While most approaches in formal methods address system correctness, ensuring robustness has remained a challenge. In this article, we present and study the logic rLTL, which provides a means to formally reason about both correctness and robustness in system design. Furthermore, we identify a large fragment of rLTL for which the verification problem can be efficiently solved, i.e., verification can be done by using an automaton, recognizing the behaviors described by the rLTL formula φ, of size at most O(3 |φ |), where |φ | is the length of φ. This result improves upon the previously known bound of O(5|φ |) for rLTL verification and is closer to the LTL bound of O(2|φ |). The usefulness of this fragment is demonstrated by a number of case studies showing its practical significance in terms of expressiveness, the ability to describe robustness, and the fine-grained information that rLTL brings to the process of system verification. Moreover, these advantages come at a low computational overhead with respect to LTL verification.

Robust Linear Temporal Logic (rLTL) was crafted to incorporate the notion of robustness into Linear-time Temporal Logic (LTL) specifications. Technically, robustness was formalized in the logic rLTL via 5 different truth values and it led to an increase in the time complexity of the associated model checking problem. In general, model checking an rLTL formula relies on constructing a generalized Büchi automaton of size 5 | φ | where | φ | denotes the length of an rLTL formula φ. It was recently shown that the size of this automaton can be reduced to 3 | φ | (and even smaller) when the formulas to be model checked come from a fragment of rLTL. In this paper, we introduce Evrostos, the first tool for model checking formulas in this fragment. We also present several empirical studies, based on models and LTL formulas reported in the literature, confirming that rLTL model checking for the aforementioned fragment incurs in a time overhead that makes the verification of rLTL practical.

We study criteria allowing to compare the conservativeness of stability certificates for switching systems. The stability certificates under consideration are Path-Complete Lyapunov functions (PCLFs), which are multiple Lyapunov functions with an underlying combinatorial structure.

A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time arbitrary switching systems. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. the converse however is not true even for discrete-time linear systems: we present such a system where a max-of-2 quadratics Lyapunov function exists while no corresponding Path-Complete Lyapunov function with 2 quadratic pieces exists. In light of this, we investigate when it is possible to decide if a Path- Complete Lyapunov function is less conservative than another. By analyzing the combinatorial and algebraic structure of the graph and the pieces respectively, we provide simple tools to decide when the existence of such a Lyapunov function implies that of another.

We study the boundedness of products of matrices associated with words in a regular language. This question naturally arises in the stability analysis of switching systems with constrained switching sequences.