Expressiveness and Closure Properties for Quantitative Languages

Abstract

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.

Using Statistical Model Checking for Measuring Systems

“…Algorithmic methods for checking the consistency between a system and its specification [6] have been generalized into measuring properties of systems [1].…”

“…Although it is known that if there is an execution that satisfies an LTL property then there is one that is ultimately periodic (see, e.g., [8]), i.e., consists of a finite prefix, and a finite part that repeats indefinitely. However, measuring non-Boolean properties may have different results where ultimately periodic sequences are not good representatives (see, e.g., [1] for measuring the limit of the average of values along a sequence). 2.…”

“…There have been several efforts to apply similar techniques for measuring more general quantitative properties based on the operations needed to compute the measurements. For instance, [1] addresses the problems associated with measurements involving operations such as limit average, maximum and discounted sum; and [7] provides a logic for expressing a generalized class of quantitative properties.…”

Compositional Branching-Time Measurements

“…Some of them are quite impressive in the size of systems that they can handle and in verification speed. A recent trend is to look at quantitative properties, for example, providing measures on how robustly a property is satisfied in probabilistic automata [9] or weighted automata [1,5]. Another approach is to ask for, in addition to the qualitative indication of the satisfaction of a property, a measurement, which is usually based on the accumulated time required to satisfy parts of the specification [2,8].…”

Making Weighted Containment Feasible: A Heuristic Based on Simulation and Abstraction

“…We say that A is simulated by B, denoted A ≤ B iff the protagonist has a winning strategy. The above definition is similar to the definition of quantitative simulation in [12,14], and has the flavor of the energy games in [4]. In these works, however, the winning condition in the game refers only to the weight along the traversed edges.…”