We introduce a linear extension of the rotor-routing model in directed graphs, akin to the sandpile model and vector addition systems, together with new rotor mechanisms that extend standard cyclic rotors. In this framework, rotor-routing is interpreted as the simultaneous movement of particles accross two coupled graphs, involving both vertex and arc-based particles. The standard, combinatorial rotor-routing of positive particles (legal routing) based on rotor-configurations, then becomes a special case of a linear equivalence. We give comprehensive reachability results characterizing legal routings among linear equivalences, expanding on previous results, and settle the algorithmic complexities associated with these problems.

Robust Markov Decision Processes (RMDPs) generalize classical MDPs that consider uncertainties in transition probabilities by defining a set of possible transition functions. An objective is a set of runs (or infinite trajectories) of the RMDP, and the value for an objective is the maximal probability that the agent can guarantee against the adversarial environment. We consider (a) reachability objectives, where given a target set of states, the goal is to eventually arrive at one of them; and (b) parity objectives, which are a canonical representation for ω-regular objectives. The qualitative analysis problem asks whether the objective can be ensured with probability 1. In this work, we study the qualitative problem for reachability and parity objectives on RMDPs without making any assumption over the structures of the RMDPs, e.g., unichain or aperiodic. Our contributions are twofold. We first present efficient algorithms with oracle access to uncertainty sets that solve qualitative problems of reachability and parity objectives. We then report experimental results demonstrating the effectiveness of our oracle-based approach on classical RMDP examples from the literature scaling up to thousands of states.

Inner and outer approximations of arbitrarily quantified reachability problems

Cut arcs , or strong bridges , are one of the most fundamental reachability notions in directed graphs. Specifically, in a strongly connected graph \(G=(V,E)\) ( \(|V|=n\) , \(|E|=m\) ), a cut arc is an arc \(e\in E\) for which there exist \(u,v\in V\) , such that all \(u\) - \(v\) walks contain \(e\) . In this paper we generalise this notion to cut paths , that is, walks \(W\) for which there exist \(u,v\in V\) , such that all \(u\) - \(v\) walks contain \(W\) as subwalk. We first prove various properties of cut paths and define their remainder structure , which we use to present a simple \(O(m)\) -time verification algorithm for a cut path. We further show that a graph contains at most \(O(n)\) maximal cut paths of length at most \(O(n)\) each, and present an optimal \(O(n^{2})\) enumeration algorithm for maximal cut paths. We apply cut paths and their remainder structure to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. Cut paths provide simple \(O(m)\) -time algorithms verifying if a walk is safe or multi-safe. Further, by simultaneous computation of remainder structures of all subwalks of a cut path in linear time, we can identify all maximal multi-safe walks in \(O(mn)\) time. This improves over the state-of-the-art algorithm running in time \(O(m^{2}+n^{3}\log n)\) .

. This paper concerns the problem of reachability of a given state for a multiagent control system in Rd. In such a system, at every time, each agent can choose their velocity which depends both on their position and on the position of the whole crowd of agents (modeled by a probability measure on Rd). The main contribution of the paper is to study the above reachability problem with a given rate of attainability through a Lyapunov method adapted to the Wasserstein space of probability measures. As a byproduct, we obtain a new comparison result for viscosity solutions of Hamilton Jacobi equations in the Wasserstein space.

Context sensitivity is a foundational technique in pointer analysis, critical and essential for improving precision but often incurring significant efficiency costs. Recent advances focus on selective context-sensitive analysis, where only a subset of program elements, such as methods or heap objects, are analyzed under context sensitivity while the rest are analyzed under context insensitivity, aiming to balance precision with efficiency. However, despite the proliferation of such approaches, existing methods are typically driven by specific code patterns, therefore lacking a comprehensive theoretical foundation for systematically identifying code scenarios that benefit from context sensitivity. This paper presents a novel and foundational theory that establishes a sound over-approximation of the ground truth, i.e., objects that really improve precision under context sensitivity. The proposed theory reformulates the identification of this upper bound into graph reachability problems over a typical Pointer Flow Graph (PFG), each of which can be efficiently solved under context insensitivity, respectively. Building on this theoretical foundation, we introduce our selective context-sensitive analysis approach, Moon. Moon performs both backward and forward traversal on a Variable Flow Graph (VFG), an optimized variant of PFG designed to facilitate efficient traversal. This traversal systematically identifies all objects that improve precision under context sensitivity. Our theoretical foundation, along with carefully designed trade-offs within our approach, allows Moon to limit the scope of objects to be selected, leading to an effective balance between its analysis precision and efficiency. Extensive experiments with Moon across 30 Java programs demonstrate that Moon achieves 37.2X and 382.0X speedups for 2-object-sensitive and 3-object-sensitive analyses, respectively with negligible precision losses of only 0.1% and 0.2%. These results highlight that the balance between efficiency and precision achieved by Moon significantly outperforms all previous approaches.

