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. 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- $\omega $-regular for deterministic limit-average and discounted-sum automata, while this set is always $\omega $-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the $\omega $-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 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 $1-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.

💡 Analysis

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. 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- $\omega $-regular for deterministic limit-average and discounted-sum automata, while this set is always $\omega $-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the $\omega $-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 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 $1-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.

📄 Content

arXiv:0905.2195v1 [cs.LO] 13 May 2009 Expressiveness and Closure Properties for Quantitative Languages Krishnendu Chatterjee1, Laurent Doyen2, and Thomas A. Henzinger2 1 IST, Austria 2 EPFL, Lausanne, Switzerland Abstract. Weighted automata are nondeterministic automata with nu- merical 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, lim- sup, liminf, limit average, or discounted sum of the transition weights. 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 0 or 1 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 L1 and L2, we consider the operations max(L1, L2), min(L1, L2), and 1−L1, which generalize the boolean oper- ations on languages, as well as the sum L1 +L2. We establish the closure properties of all classes of quantitative languages with respect to these four operations. 1 Introduction A boolean language L can be viewed as a function that assigns to each word w a boolean value, namely, L(w) = 1 if the word w belongs to the language, and L(w) = 0 otherwise. Boolean languages model the computations of reactive programs. The verification problem “does the program A satisfy the specifica- tion B?” then reduces to the language-inclusion problem “is LA ⊆LB?”, or equivalently, “is LA(w) ≤LB(w) for all words w?”, where LA represents all behaviors of the program, and LB contains all behaviors allowed by the spec- ification. When boolean languages are defined by finite automata, this elegant framework is called the automata-theoretic approach to model-checking [VW86]. In a natural generalization of this framework, a cost function assigns to each word a real number instead of a boolean value. For instance, the value of a word (or behavior) can be interpreted as the amount of some resource (e.g., memory consumption, or power consumption) that the program needs to produce it, and a specification may assign a maximal amount of available resource to each behavior, or bound the long-run average available use of the resource. Weighted automata over semirings (i.e., finite automata with transition weights in a semiring structure) have been used to define cost functions, called formal power series for finite words [Sch61,KS86] and ω-series for infinite words [CK94,DK03,´EK04]. In [CDH08], we study new classes of cost functions using operations over rational numbers that do not form a semiring. We call them quantitative languages. We set the value of a (finite or infinite) word w as the maximal value of all runs over w (if the automaton is nondeterministic, then there may be many runs over w), and the value of a run r is a function of the (finite or infinite) sequence of weights that appear along r. We consider several functions, such as Max and Sum of weights for finite runs, and Sup, LimSup, LimInf, limit average, and discounted sum of weights for infinite runs. For example, peak power consumption can be modeled as the maximum of a sequence of weights representing power usage; energy use can be modeled as the sum; average response time as the limit average [CCH+05,CdAHS03]. Quanti- tative languages can also be used to specify and verify reliability requirements: if a special symbol ⊥is used to denote failure and has weight 1, while the other symbols have weight 0, one can use a limit-average automaton to specify a bound on the rate of failure in the long run [CGH+08]. The discounted sum can be used to specify that failures happening later are less important than those happening soon [dAHM03]. The quantitative language-inclusion problem “Given two automata A and B, is LA(w) ≤LB(w) for all words w?” can then be used to check, say, if for each behavior, the peak power used by the system lies below the bound given by the specification; or if for each behavior, the long-run average response time of the system lies below the specified average response requirement.s In [CDH08], we showed that the quantitative language-inclusion problem is PSPACE-complete for Sup-, LimSup-, and LimInf-automata, while the decidability is unknown for (nondeterministic) limit-average and discounted-sum automata. We also com- pared the expressive power of the different classes of quantitative languages and showed that nondeterministic automata are strictly more expressive in the case of limit-average and discounted-sum. I

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