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<Paper uid="J94-3001">
  <Title>Regular Models of Phonological Rule Systems</Title>
  <Section position="7" start_page="375" end_page="376" type="concl">
    <SectionTitle>
8. Conclusion
</SectionTitle>
    <Paragraph position="0"> Our aim in this paper has been to provide the core of a mathematical framework for phonology. We used systems of rewriting rules, particularly as formulated in SPE, to give concreteness to our work and to the paper. However, we continually sought solutions in terms of algebraic abstractions of sufficiently high level to free them from any necessary attachment to that or any other specific theory. If our approach proves useful, it will only be because it is broad enough to encompass new theories and new variations on old ones. If we have chosen our abstractions well, our techniques will extend smoothly and incrementally to new formal systems. Our discussion of two-level rule systems illustrates how we expect such extensions to unfold. These techniques may even extend to phonological systems that make use of matched pairs of brackets. Clearly, context-free mechanisms are sufficient to enforce dependencies between corresponding brackets, but further research may show that accurate phonological description does not exploit the power needed to maintain the balance between particular pairs, and thus that only regular devices are required for the analysis and interpretation of such systems.</Paragraph>
    <Paragraph position="1"> An important goal for us was to establish a solid basis for computation in the domain of phonological and orthographic systems. With that in mind, we developed a well-engineered computer implementation of the calculus of regular languages and relations, and this has made possible the construction of practical language processing systems. The common data structures that our programs manipulate are clearly states, transitions, labels, and label pairs--the building blocks of finite automata and transducers. But many of our initial mistakes and failures arose from attempting also to think in terms of these objects. The automata required to implement even the simplest examples are large and involve considerable subtlety for their construction. To view them from the perspective of states and transitions is much like predicting weather patterns by studying the movements of atoms and molecules or inverting a matrix with a Turing machine. The only hope of success in this domain lies in developing an appropriate set of high-level algebraic operators for reasoning about languages and relations and for justifying a corresponding set of operators and automata for computation.</Paragraph>
    <Paragraph position="2"> From a practical point of view, the result of the work reported here has been a set of powerful and sometimes quite complex tools for compiling phonological grammars in a variety of formalisms into a single representation, namely a finite-state transducer.</Paragraph>
    <Paragraph position="3"> This representation has a number of remarkable advantages: (1) The program required to interpret this representation is simple almost to the point of triviality, no matter how intricate the original grammars might have been. (2) That same program can be used to generate surface or textual forms from underlying lexical representations or  Ronald M. Kaplan and Martin Kay Regular Models of Phonological Rule Systems to analyze text into a lexical string; the only difference is in which of the two symbols on a transition is regarded as the input and which the output. (3) The interpreter is constant even under radical changes in the theory and the formalism that informed the compiler. (4) The compiler consists almost entirely of an implementation of the basic calculus. Given the operators and data types that this makes available, only a very few lines of code make up the compiler for a particular theory.</Paragraph>
    <Paragraph position="4"> Reflecting on the way the relation for a rewriting rule is constructed from simpler relations, and on how these are composed to create a single relation for a complete grammar, we come naturally to a consideration of how that relation should comport with the other parts of a larger language-processing system. We can show, for example, that the result of combining together a list of items that have exceptional phonological behavior with a grammar-derived relation for general patterns is still a regular relation with an associated transducer. If E is a relation for a finite list of exceptional input-output pairs and P is the general phonological relation, then the combination is given by</Paragraph>
    <Paragraph position="6"> This relation is regular because E is regular (as is any finite list of pairs); it suppresses the general mapping provided by P for the exceptional items, allowing outputs for them to come from E only. As another example, the finite list of formatives in a lexicon L can be combined with a regular phonology (perhaps with exceptions already folded in) by means of the composition Id(L) o P. This relation enshrines not only the phonological regularities of the language but its lexical inventory as well, and its corresponding transducer would perform phonological recognition and lexical lookup in a single sequence of transitions. This is the sort of arrangement that Karttunen et al.</Paragraph>
    <Paragraph position="7"> (1992) discuss. Finally, we know that many language classes are closed under finite-state transductions or composition with regular relations--the images of context-free languages, for example, are context-free. It might therefore prove advantageous to seek ways of composing phonology and syntax to produce a new system with the same formal properties as syntax alone.</Paragraph>
  </Section>
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