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

We look into the problems of comparing nondeterministic discounted-sum automata on finite and infinite words. That is, the problems of checking for automata $A$ and $B$ whether or not it holds that for all words $w$, $A(w)=B(w), A(w) \leq B(w)$, or $A(w)<B(w)$. 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 $N$ and deterministic automaton $D$, each with a single, possibly different, rational discount factor, whether or not $N(w) = D(w)$, $N(w) \geq D(w)$, or $N(w) > D(w)$ for all words $w$.