Natural Language & Linguistic Theory

, Volume 29, Issue 1, pp 3–70 | Cite as

Ranking and necessity: the Fusional Reduction Algorithm

  • Adrian BrasoveanuEmail author
  • Alan Prince
Open Access


Understanding a linguistic theory within OT requires an exact characterization of the ranking conditions necessitated by data. These conditions determine the formal shape of the grammar while providing the crucial link between the data and its interpretation. We introduce an algorithm (‘Fusional Reduction’, FRed) which calculates the necessary and sufficient ranking conditions inherent in any collection of candidates and presents them in a maximally concise and informative way. The algorithm, stemming from the original proposal of Brasoveanu 2003, is set within the fusional ERC theory of Prince 2002a. In this context, the Most Informative Basis and the Skeletal Basis emerge as the two important types of reduced representations of ranking structure. We examine their properties and show how FRed produces them from data. Fine-grained FRed is compared with broad-stroke RCD (Tesar and Smolensky 1993, Tesar 1995 et seq.), and RCD is reinterpreted and embraced within FRed as a simplified, information-losing sub-case. Finally, FRed is compared with other related algorithms in structure, worst-case complexity, and relevance to the analytical enterprise. This paper revises Brasoveanu and Prince 2005, 2007; Prince and Brasoveanu 2010 gives a more formal perspective, with proof of the theorems.


