Hyper-optimization for Deterministic Tree Automata

A recent minimization technique, called hyper-minimization, permits reductions of language representations beyond the limits imposed by classical semantics-preserving minimization. Naturally, the semantics is not preserved by hyper-minimization; rather the reduced representation, which is called hyper-minimal, can accept a language that has a nite symmetric di erence to the language of the original representation. It was demonstrated that hyper-minimization for (bottom-up) deterministic tree automata (dtas), which represent the recognizable tree languages, can be achieved in time O(m · logn), where m is the size of the dta and n is the number of its states. In this contribution, this result is complemented by two results on the quantity of the errors. It is shown that optimal hyper-minimization for dtas (i.e., computing a hyper-minimal dta that commits the least number of errors of all hyperminimal dtas) can be achieved in time O(m ·n). In the same time bound also the number of errors of any hyper-minimal dta can be computed.