Approximating deterministic lattice automata

Traditional automata accept or reject their input, and are therefore Boolean. Lattice automata generalize the traditional setting and map words to values taken from a lattice. In particular, in a fully-ordered lattice, the elements are 0,1,...,n - 1, ordered by the standard ≤ order. Lattice automata, and in particular lattice automata defined with respect to fully-ordered lattices, have interesting theoretical properties as well as applications in formal methods. Minimal deterministic automata capture the combinatorial nature and complexity of a formal language. Deterministic automata have many applications in practice. In [13], we studied minimization of deterministic lattice automata. We proved that the problem is in general NP-complete, yet can be solved in polynomial time in the case the lattices are fully-ordered. The multi-valued setting makes it possible to combine reasoning about lattice automata with approximation. An approximating automaton may map a word to a range of values that are close enough, under some pre-defined distance metric, to its exact value. We study the problem of finding minimal approximating deterministic lattice automata defined with respect to fully-ordered lattices. We consider approximation by absolute distance, where an exact value x can be mapped to values in the range [x - t,x + t], for an approximation factor t, as well as approximation by separation, where values are mapped into t classes. We prove that in both cases the problem is in general NP-complete, but point to special cases that can be solved in polynomial time.