Input-Output Mismatches in ot∗

Bidirectional Optimality Theory allows us to see a wide range of problems which would previously have been considered unrelated from a new perspective, the perspective of asymmetric relationships between input and output. For interpretation, the input is a form and the output a meaning, and for production the input is a meaning and the output is a form. A mismatch is any case where there is no isomorphism between the space of meanings and the space of forms, say because one form has no meaning, or multiple meanings, or because a meaning is inexpressible, or may be expressed in multiple ways. Is there such a thing as a perfect language, one that would lack any mismatch of this sort? Certainly there are subsystems of natural and formal languages that, if taken in isolation, would be perfectly symmetric. For example, the Arabic notation for integers (assuming that initial zeroes are ill-formed) stands in a one to one relationship with the abstract semantic space of integers. But even formal languages are commonly not perfect in this very strong sense. For example, in first order logic there may be multiple constants referring to the same individual, and more generally there are an infinite number of ways of expressing any proposition that can be expressed at all. There may also be objects in the model for which there is no corresponding constant, or facts that are true in a given model or frame and yet inexpressible in first order logic. As far as form-meaning symmetry goes, the only way that first order logic scores qualitatively over natural language is that the former is (when properly notated, and interpreted with respect to a specific model) unambiguous: for any form there is exactly one meaning. Along with ambiguity, we will be considering optionality, ineffability, uninterpretability, blocking and freezing. All of these involve a mismatch between form and meaning, and we will study how various versions of ot handle these mismatches. Initially, we will be considering simpler, relatively standard ot architectures. The first two of these are unidirectional. What we will term naive ot production is the approach seen in most ot syntax papers, and is close to the model that is used in ot phonology. To recap what we assume is already familiar to most readers of this article, naive ot production starts with some representation of meaning as input, and a set of candidate outputs provided by a function referred to as gen. A set of linearly ranked constraints is then used to select between candidate surface forms. The second unidirectional approach, not surprisingly,