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Christian List
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"Optimality Theory and the Problem of Constraint Aggregation", with Daniel Harbour, forthcoming in Natural Language Semantics and Philosophy, MIT Working Papers in Linguistics and Philosophy, 1, 2000
Abstract
1. The Issue
Optimality Theory (Prince & Smolensky, 1993) claims that "Universal Grammar consists largely of a set of constraints on representational well-formedness" and that grammaticality is a matter of structural well-formedness. As constraints make "sharply conflicting claims about the well-formedness of most representations", grammars require a means of resolving such conflicts in order to determine a given input’s "surface representation", the analysis "most harmonic" with the well-formedness constraints. Prince & Smolensky claim that conflicts are resolved by "rank[ing] constraints in a strict dominance hierarchy", a position adopted in most optimality-theoretic research. This paper questions that assumption and presents a mathematical answer to the question of whether conflict resolution can be theoretically modelled in other ways. Crucial to this enterprise are two realisations: that strict dominance hierarchies are only one of many different ways to construct a constraint-based phonology; and that the changes in phonological theory that gave increasing prominence to constraints in no way commit one to the use of strict dominance hierarchies. That is, Optimality Theory is only one of many possible constraint-based phonologies.
We pay close attention to the nature of constraint aggregation (to be defined), an issue largely ignored in optimality-theoretic research but one which, we argue, is essential to the enterprise of devising explanatorily adequate constraint-based grammars. Our discussion of aggregation has several interesting consequences. It allows us to ask whether problems such as opacity are merely the result of assumptions particular to Optimality Theory, or whether they would be problems for any constraint-based theory. It also facilitates rigorous comparison of constraint-based theories and leads to empirical insights.
2. The Methodology
A rigorous analysis of the nature of constraint conflict resolution requires a certain amount of formalism. The formal methods of the paper are drawn from social choice theory, a subdiscipline of mathematical decision theory. In particular, we exploit the formal similarity between the problems that social choice theory and Optimality Theory address. The nature of these parallels is discussed immediately below. Crucially, we formalise three different dimensions of the problem: measurability, comparability, and aggregation.
We aim to show that the problems Optimality Theory and social choice theory address are formally similar. This motivates use of the methods of the latter in constraint-based phonology. In particular, it enables us to examine the important issue of constraint aggregation.
Committee decisions or elections exemplify social-choice-theoretic problems. The ‘head of the committee’ must determine, for every voter-candidate pair, a measure (number) of that voter’s (dis)preference for that candidate (option), and, on the basis of all voter-candidate measurements, must rank the candidates from ‘best’ to ‘worst’. To do this, the committee head must answer three types of questions. Questions of measurability concern how much significance we attach to the numbers that measure (dis)preference. Are they significant on some absolute scale? Are only their ratios or differences significant? Is the way in which they rank-order the different policy options all that is meaningful? Questions of interpersonal comparability concern whether one can compare the numbers representing different voter’s preferences for the same or different candidate options. Is it meaningful to ask whether, say, Candidate x is more strongly (dis)preferred by Voter 1 than Candidate y is by Voter 2? Questions of aggregation concern how the candidates’ ranking is determined given all voter-candidate measurements. Are all voters equal, or do some have priority?
To see the relevance of social-choice-theoretic methods to the problem of conflict resolution, recall how Optimality Theory determines the output for any given input. The input is fed into the generator (Gen), which produces the set of possible outputs for that input. These are the candidates in the social-choice-theoretic problem. Then the degree to which each candidate violates each constraint is measured. The next step is the determination of the optimal candidate on the basis of the measured information. An aggregation function takes measures (‘scores’) of constraint violation as its input and generates rankings of candidates as its output (selecting a unique winner is mathematically equivalent). Clearly, we can ask the same questions here as in social choice theory. Optimality Theory already crucially involves measurement and aggregation, so these issues are familiar, but issues of comparability are less familiar. However, we argue that appropriate comparability assumptions appear necessary for dissolving several cases of opacity.
Kenneth Arrow’s famous impossibility theorem (1951) implies that the nature of aggregation is strongly restricted by measurability and comparability assumptions. We argue for three assumptions, based on Arrow’s social-choice-theoretic axioms, concerning the nature of aggregation. Informally paraphrased, they are:
- Universal Domain: The aggregation function is defined for all logically possible inputs (approximately equivalently: for all logically possible tableaux).
- Strong Pareto Principle: Let x and y be candidate outputs. If x violates no constraint more than y does, then y cannot be higher ranked (more "globally harmonic") than x. And if there is a constraint that x violates less than y does, then x is higher ranked than y. (In this case, x is Pareto-superior to y.)
Under varying measurability and comparability assumptions, we state ‘impossibility’ theorems concerning the nature of aggregation in different constraint-based grammars. In particular, we show that Optimality Theory makes the most minimal assumptions concerning the measurability and comparability. This minimality underlines one of the advantages of considering alternative aggregation procedures: we can ask rigorously whether any of Optimality Theory’s problems, such as opacity, are dissolved by changing measurability and comparability assumptions.
3. Applications and Advantages of Different Aggregation Procedures: Some
Examples
The applications and advantages of the formalism outlined above range from empirical to theoretical matters in phonology and even extend to the nature of economy of derivation in minimalist syntax (Chomsky, 1995).
Not all work in constraint-based phonology assumes strict dominance hierarchies. Alternatives include weighted combinations of constraints (Flemming, 1997; Burzio, 1999), constraint co-ranking (Itô & Mester, 1997) and constraint conjunction (Smolensky, 1995; Kirchner, 1996; Crowhurst & Hewitt, 1999). We show that in all these cases the assumed nature of measurability and comparability is really at stake. Moreover, we solve an empirical problem with Itô & Mester’s account by making these underlying assumptions explicit.
On the basis of the Strong Pareto Principle, we distinguish two types of opacity. Pareto opacity, where the selected candidate is Pareto-superior to the real output, is, we argue, unlikely to arise and would present grave problems for most constraint-based phonologies, including Optimality Theory. NonPareto opacity, where the selected candidate performs worse than the real output on at least one constraint, can in some instances be solved by modifying measurability and cross-constraint comparability assumptions (e.g.: Kirchner’s, 1996). We underline the usefulness of the concept of Pareto superiority by showing that it can often be used to solve cases of opacity (e.g.: glottal stop deletion in Tiberian segholates). However, some cases of NonPareto opacity are impervious to changes in assumptions. One example is Bedouin Arabic a-raising, where the set of violation scores for the optimal candidate of one selection process can be too similar (in a technical sense) to the set of violation scores for a suboptimal candidate of another selection process. This makes it impossible for any aggregation function using only violation scores to determine the right outcome in both cases.
Finally, we point out that, in order for pure constraint-based phonology to work, one must accept what we call evaluationism: the constraint violation scores of any two candidates must always contain sufficient information to rank them in an order of global harmony. We briefly explore the implications of different answers to the question of whether this information is sufficient, and we suggest that, if there is more to grammaticality than structural well-formedness, then, whatever role constraints play in phonology, it is not captured by Optimality Theory, nor can it be by any other pure, constraint-based theory.