Having defined GEN and the five syllabification constraints we are now in a position to address the main issue: how are optimality constraints applied?
Given that GEN denotes a relation and that the constraints can be thought of as identity relations on sets, the simplest idea is to proceed in the same way as with the rewrite rules in Figure 2. We could compose GEN with the constraints to yield a transducer that maps each input to its most optimal realization, letting the ordering of the constraints in the cascade implement their ranking (Figure 9).
But it is immediately obvious that composition does not work here as intended. The 6-state transducer illustrated in Figure 9 works fine on inputs such as panama yielding O[p]N[a]O[n]N[a]O[m]N[a] but it fails to produce any output on inputs like america that fail on some constraint. Only strings that have a perfect output candidate survive this merciless cascade. We need to replace composition with some new operation to make this schema work correctly.