Safety and Co-safety Comparator Automata for Discounted-Sum Inclusion

Abstract

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 Büchi automata that read two infinite 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 Büchi operations. Motivated by the connections between discounted-sum and Büchi automata, this paper undertakes an investigation of language-theoretic properties of DS-comparators in order to mitigate the challenges of Büchi 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 Büchi complementation, yielding tighter worst-case complexity and improved empirical scalability for DS-inclusion.

On Satisficing in Quantitative Games

“…We begin with formal definitions: Proof. The proof is identical to that for threshold value v = 0 from [9].…”

“…This section extends the existing literature on comparators with threshold value v = 0 [6,5,9] to permit non-zero thresholds. The properties we investigate are of safety/co-safety and ω-regularity.…”

“…. , μ} that accepts A ∈ Σ ω iff DS (A, d) R 0 holds [5,9]. The comparator automata with upper bound μ, relation R, discount factor d is the automaton that accepts the corresponding comparison language [6].…”

“…The comparator automata with upper bound μ, relation R, discount factor d is the automaton that accepts the corresponding comparison language [6]. Depending on R, these languages are safety or co-safety [9]. A comparison language is said to be ω-regular if its automaton is a Büchi automaton.…”

“…In existing literature, such automata are called comparator automata (comparators, in short) when the threshold value v = 0 [6,7]. They are known to have a compact safety or co-safety automaton representation [9,19], which could be used to reduce the satisficing problem with zero threshold value. To solve satisficing for arbitrary threshold values v ∈ Q, we extend existing results on comparators to permit arbitrary but fixed threshold values v ∈ Q.…”