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Formalizing and reasoning about quality
, 2012
"... Abstract. Traditional formal methods are based on a Boolean satisfaction notion: a reactive system satisfies, or not, a given specification. We generalize formal methods to also address the quality of systems. As an adequate specification formalism we introduce the linear temporal logic LTL[F]. The ..."
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Abstract. Traditional formal methods are based on a Boolean satisfaction notion: a reactive system satisfies, or not, a given specification. We generalize formal methods to also address the quality of systems. As an adequate specification formalism we introduce the linear temporal logic LTL[F]. The satisfaction value of an LTL[F] formula is a number between 0 and 1, describing the quality of the satisfaction. The logic generalizes traditional LTL by augmenting it with a (parameterized) set F of arbitrary functions over the interval [0, 1]. For example, F may contain the maximum or minimum between the satisfaction values of subformulas, their product, and their average. The classical decision problems in formal methods, such as satisfiability, model checking, and synthesis, are generalized to search and optimization problems in the quantitative setting. For example, model checking asks for the quality in which a specification is satisfied, and synthesis returns a system satisfying the specification with the highest quality. Reasoning about quality gives rise to other natural questions, like the distance between specifications. We formalize these basic questions and study them for LTL[F]. By extending the automatatheoretic approach for LTL to a setting that takes quality into an account, we are able to solve the above problems and show that reasoning about LTL[F] has roughly the same complexity as reasoning about traditional LTL. 1
Quantitative interprocedural analysis
 In POPL
, 2015
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Discounting in LTL
"... Abstract. 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 ..."
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Abstract. 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 modelchecking problem. Further augmenting the logic with unary propositional quality operators preserves decidability, whereas adding an averageoperator makes the modelchecking problem undecidable. We also discuss the complexity of the problem, as well as various extensions. 1
Quantitative monadic secondorder logic
 In Proceedings of LICS’13
, 2013
"... Abstract—While monadic secondorder logic is a prominent logic for specifying languages of finite words, it lacks the power to compute quantitative properties, e.g. to count. An automata model capable of computing such properties are weighted automata, but logics equivalent to these automata have o ..."
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Abstract—While monadic secondorder logic is a prominent logic for specifying languages of finite words, it lacks the power to compute quantitative properties, e.g. to count. An automata model capable of computing such properties are weighted automata, but logics equivalent to these automata have only recently emerged. We propose a new framework for adding quantitative properties to logics specifying Boolean properties of words. We use this to define Quantitative Monadic SecondOrder Logic (QMSO). In this way we obtain a simple logic which is equally expressive to weighted automata. We analyse its evaluation complexity, both data and combined complexity, and show completeness results for combined complexity. We further refine the analysis of this logic and obtain fragments that characterise exactly subclasses of weighted automata defined by the level of ambiguity allowed in the automata. In this way, we define a quantitative logic which has good decidability properties while being resonably expressive and enjoying a simple syntactical definition. I.
Finitememory strategy synthesis for robust multidimensional meanpayoff objectives
 In CSLLICS
, 2014
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Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs
"... We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the meanpayoff property, the ratio property, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative pr ..."
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We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the meanpayoff property, the ratio property, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with constant treewidth, and it is wellknown that the controlflow graphs of most programs have constant treewidth. Let n denote the number of nodes of a graph, m the number of edges (for constant treewidth graphs m = O(n)) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for constant treewidth graphs we present an algorithm that approximates the meanpayoff value within a multiplicative factor of in time O(n · log(n/)) and linear space, as compared to the classical algorithms that require quadratic time. Second, for the ratio property we present an algorithm that for constant treewidth graphs works in time O(n · log(a · b)) = O(n · log(n ·W)), when the output is a b, as compared to the previously best known algorithm with running time O(n2 · log(n ·W)). Third, for the minimum initial credit problem we show that (i) for general graphs the problem can be solved in O(n2 ·m) time and the associated decision problem can be solved inO(n ·m) time, improving the previous known O(n3 · m · log(n · W)) and O(n2 · m) bounds, respectively; and (ii) for constant treewidth graphs we present an algorithm that requires O(n · logn) time, improving the previous known O(n4 · log(n ·W)) bound. We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks.