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   Discounted-sum inclusion (DS-inclusion, in short) formalizes the goal of comparing quantitative dimensions of systems such as cost, resource consumption, and the like, when the mode of aggregation for the quantitative dimension is discounted-sum aggregation. Discounted-sum comparator automata, or DS-comparators in short, are Buechi automata that read two in nite sequences of weights synchronously and relate their discounted-sum. Recent empirical investigations have shown that while DS-comparators enable competitive algorithms for DS-inclusion, they still suffer from the scalability bottleneck of Bueuchi operations. Motivated by the connections between discounted-sum and Buechi automata, this paper undertakes an investigation of language-theoretic properties of DS-comparators in order to mitigate the challenges of Buechi DS-comparators to achieve improved scalability of DS-inclusion. Our investigation uncovers that DS-comparators possess safety and co-safety language-theoretic properties. As a result, they enable reductions based on subset construction-based methods as opposed to higher complexity Buechi complementation, yielding tighter worst-case complexity and improved empirical scalability for DS-inclusion. 

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