Weighted automata and weighted logics with discounting

Droste, Manfred; Rahonis, George

doi:10.1016/j.tcs.2009.03.029

Cited by 28 publications

(8 citation statements)

References 45 publications

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“…The maximum of two quantitative languages defined by nondeterministic automata can be obtained by an initial nondeterministic choice between the two automata. This observation was also made in [DR07] for discountedsum automata. For deterministic automata, a synchronized product can be used for Sup and LimSup, while for LimInf we use the fact that NLinf is determinizable with an exponential blow-up [CDH08].…”

Section: Closure Propertiessupporting

confidence: 67%

“…Weighted automata with discounted sum have been considered in [DR07], with multiple discount factors and a boolean acceptance condition (Muller or Büchi); they are shown to be equivalent to a weighted monadic second-order logic with discounting. Several other works have considered quantitative generalizations of languages, over finite words [DG07], over trees [DKR08], or using finite lattices [GC03], but none of these works has addressed the expressiveness questions and closure properties for quantitative languages that are studied here.…”

Section: Introductionmentioning

confidence: 99%

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Expressiveness and Closure Properties for Quantitative Languages

Chatterjee¹,

Doyen²,

Henzinger³

2010

Logical Methods in Computer Science

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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 0 or 1 and show that they are as expressive as general weighted automata in the limit-average case, but not in the discountedsum 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 operations 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.

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Section: Closure Propertiessupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Expressiveness and Closure Properties for Quantitative Languages

Chatterjee¹,

Doyen²,

Henzinger³

2010

Logical Methods in Computer Science

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“…Their proof is based on a notion of recoverable gap, similar to that of delays. Finally in [17], the relation between discounted weighted automata over a semiring and weighted logics is studied.…”

Section: Introductionmentioning

confidence: 99%

Quantitative Languages Defined by Functional Automata

Filiot

Gentilini

Raskin

2015

Logical Methods in Computer Science

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Abstract. A weighted automaton is functional if any two accepting runs on the same finite word have the same value. In this paper, we investigate functional weighted automata for four different measures: the sum, the mean, the discounted sum of weights along edges and the ratio between rewards and costs. On the positive side, we show that functionality is decidable for the four measures. Furthermore, the existential and universal threshold problems, the language inclusion problem and the equivalence problem are all decidable when the weighted automata are functional. On the negative side, we also study the quantitative extension of the realizability problem and show that it is undecidable for sum, mean and ratio. We finally show how to decide whether the language associated with a given functional automaton can be defined with a deterministic one, for sum, mean and discounted sum. The results on functionality and determinizability are expressed for the more general class of functional group automata. This allows one to formulate within the same framework new results related to discounted sum automata and known results on sum and mean automata. Ratio automata do not fit within this general scheme and different techniques are required to decide functionality.

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“…The richness of today's systems, however, justifies specification formalisms that are multi-valued. The multi-valued setting arises directly in systems with quantitative aspects (multi-valued / probabilistic / fuzzy) [9,10,11,16,23], but is applied also with respect to Boolean systems, where it origins from the semantics of the specification formalism itself [1,7].…”

Section: Introductionmentioning

confidence: 99%

Discounting in LTL

Almagor

Boker

Kupferman

2014

Tools and Algorithms for the Construction and Analysis of Systems

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In recent years, there is growing need and interest in formalizing and reasoning about the quality of software and hardware systems. As opposed to traditional verification, where one handles the question of whether a system satisfies, or not, a given specification, reasoning about quality addresses the question of how well the system satisfies the specification. One direction in this effort is to refine the "eventually" operators of temporal logic to discounting operators: the satisfaction value of a specification is a value in [0, 1], where the longer it takes to fulfill eventuality requirements, the smaller the satisfaction value is. In this paper we introduce an augmentation by discounting of Linear Temporal Logic (LTL), and study it, as well as its combination with propositional quality operators. We show that one can augment LTL with an arbitrary set of discounting functions, while preserving the decidability of the model-checking problem. Further augmenting the logic with unary propositional quality operators preserves decidability, whereas adding an average-operator makes some problems undecidable. We also discuss the complexity of the problem, as well as various extensions.LTL disc [D] is actually a family of logics, each parameterized by a set D of discounting functions -strictly decreasing functions from AE to [0, 1] that tend to 0 (e.g., linear decaying, exponential decaying, etc.). LTL disc [D] includes a discounting-"until" (U η ) operator, parameterized by a function η ∈ D. We solve the model-checking threshold problem for LTL disc [D]: given a Kripke structure K, an LTL disc [D] formula ϕ and a threshold t ∈ [0, 1], the algorithm decides whether the satisfaction value of ϕ in K is at least t.In the Boolean setting, the automata-theoretic approach has proven to be very useful in reasoning about LTL specifications. The approach is based on translating LTL formulas to nondeterministic Büchi automata on infinite words [28]. Applying this approach to the discounted setting, which gives rise to infinitely many satisfaction values, poses a big algorithmic challenge: model-checking algorithms, and in particular those that follow the automata-theoretic approach, are based on an exhaustive search, which cannot be simply applied when the domain becomes infinite. A natural relevant extension to the automata-theoretic approach is to translate formulas to weighted automata [22]. Unfortunately, these extensively-studied models are complicated and many problems become undecidable for them [15]. We show that for threshold problems, we can translate LTL disc [D] formulas into (Boolean) nondeterministic Büchi automata, with the property that the automaton accepts a lasso computation iff the formula attains a value above the threshold on that computation. Our algorithm relies on the fact that the language of an automaton is non-empty iff there is a lasso witness for the non-emptiness. We cope with the infinitely many possible satisfaction values by using the discounting behavior of the eventualities and the given thre...

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Weighted automata and weighted logics with discounting

Cited by 28 publications

References 45 publications

Expressiveness and Closure Properties for Quantitative Languages

Expressiveness and Closure Properties for Quantitative Languages

Quantitative Languages Defined by Functional Automata

Discounting in LTL

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