Aczel-Mendler bisimulations are a coalgebraic extension of a variety of computational relations between systems. It is usual to assume that the underlying category satisfies some form of the axiom of choice, so that the collection of bisimulations enjoys desirable properties, such as closure under composition. In this paper, we accommodate the definition in general regular categories and toposes. We show that this general definition: 1) is closed under composition without using the axiom of choice, 2) coincides with other types of coalgebraic formulations under milder conditions, 3) coincides with the usual definition when the category satisfies the regular axiom of choice. In particular, the case of toposes heavily relies on power-objects, for which we recover some favourable properties along the way. Finally, we describe several examples in Stone spaces, toposes for name-passing, and modules over a ring.

We introduce a goal-aware extension of responsibility-sensitive safety (RSS), a recent methodology for rule-based safety guarantee for automated driving systems (ADS). Making RSS rules guarantee goal achievement—in addition to collision avoidance as in the original RSS—requires complex planning over long sequences of manoeuvres. To deal with the complexity, we introduce a compositional reasoning framework based on program logic, in which one can systematically develop RSS rules for smaller subscenarios and combine them to obtain RSS rules for bigger scenarios. As the basis of the framework, we introduce a program logic dFHL that accommodates continuous dynamics and safety conditions. Our framework presents a dFHL-based workflow for deriving goal-aware RSS rules; we discuss its software support, too. We conducted experimental evaluation using RSS rules in a safety architecture. Its results show that goal-aware RSS is indeed effective in realising both collision avoidance and goal achievement.

Aczel-Mendler bisimulations are a coalgebraic extension of a variety of computational relations between systems. It is usual to assume that the underlying category satisfies some form of the axiom of choice, so that the collection of bisimulations enjoys desirable properties, such as closure under composition. In this paper, we accommodate the definition in general regular categories and toposes. We show that this general definition: 1) is closed under composition without using the axiom of choice, 2) coincides with other types of coalgebraic formulations under milder conditions, 3) coincides with the usual definition when the category satisfies the regular axiom of choice. In particular, the case of toposes heavily relies on power-objects, for which we recover some favourable properties along the way. Finally, we describe several examples in Stone spaces, toposes for name-passing, and modules over a ring.

Abstract In the open map approach to bisimilarity, the paths and their runs in a given state-based system are the first-class citizens, and bisimilarity becomes a derived notion. While open maps were successfully used to model bisimilarity in non-deterministic systems, the approach fails to describe quantitative system equivalences such as probabilistic bisimilarity. In the present work, we see that this is indeed impossible and we thus generalize the notion of open maps to also accommodate weighted and probabilistic bisimilarity. Also, extending the notions of strong path and path bisimulations into this new framework, we show that branching bisimilarity can be captured by this extended theory and that it can be viewed as the history preserving restriction of weak bisimilarity.

Safety architectures play a crucial role in the safety assurance of automated driving vehicles (ADVs). They can be used as safety envelopes of black-box ADV controllers, and for graceful degradation from one ODD to another. Building on our previous work on the formalization of responsibility-sensitive safety (RSS), we introduce a novel program logic that accommodates assume-guarantee reasoning and fallback-like constructs. This allows us to formally define and prove the safety of existing and novel safety architectures. We apply the logic to a pull over scenario and experimentally evaluate the resulting safety architecture.

In this paper, we develop an Isabelle/HOL library of order-theoretic fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often with only antisymmetry or attractivity, a mild condition implied by either antisymmetry or transitivity. In particular, we generalize various theorems ensuring the existence of a quasi-fixed point of monotone maps over complete relations, and show that the set of (quasi-)fixed points is itself complete. This result generalizes and strengthens theorems of Knaster-Tarski, Bourbaki-Witt, Kleene, Markowsky, Pataraia, Mashburn, Bhatta-George, and Stouti-Maaden.

Abstract Bisimulation and bisimilarity are fundamental notions in comparing state-based systems. Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions. In this paper we enhance a categorical framework for such extended (bi)simulation notions. We use coalgebras as system models and fibrations for organizing predicates—following the seminal work by Hermida and Jacobs. Endofunctor liftings are crucial predicate-forming ingredients; the first contribution of this work is to extend several extant lifting techniques from particular fibrations to $\textbf {CLat}_\wedge $-fibrations over $\textbf {Set}$. The second contribution of this work is to introduce endolifting morphisms as a mechanism for comparing predicates between fibrations. We apply these techniques by deriving some known properties of the Hausdorff pseudometric and approximate bisimulation in control theory.

