Expressiveness and Closure Properties for Quantitative Languages

Abstract

Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages~$L$ that assign to each word~$w$ a real number~$L(w)$. In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit-average, or discounted-sum of the transition weights. The value of a word $w$ is the supremum of the values of the runs over $w$. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-$ω$-regular for deterministic limit-average and discounted-sum automata, while this set is always $ω$-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the $ω$-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights Weighted automata are nondeterministic automata with numerical weights ontransitions. They can define quantitative languages~$L$ that assign to eachword~$w$ a real number~$L(w)$. In the case of infinite words, the value of arun is naturally computed as the maximum, limsup, liminf, limit-average, ordiscounted-sum of the transition weights. The value of a word $w$ is thesupremum of the values of the runs over $w$. We study expressiveness andclosure questions about these quantitative languages. We first show that the set of words with value greater than a threshold canbe non-$ω$-regular for deterministic limit-average and discounted-sumautomata, while this set is always $ω$-regular when the threshold isisolated (i.e., some neighborhood around the threshold contains no word). Inthe latter case, we prove that the $ω$-regular language is robust againstsmall perturbations of the transition weights. We next consider automata with transition weights $0$ or $1$ and show thatthey are as expressive as general weighted automata in the limit-average case,but not in the discounted-sum case. Third, for quantitative languages $L_1$ and~$L_2$, we consider the operations$\max(L_1,L_2)$, $\min(L_1,L_2)$, and $1-L_1$, which generalize the booleanoperations on languages, as well as the sum $L_1 + L_2$. We establish theclosure properties of all classes of quantitative languages with respect tothese four operations.$ or $ and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages $L_1$ and~$L_2$, we consider the operations $\max(L_1,L_2)$, $\min(L_1,L_2)$, and -L_1$, which generalize the boolean operations on languages, as well as the sum $L_1 + L_2$. We establish the closure properties of all classes of quantitative languages with respect to these four operations.