On the Comparison of Discounted-Sum Automata with Multiple Discount Factors

Abstract

AbstractWe look into the problems of comparing nondeterministic discounted-sum automata on finite and infinite words. That is, the problems of checking for automata $${\mathcal {A}}$$ A and $${\mathcal {B}}$$ B whether or not it holds that for all words w, $${\mathcal {A}}(w)={\mathcal {B}}(w), {\mathcal {A}}(w)\le {\mathcal {B}}(w)$$ A ( w ) = B ( w ) , A ( w ) ≤ B ( w ) , or $${\mathcal {A}}(w)<{\mathcal {B}}(w)$$ A ( w ) < B ( w ) .These problems are known to be decidable when both automata have the same single integral discount factor, while decidability is open in all other settings: when the single discount factor is a non-integral rational; when each automaton can have multiple discount factors; and even when each has a single integral discount factor, but the two are different.We show that it is undecidable to compare discounted-sum automata with multiple discount factors, even if all are integrals, while it is decidable to compare them if each has a single, possibly different, integral discount factor. To this end, we also provide algorithms to check for given nondeterministic automaton $${\mathcal {N}}$$ N and deterministic automaton $${\mathcal {D}}$$ D , each with a single, possibly different, rational discount factor, whether or not $${\mathcal {N}}(w) = {\mathcal {D}}(w)$$ N ( w ) = D ( w ) , $${\mathcal {N}}(w) \ge {\mathcal {D}}(w)$$ N ( w ) ≥ D ( w ) , or $${\mathcal {N}}(w) > {\mathcal {D}}(w)$$ N ( w ) > D ( w ) for all words w.