Violation Semirings in Optimality Theory

  • Jason RiggleEmail author


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.


Optimality Theory Complexity Phonology 


  1. Bistarelli S., Montanari U., Rossi F. (1997) Semiring-based constraint satisfaction and optimization. Journal of the ACM 44(2): 201–236CrossRefGoogle Scholar
  2. 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.Google Scholar
  3. Cormen T.H., Leiserson C.E., Rivest R.L. (1990) Introduction to algorithms. MIT Press, Cambridge, MAGoogle Scholar
  4. Dijkstra E.W. (1959) A note on two problems in connexion with graphs. Numerische Mathematik 1: 269–271CrossRefGoogle Scholar
  5. 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.Google Scholar
  6. 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.Google Scholar
  7. 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.Google Scholar
  8. 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.Google Scholar
  9. Ellison, M. T. (1994). Phonological derivation in optimality theory. In Proceedings of the 15th international conference on computational linguistics (COLING), Kyoto, pp. 1007–1013.Google Scholar
  10. Fink E. (1992) A survey of sequential and systolic algorithms for the algebraic path problem. Carnegie Mellon University, PittsburghGoogle Scholar
  11. Frank R., Satta G. (1998) Optimality theory and the generative complexity of constraint violability. Computational Linguistics 24(2): 307–315Google Scholar
  12. Gerdemann, D., van Noord, G. (2000). Approximation and exactness in finite state optimality theory. In Coling Workshop finite state phonology, Luxembourg.Google Scholar
  13. Goldsmith J. (1993) Harmonic phonology. University of Chicago Press, Chicago, pp 221–269Google Scholar
  14. Heinz J., Kobele G., Riggle J. (2009) Evaluating the complexity of Optimality Theory. Linguistic Inquiry 40: 277–288 ROA 968-0508CrossRefGoogle Scholar
  15. 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.Google Scholar
  16. Idsardi W.J. (2006) A simple proof that optimality theory is computationally intractable. Linguistic Inquiry 37(2): 271–275CrossRefGoogle Scholar
  17. Kaplan R.M., Martin K. (1994) Regular models of phonological rule systems. Computational Linguistics 20(3): 331–378Google Scholar
  18. 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 1–12Google Scholar
  19. 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.Google Scholar
  20. Klein D., Manning C.D. (2004) Parsing and hypergraphs. Kluwer, Norwell, MA, pp 351–372Google Scholar
  21. Knuth D.E. (1969) Seminumerical algorithms. Addison-Wesley, Reading, MAGoogle Scholar
  22. 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.Google Scholar
  23. 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 99–116Google Scholar
  24. 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.Google Scholar
  25. Mohri M. (2002) Semiring frameworks and algorithms for shortest-distance problems. Journal of Automata, Languages and Combinatorics 7(3): 321–350Google Scholar
  26. Pater, J., Bhatt, R., & Potts, C. (2007a). Linguistic optimization. Ms., UMASS Amherst.Google Scholar
  27. Pater J., Potts C., Bhatt R. (2007b) Harmonic grammar with linear programming. University of Massachusetts, AmherstGoogle Scholar
  28. Prince, A., & Smolensky, P. (1993/2004). Optimality theory: Constraint interaction in generative grammar. Boulder, CA: Rutgers University.Google Scholar
  29. Riggle, J. (2004). Generation, recognition, and learning in finite state optimality theory. Ph.D. thesis, University of California, Los Angeles.Google Scholar
  30. 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.Google Scholar
  31. 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, MAGoogle Scholar
  32. Takaoka, T. (1996). Shortest path algorithms for nearly acyclic directed graphs. In Workshop on graph-theoretic concepts in computer science, pp. 367–374.Google Scholar
  33. Tesar B., Smolensky P. (2000) Learnability in optimalty theory. MIT Press, CambridgeGoogle Scholar
  34. Wareham, H. T. (1998). Systematic parameterized complexity analysis in computational phonology. Ph.D. thesis, University of Victoria.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA

Personalised recommendations