Optimality Theory Ranking ERC Fusional Reduction RCD 


  1. ROA = Rutgers Optimality Archive.
  2. Anderson, Alan Ross, and Nuel D. Belnap Jr., eds. 1975. Entailment: the logic of relevance and necessity, Vol. 1. Princeton: Princeton University Press. Google Scholar
  3. Baković, Eric. 2000. Harmony, dominance, and control. PhD dissertation, Rutgers University, New Brunswick. ROA-360: 25–50. Google Scholar
  4. Brasoveanu, Adrian. 2003. Minimal fusion normal form. Manuscript, Rutgers University.
  5. Brasoveanu, Adrian, and Alan Prince. 2005. Ranking and necessity, Part I (first version). ROA-794. Google Scholar
  6. Brasoveanu, Adrian, and Alan Prince. 2007. Fusional reduction and the logic of ranking arguments in OT. Handout. In 31st Penn linguistics colloquium. University of Pennsylvania. Google Scholar
  7. Gigerenzer, Gerd, and Daniel G. Goldstein. 1996. Reasoning the fast and frugal way: models of bounded rationality. Psychological Review 103(4): 650–669. CrossRefGoogle Scholar
  8. Gigerenzer, Gerd, Peter M. Todd, and the ABC Research Group. 1999. Simple heuristics that make us smart. New York: Oxford University Press. Google Scholar
  9. Grimshaw, Jane. 1997. Projection, heads, and optimality. Linguistic Inquiry 28(4): 373–422. ROA-68. Google Scholar
  10. Hammer, Peter, and Alexander Kogan. 1993. Optimal compression of propositional Horn knowledge bases: complexity and approximation. Artificial Intelligence 64: 131–145. CrossRefGoogle Scholar
  11. Hayes, Bruce. 2004. Phonological acquisition in optimality theory: the early stages. In Constraints in Phonological Acquisition, eds. René Kager, Joe Pater, and Wim Zonneveld, 158–203. Cambridge: Cambridge University Press. Also as ROA-327. Google Scholar
  12. Hayes, Bruce, Bruce Tesar, and Kai Zuraw. 2004. OTSoft.
  13. Lombardi, Linda. 1999. Positional faithfulness and voicing assimilation in optimality theory. Natural Language & Linguistic Theory 17: 267–302. CrossRefGoogle Scholar
  14. Merchant, Nazarré. 2008. Discovering underlying forms: contrast pairs and ranking. PhD dissertation, Rutgers University, New Brunswick. ROA-964. Google Scholar
  15. Meyer, Robert K. 1975. Chapters 29.3 and 29.12. In Vol. 1 of Entailment: the logic of relevance and necessity, eds. Alan Ross Anderson and Nuel D. Belnap Jr. Princeton: Princeton University Press. Google Scholar
  16. Parks, Zane R. 1972. A note on R-mingle and Sobociński’s three-valued logic. Notre Dame Journal of Formal Logic 13: 227–228. CrossRefGoogle Scholar
  17. Prince, Alan. 1998. A proposal for the reformation of tableaux. ROA-288. Google Scholar
  18. Prince, Alan. 2000. Comparative Tableaux. ROA-376. Google Scholar
  19. Prince, Alan. 2002a. Entailed ranking arguments. ROA-500. Google Scholar
  20. Prince, Alan. 2002b. Arguing optimality. ROA-562. Google Scholar
  21. Prince, Alan. 2006a. Lectures on optimality theory, Università degli Studi di Verona. Lecture 2. Google Scholar
  22. Prince, Alan. 2006b. Implication & impossibility in grammatical systems. ROA-880. Google Scholar
  23. Prince, Alan. 2006c. No more than Necessary: beyond the Four Rules, and a bug report. ROA-882. Google Scholar
  24. Prince, Alan. 2008a. The proper treatment of ranking in OT. Manuscript, Rutgers University. Google Scholar
  25. Prince, Alan. 2008b. ERC minimization is tractable. Manuscript, Rutgers University. Google Scholar
  26. Prince, Alan. 2008c. OTWorkplace. Freeware.
  27. Prince, Alan. 2009. RCD–the movie. ROA-1057. Google Scholar
  28. Prince, Alan, and Adrian Brasoveanu. 2010. The formal structure of ranking arguments in OT. Manuscript, Rutgers University and UC Santa Cruz. Google Scholar
  29. Prince, Alan, and Paul Smolensky. 2004. Optimality theory: constraint interaction in generative grammar. Oxford: Blackwell. Revised from ROA-537 (1993 version). Google Scholar
  30. Prince, Alan, and Bruce Tesar. 2004. Learning phonotactic distributions. In Constraints in phonological acquisition, eds. René Kager, Joe Pater, and Wim Zonneveld, 245–291. Cambridge: Cambridge University Press. Also as ROA-353 and RUCCS-TR-54. Google Scholar
  31. Prince, Alan, and Bruce Tesar. 2008. RUBOT. Freeware, Dept of Linguistics, Rutgers University. Google Scholar
  32. Riggle, Jason. 2007. Efficiently Computing OT Typologies. Talk at Linguistic Society of America Annual Meeting. Anaheim, CA. Abstract available at
  33. Samek-Lodovici, Vieri. 1992. Universal constraints and morphological gemination: a crosslinguistic study. Revised as “A Unified Analysis of Cross-linguistic Morphological Gemination,” 1996: ROA-149. Google Scholar
  34. Samek-Lodovici, Vieri, and Alan Prince. 1999. Optima. RuCCS-TR-57. ROA-363. Google Scholar
  35. Samek-Lodovici, Vieri, and Alan Prince. 2005. Fundamental Properties of Harmonic Bounding. RUCCS-TR-71: Corrected 2005 as ROA-785.
  36. Sobociński, Bolesław. 1952. Axiomatization of a partial system of three-valued calculus of propositions. The Journal of Computing Systems 1: 23–55. Google Scholar
  37. Tesar, Bruce. 1995. Computational optimality theory. PhD Dissertation, University of Colorado at Boulder. ROA-90. Google Scholar
  38. Tesar, Bruce. 1997a. Multi-recursive constraint demotion. ROA-197. Google Scholar
  39. Tesar, Bruce. 1997b. Using the mutual inconsistency of structural descriptions to overcome ambiguity in language learning. In Proceedings of the North East Linguistic Society 28, eds. Pius N. Tamanji and Kiyomi Kusumoto, 469–483. Amherst: GLSA, University of Massachusetts. Google Scholar
  40. Tesar, Bruce, and Alan Prince. 2005. Using phonotactics to learn phonological alternations. In Proceedings of the thirty-ninth conference of the Chicago Linguistics Society, Vol. II. The Panels. ROA-620. Google Scholar
  41. Tesar, Bruce, and Paul Smolensky. 1993. The learnability of optimality theory: an algorithm and some basic complexity results. ROA-2. Google Scholar
  42. Tesar, Bruce, and Paul Smolensky. 2000. Learnability in optimality theory. Cambridge: MIT Press. Also ROA-156. Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.UC Santa CruzSanta CruzUSA
  2. 2.Rutgers UniversityNew BrunswickUSA

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