Counting Rankings

In this paper, I present a recursive algorithm that calculates the number of rankings that are consistent with a set of data (i.e. optimal candidates) in the framework of Optimality Theory. The ability to compute this quantity, which I call the r-volume, makes possible a simple and effective Bayesian heuristic in learning – all else equal, choose the candidate preferred by the highest number of constraint rankings consistent with previous observations. Using this heuristic, I formulate a learning algorithm that is guaranteed to make fewer than k log2 k errors while learning rankings of k constraints. This log-linear mistake bound is an improvement over the quadratic mistake bound of Recursive Constraint Demotion and is within a logarithmic factor of the best possible mistake bound for OT learning. I conclude with an illustration of learning syllable structure grammars to contrast the learning curves with the mistake bounds and computational requirements of several ranking algorithms.