We aim at measuring the influence of the nondeterministic choices of a part of a system on its ability to satisfy a specification. For this purpose, we apply the concept of Shapley values to verification as a means to evaluate how important a part of a system is. The importance of a component is measured by giving its control to an adversary, alone or along with other components, and testing whether the system can still fulfill the specification. We study this idea in the framework of modelchecking with various classical types of linear-time specification, and propose several ways to transpose it to branching ones. We also provide tight complexity bounds in almost every case.

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Let n be a natural number, and let \scrM be a set of n\times n-matrices over the nonnegative integers such that the joint spectral radius of \scrM is at most one. We show that if the zero matrix 0 is a product of matrices in \scrM , then there are M 1 , . . . , M n 5 \in \scrM with M 1 \cdot \cdot \cdot M n 5 = 0. This result has applications in automata theory and the theory of codes. Specifically, if X \subset \Sigma \ast is a finite incomplete code, then there exists a word w \in \Sigma \ast of length polynomial in \sum x\in X | x| such that w is not a factor of any word in X \ast . This proves a weak version of Restivo's conjecture.

Runtime monitoring is commonly used to detect the violation of desired properties in safety critical cyber-physical systems by observing its executions. Bauer et al. introduced an influential framework for monitoring Linear Temporal Logic (LTL) properties based on a three-valued semantics for a finite execution: the formula is already satisfied by the given execution, it is already violated, or it is still undetermined, i.e., it can still be satisfied and violated by appropriate extensions of the given execution. However, a wide range of formulas are not monitorable under this approach, meaning that there are executions for which satisfaction and violation will always remain undetermined no matter how it is extended. In particular, Bauer et al. report that 44% of the formulas they consider in their experiments fall into this category. Recently, a robust semantics for LTL was introduced to capture different degrees by which a property can be violated. In this paper we introduce a robust semantics for finite strings and show its potential in monitoring: every formula considered by Bauer et al. is monitorable under our approach. Furthermore, we discuss which properties that come naturally in LTL monitoring—such as the realizability of all truth values—can be transferred to the robust setting. We show that LTL formulas with robust semantics can be monitored by deterministic automata, and provide tight bounds on the size of the constructed automaton. Lastly, we report on a prototype implementation and compare it to the LTL monitor of Bauer et al. on a sample of examples.