1 Introduction

Formal methods are mathematical techniques, originally proposed by the computer science community, to rigorously analyze software systems. In recent years we have witnessed an increase in the use of techniques originating in this area to solve control problems. Similarly, the idea of synthesizing a controller that enforces the desired specifications is becoming an alternative to the verification paradigm prevalent in the formal methods area. There is now a growing body of literature at the intersection of these two disciplines, formal methods and control theory, and the purpose of this special issue is to present the latest developments in this area.

2 Issue at a glance

The call for papers attracted twelve submissions. After a thorough review process, six full papers and two short papers were selected to appear in the special issue. The topics covered include reactive synthesis, abstraction based hierarchical control, synthesis for timed automata, and verification and falsification of linear and hybrid systems.

Reactive synthesis involves the algorithmic generation of systems or controllers that can interact with their environment in the presences of uncontrollable events or adversarial agents. Ehlers et al. compare and contrast supervisory control in discrete event systems and reactive synthesis in formal methods while providing a comprehensive introduction to both topics. They highlight several similarities and differences in the approaches taken by the discrete event systems and the formal methods communities. Schmuck et al. present an algorithm for hierarchical reactive controller synthesis that solves a reactive synthesis problem in a compositional manner. Their modeling formalism provides a means to decompose a reactive synthesis problem into consistent layers of abstractions. They present an example, which involves an autonomous robot in a building environment, demonstrating the scalability of the approach in comparison with a monolithic solution.

Many complex control tasks require discrete (non-smooth) decision making while the underlying dynamics are continuous. A key technique to design and analyze controllers to achieve such complex tasks, in particular those captured by temporal logics, is to create an abstraction of the continuous system to be controlled. The paper by Nilsson et al. proposes a new abstraction structure called augmented finite transition systems that captures transience properties of the underlying dynamics. In addition, they propose an incremental synthesis algorithm, which uses preorders among augmented finite transition systems, for abstraction refinement in order to systematically and adaptively increase the discrete state-space size only when needed based on the specification and dynamics. The paper by Zamani et al. addresses the scalability problem for abstraction based control synthesis for stochastic systems. Instead of discretizing the state-space, they propose to discretize input sequences and characterize several properties of this new type of abstraction. DeCastro et al. address the problem of generating explanations of unrealizability of a high-level mission specification leveraging the information about the abstraction of the dynamics and the assumed behavior of the environment. They further develop a visualization tool to present these unrealizability certificates to a user who can then decide on potential revisions to the specification guided by their algorithm.

Bin Waez et al. revisit the controller synthesis paradigm in the context of timed automata which are finite-state automata equipped with clocks used to describe timing properties of software systems. Through a case study, Bin Waez et al. motivate the use of a variant to timed automata, termed timed process automata, and argue that this model offers two essential features for industrial systems: (i) compositional modeling with reusable designs for different contexts, and (ii) state-space reduction technique. In the context of process timed automata, they show how to reduce the verification of safety and reachability properties to solving timed games. In addition, the authors also discuss the use of compositional reasoning and aggressive abstractions as state-space reduction techniques.

Verification, the task of algorithmically generating correctness certificates, and falsification, the task of automatically identifying bugs, are crucial steps before a safety-critical control system can be deployed. Therefore, scalable verification and falsification techniques are of great potential. The paper by Tran et al. proposes balanced truncation as a means of reducing the dimensionality of high-dimensional linear systems to enable formal verification based on reachability. To this end, they provide methods for the computation of error bounds between the concrete system’s output and the reduced-order system’s output. These bounds are then used as margins to perform analysis on the reduced order system. The approach is evaluated on a number of computational benchmark problems. The paper by Rawlings and Ydstie focuses on discovering errors in the discrete logic controlling a hybrid plant when the requirements are given in a fragment of computational tree logic containing global safety and existential reachability specifications on the discrete states. For errors in the discrete logic the falsification problem is reduced to a supervisory control problem based on the discrete transition system induced by the discrete states of the overall hybrid system.

Most of the papers also include links to open-source software repositories where the software implementing the proposed algorithms can be accessed.