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

Abstract

We introduce E-equivalence, which is a straightforward generalization of almost-equivalence. While almost-equivalence asks for ordinary equivalence up to a finite number of exceptions, in E-equivalence these exceptions or errors must belong to a (regular) set E. The computational complexity of minimization problems and their variants w.r.t. almost- and E-equivalence are studied. Roughly speaking, whenever nondeterministic finite automata (NFAs) are involved, most minimization problems, and their equivalence problems they are based on, become PSPACE-complete, while for deterministic finite automata (DFAs) the situation is more subtle. For instance, hyper-minimizing DFAs is NL-complete, but E-minimizing DFA s is NP-complete, even for finite E. The obtained results nicely fit to the known ones on ordinary minimization for finite automata. Moreover, since hyper-minimal and E-minimal automata are not necessarily unique (up to isomorphism as for minimal DFAs), we consider the problem of counting the number of these minimal automata. It turns out that counting hyper-minimal DFAs can be done in FP, while counting E-minimal DFA s is #P-hard, and belongs to the counting class #·coNP.