We propose a symbolic self-triggered controller synthesis procedure for non-deterministic continuous-time nonlinear systems without stability assumptions. The goal is to compute a controller that satisfies two objectives. The first objective is represented as a specification in a fragment of LTL, which we call 2-LTL. The second one is an energy objective, in the sense that control inputs are issued only when necessary, which saves energy. To this end, we first quantise the state and input spaces, and then translate the controller synthesis problem to the computation of a winning strategy in a mean-payoff parity game. We illustrate the feasibility of our method on the example of a navigating nonholonomic robot.

In the field of quality assurance of hybrid systems, Platzer’s differential dynamic logic (dL) is widely recognized as a deductive verification method with solid mathematical foundations and sophisticated tool support. Motivated by case studies provided by our industry partner, we study a relational extension of dL, aiming to formally prove statements such as “an earlier engagement of the emergency brake yields a smaller collision speed.” A main technical challenge is to combine two dynamics, so that the powerful inference rules of dL (such as the differential invariant rules) can be applied to such relational reasoning, yet in such a way that we relate two different time points. Our contributions are a semantical theory of time stretching , and the resulting synchronization rule that expresses time stretching by the syntactic operation of Lie derivative. We implemented this rule as an extension of KeYmaera X , by which we successfully verified relational properties of a few models taken from the automotive domain.

Abstract In this paper, we give a new definition of partial Higher Dimension Automata using lax functors. This definition is simpler and more natural from a categorical point of view, but also matches more clearly the intuition that pHDA are Higher Dimensional Automata with some missing faces. We then focus on trees. Originally, for example in transition systems, trees are defined as those systems that have a unique path property. To understand what kind of unique property is needed in pHDA, we start by looking at trees as colimits of paths. This definition tells us that trees are exactly the pHDA with the unique path property modulo a notion of homotopy, and without any shortcuts. This property allows us to prove two interesting characterisations of trees: trees are exactly those pHDA that are an unfolding of another pHDA; and trees are exactly the cofibrant objects, much as in the language of Quillen’s model structure. In particular, this last characterisation gives the premisses of a new understanding of concurrency theory using homotopy theory.

Abstract There are different categorical approaches to variations of transition systems and their bisimulations. One is coalgebra for a functor G, where a bisimulation is defined as a span of G-coalgebra homomorphism. Another one is in terms of path categories and open morphisms, where a bisimulation is defined as a span of open morphisms. This similarity is no coincidence: given a functor G, fulfilling certain conditions, we derive a path-category for pointed G-coalgebras and lax homomorphisms, such that the open morphisms turn out to be precisely the G-coalgebra homomorphisms. The above construction provides path-categories and trace semantics for free for different flavours of transition systems: (1) non-deterministic tree automata (2) regular nondeterministic nominal automata (RNNA), an expressive automata notion living in nominal sets (3) multisorted transition systems. This last instance relates to Lasota’s construction, which is in the converse direction.

Hybrid Systems and their Verification. With the ever increasing degree of digitalisation and automation, cyber-physical systems (CPS) are becoming exceedingly common in industry. This trend is accompanied by a similar increase in the research efforts directed towards CPS. The biggest concern is sparked by many safety-critical applications involving CPS, such as automated driving. The quality assurance of CPS thus poses a pressing socio-economical challenge.

In this paper, we develop an Isabelle/HOL library of order-theoretic concepts, such as various completeness conditions and fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often without any property of ordering, thus complete non-orders. In particular, we generalize the Knaster - Tarski theorem so that we ensure the existence of a quasi-fixed point of monotone maps over complete non-orders, and show that the set of quasi-fixed points is complete under a mild condition - attractivity - which is implied by either antisymmetry or transitivity. This result generalizes and strengthens a result by Stauti and Maaden. Finally, we recover Kleene’s fixed-point theorem for omega-complete non-orders, again using attractivity to prove that Kleene’s fixed points are least quasi-fixed points.

In this paper, we propose a categorical framework for bisimulations and unfoldings that unifies the classical approach from Joyal and al. via open maps and unfoldings. This is based on a notion of categories accessible with respect to a subcategory of path shapes, i.e., for which one can define a nice notion of trees as glueing of paths. We prove that transitions systems and pre sheaf models are a particular case of our framework. We also prove that in our framework, several characterizations of bisimulation coincide, in particular an "operational one" akin to the standard definition in transition systems. Also, accessibility is preserved by coreflexions. We then design a notion of unfolding, which has good properties in the accessible case: its is a right adjoint and is a universal covering, i.e., initial among the morphisms that have the unique lifting property with respect to path shapes. As an application, we prove that the universal covering of a groupoid, a standard construction in algebraic topology, coincides with an unfolding, when the category of path shapes is well chosen.

The homotopy hypothesis was originally stated by Grothendieck: topological spaces should be "equivalent" to (weak) infinite-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g., for rewriting, distributed systems, (homotopy) type theory etc. But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis' directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (infinite,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology.