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Violation Semirings in Optimality Theory

Abstract

This paper provides a brief algebraic characterization of constraint violations in Optimality Theory (OT). I show that if violations are taken to be multisets over a fixed basis set Con then the merge operator on multisets and a ‘min’ operation expressed in terms of harmonic inequality provide a semiring over violation profiles. This semiring allows standard optimization algorithms to be used for OT grammars with weighted finite-state constraints in which the weights are violation-multisets. Most usefully, because multisets are unordered, the merge operation is commutative and thus it is possible to give a single graph representation of the entire class of grammars (i.e. rankings) for a given constraint set. This allows a neat factorization of the optimization problem that isolates the main source of complexity into a single constant γ denoting the size of the graph representation of the whole constraint set. I show that the computational cost of optimization is linear in the length of the underlying form with the multiplicative constant γ. This perspective thus makes it straightforward to evaluate the complexity of optimization for different constraint sets.

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References

  • Bistarelli S., Montanari U., Rossi F. (1997) Semiring-based constraint satisfaction and optimization. Journal of the ACM 44(2): 201–236

    Article  Google Scholar 

  • Charniak, E., & Johnson, M. (2005). Coarse-to-fine-grained n-best parsing and discriminative reranking. In In Proceedings of the 43rd ACL, Ann Arbor, MI, pp. 173–180.

  • Cormen T.H., Leiserson C.E., Rivest R.L. (1990) Introduction to algorithms. MIT Press, Cambridge, MA

    Google Scholar 

  • Dijkstra E.W. (1959) A note on two problems in connexion with graphs. Numerische Mathematik 1: 269–271

    Article  Google Scholar 

  • Eisner, J. (1997). Efficient generation in primitive optimality theory. In Proceedings of the 35th Annual Meeting of the Association for Computational Linguistics (ACL), Madrid, pp. 313–320.

  • Eisner, J. (2000). Easy and hard constraint ranking in optimality theory: Algorithms and complexity. In J. Eisner, L. Karttunen, & A. Thériault (Eds.), Finite-state phonology: Proceedings of the 5th workshop of the ACL special interest group in computational phonology (SIGPHON) (pp. 22–33). Luxembourg.

  • Eisner, J. (2001). Expectation semirings: Flexible EM for finite-state transducers. In G. van Noord (Ed.), Proceedings of the ESSLLI workshop on finite-state methods in natural language processing (FSMNLP) (5 pp). Extended abstract.

  • Eisner, J. (2003). Simpler and more general minimization for weighted finite-state automata. In Proceedings of the joint meeting of the human language technology conference and the North American chapter of the association for computational linguistics (HLT-NAACL), Edmonton, pp. 64–71.

  • Ellison, M. T. (1994). Phonological derivation in optimality theory. In Proceedings of the 15th international conference on computational linguistics (COLING), Kyoto, pp. 1007–1013.

  • Fink E. (1992) A survey of sequential and systolic algorithms for the algebraic path problem. Carnegie Mellon University, Pittsburgh

    Google Scholar 

  • Frank R., Satta G. (1998) Optimality theory and the generative complexity of constraint violability. Computational Linguistics 24(2): 307–315

    Google Scholar 

  • Gerdemann, D., van Noord, G. (2000). Approximation and exactness in finite state optimality theory. In Coling Workshop finite state phonology, Luxembourg.

  • Goldsmith J. (1993) Harmonic phonology. University of Chicago Press, Chicago, pp 221–269

    Google Scholar 

  • Heinz J., Kobele G., Riggle J. (2009) Evaluating the complexity of Optimality Theory. Linguistic Inquiry 40: 277–288 ROA 968-0508

    Article  Google Scholar 

  • Hopcroft, J. E., & Ullman, J. D. (1979). Introduction to automata theory, languages, and computation. Reading, MA: Addison-Wesley, 78067950. Hopcroft, J. E., & Ullman, J. D. Addison-Wesley series in computer science. Includes index. Bibliography, pp. 396–410.

  • Idsardi W.J. (2006) A simple proof that optimality theory is computationally intractable. Linguistic Inquiry 37(2): 271–275

    Article  Google Scholar 

  • Kaplan R.M., Martin K. (1994) Regular models of phonological rule systems. Computational Linguistics 20(3): 331–378

    Google Scholar 

  • Karttunen L. (1998) The proper treatment of optimality in computational phonology. In: Oflazer K., Karttunen L. (eds) Finite state methods in natural language processing. Bilkent University, Ankara, Turkey, pp 1–12

    Google Scholar 

  • Kempe, A., Champarnaud, J.-M., & Eisner, J. (2004). A note on join and auto-intersection of n-ary rational relations. In L. Cleophas & B. Watson, (Eds.), Proceedings of the Eindhoven FASTAR Days (Computer Science Technical Report 04-40) (pp. 64–78). The Netherlands: Department of Mathematics and Computer Science, Technische Universiteit Eindhoven.

  • Klein D., Manning C.D. (2004) Parsing and hypergraphs. Kluwer, Norwell, MA, pp 351–372

    Google Scholar 

  • Knuth D.E. (1969) Seminumerical algorithms. Addison-Wesley, Reading, MA

    Google Scholar 

  • Koskenniemi, K. (1984). A general computational model for word-form recognition and production. In COLING-84 (pp. 178–181). Stanford University, California, USA: Association for Computational Linguistics.

  • Koskenniemi K. (1997) Representations and finite-state components in natural language. In: Roche E., Schabes Y. (eds) Finite-State Language Processing. MIT Press, Cambrige, pp 99–116

    Google Scholar 

  • Legendre, G., Miyata, Y., & Smolensky, P. (1990). Harmonic grammar – A formal multi-level connectionist theory of linguistic well-formedness: Theoretical foundations. In Proceedings of the twelfth annual conference of the cognitive science society (pp. 388–395). Cambridge, MA: Lawrence Erlbaum.

  • Mohri M. (2002) Semiring frameworks and algorithms for shortest-distance problems. Journal of Automata, Languages and Combinatorics 7(3): 321–350

    Google Scholar 

  • Pater, J., Bhatt, R., & Potts, C. (2007a). Linguistic optimization. Ms., UMASS Amherst.

  • Pater J., Potts C., Bhatt R. (2007b) Harmonic grammar with linear programming. University of Massachusetts, Amherst

    Google Scholar 

  • Prince, A., & Smolensky, P. (1993/2004). Optimality theory: Constraint interaction in generative grammar. Boulder, CA: Rutgers University.

  • Riggle, J. (2004). Generation, recognition, and learning in finite state optimality theory. Ph.D. thesis, University of California, Los Angeles.

  • Simon, I. (1988). Recognizable sets with multiplicities in the tropical semiring. In MFCS ’88: Proceedings of the mathematical foundations of computer science 1988 (pp. 107–120). London, UK: Springer-Verlag.

  • Smolensky P., Legendre G. (2006) The harmonic mind: From neural computation to optimality-theoretic grammar, Vol. I: Cognitive architecture (Bradford Books). The MIT Press, Cambridge, MA

    Google Scholar 

  • Takaoka, T. (1996). Shortest path algorithms for nearly acyclic directed graphs. In Workshop on graph-theoretic concepts in computer science, pp. 367–374.

  • Tesar B., Smolensky P. (2000) Learnability in optimalty theory. MIT Press, Cambridge

    Google Scholar 

  • Wareham, H. T. (1998). Systematic parameterized complexity analysis in computational phonology. Ph.D. thesis, University of Victoria.

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Correspondence to Jason Riggle.

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Riggle, J. Violation Semirings in Optimality Theory. Res on Lang and Comput 7, 1 (2009). https://doi.org/10.1007/s11168-009-9063-0

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Keywords

  • Optimality Theory
  • Complexity
  • Phonology