From Equivalence to Almost-Equivalence, and Beyond—Minimizing Automata with Errors

Minimal and Hyper-Minimal Biautomata

“…For deterministic finite automata the situation is similar as in the case of classical minimization: efficient hyperminimization algorithms with running time O(n log n) are known [5,10], and it was shown in [6] that the hyper-minimization problem for DFAs is NL-complete. On the one hand, since classical DFA minimization methods also work well for DBiAs, one could expect that hyper-minimization of DBiAs is as easy as for ordinary DFAs.…”

“…Checking whether A and B are almost-equivalent can be done by testing whether the contained DFAs A fwd and B fwd are almost-equivalent. It is shown in [6] that this question for DFAs can be decided in NL. Therefore, the hyper-minimization problem for biautomata can be decided by a non-deterministic Turing machine in polynomial time.…”

“…Therefore, the hyper-minimization problem for biautomata can be decided by a non-deterministic Turing machine in polynomial time. ⊓ ⊔ In [6] another form of equivalence of languages and automata was considered, namely E-equivalence. Given an error language E ⊆ Σ * , two languages L and L ′ over Σ are E-equivalent, for short L ∼ E L ′ , if L △ L ′ ⊆ E, and two automata A and A ′ are E-equivalent, if L(A) ∼ E L(A ′ ).…”

“…First, all of the above mentioned computational complexity results remain valid for hyper-minimization. Thus, computing a hyper-minimal DFA can be done in O(n log n) time [10] and the hyper-minimization problem is NL-complete [6]. In fact it is known that minimization for DFAs linearly reduces to hyper-minimization [10].…”

“…In fact it is known that minimization for DFAs linearly reduces to hyper-minimization [10]. Moreover, the intractability result for NFAs remains, that is, hyper-minimization for NFAs is PSPACE-complete [6], just as it is for ordinary NFA minimization. What can be said about the structural properties of hyperminimal finite state machines?…”