# 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

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