# Violation Semirings in Optimality Theory

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## 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.

## Keywords

Optimality Theory Complexity Phonology## References

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