the linguists Alan Prince and Paul Smolensky in 1991, and later expanded by Prince and John J. McCarthy. Although much of the interest in OT has been associated with its use in phonology, the area to which OT was first applied, the theory is also applicable to other subfields of linguistics (e.g. syntax and semantics). Optimality theory is like other theories of generative grammar in its focus on the investigation of universal principles, linguistic typology and language acquisition. 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 OT 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]). Gen is free to generate any number of output candidates, however much they deviate from the input. This is called "freedom of analysis". The grammar (ranking of constraints) of the language determines which of the 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. Markedness constraints motivate changes from the underlying form, and faithfulness constraints prevent every input from being realized as some completely unmarked form (such as [ba]). 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 total orders on the constraints of Con 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. Since there is no way to distinguish these two rankings they are said to belong to the same grammar. A grammar in OT is equivalent to an antimatroid. 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.

Faithfulness constraints McCarthy and Prince (1995) propose three basic families of faithfulness constraints:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Optimality theory

Start with the simplest possible case. Write down what Optimality theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Optimality theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Optimality theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Optimality theory

In research: Optimality theory appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Optimality theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Optimality theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linguistics, Mathematical linguistics, Optimality Theory, so understanding it makes those chapters shorter.
In everyday life: Look for Optimality theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Optimality theory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Optimality theory means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Optimality theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Optimality theory in simple terms?

Why does Optimality theory matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Optimality theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Optimality theory.