Nowadays, internet connectivity suffers from instability and slowness due to optical fiber cable attacks across the seas and oceans. The optimal solution to this problem is using the Low Earth Orbit (LEO) satellite network, which can resolve the problem of internet connectivity and reachability, and it has the power to bring real-time, reliable, low-latency, high-bandwidth, cost-effective internet access to many urban and rural areas in any region of the Earth. However, satellite orbital placement (SOP) and navigation should be carefully designed to reduce signal impairments. The challenges of orbital satellite placement for LEO include constellation development, satellite parameter optimization, bandwidth optimization, consideration of signal impairment, and coverage optimization. This paper presents a comprehensive review of SOP and coverage optimization, examines prevalent issues affecting LEO internet connectivity, evaluates existing solutions, and proposes novel solutions to address these challenges. Furthermore, it recommends a machine learning solution for coverage optimization and SOP that can be used to efficiently enhance internet reliability and reachability for LEO satellite networks. This survey will open the gate for developing an optimal solution for global internet connectivity and reachability.

Abstract reachability problems with constraints of asymptotic nature (CAN) are considered; for these problems, the result (analog of reachability set) is defined every time in the form of attraction set (AS) in topological space. The CANs themselves are generated by non-empty families of subsets of the original set of ordinary (available for implementation) solutions. Among these families, filters are distinguished: the family of all possible AS is realized by adding an empty set to a similar family of AS corresponding to each CAN generated by a filter; in addition, a unique attraction element is assigned to ultrafilters each time. This allows us to establish a number of important properties of the family of all AS generated by filters. So, in particular, it is established that given family is closed under finite unions; conditions are indicated under which a finite union of filters generates an AS of the above-mentioned family. A family of singletons, which are AS generated by filters, is indicated. The very appearance of non-empty non-singletonic AS can be interpreted in terms of the non-maximality of the filter generated the CAN: nonempty ASs that are not singletons necessarily correspond to CAN generated by filters that are not ultrafilters.

The decidability of the reachability problem for finitary PCF has been used as a theoretical basis for fully automated verification tools for functional programs. The reachability problem, however, often becomes undecidable for a slight extension of finitary PCF with side effects, such as exceptions, algebraic effects, and references, which hindered the extension of the above verification tools for supporting functional programs with side effects. In this paper, we first give simple proofs of the undecidability of four extensions of finitary PCF, which would help us understand and analyze the source of undecidability. We then focus on an extension with references, and give a decidable fragment using a type system. To our knowledge, this is the first non-trivial decidable fragment that features higher-order recursive functions containing reference cells.

Context-free language (CFL) reachability is a standard approach in static analyses, where the analysis question (e.g., is there a dataflow from x to y ?) is phrased as a language reachability problem on a graph G wrt a CFL L . However, CFLs lack the expressiveness needed for high analysis precision. On the other hand, common formalisms for context-sensitive languages are too expressive, in the sense that the corresponding reachability problem becomes undecidable. Are there useful context-sensitive language-reachability models for static analysis? In this paper, we introduce Multiple Context-Free Language (MCFL) reachability as an expressive yet tractable model for static program analysis. MCFLs form an infinite hierarchy of mildly context sensitive languages parameterized by a dimension d and a rank r . Larger d and r yield progressively more expressive MCFLs, offering tunable analysis precision. We showcase the utility of MCFL reachability by developing a family of MCFLs that approximate interleaved Dyck reachability, a common but undecidable static analysis problem. Given the increased expressiveness of MCFLs, one natural question pertains to their algorithmic complexity, i.e., how fast can MCFL reachability be computed? We show that the problem takes O ( n 2 d +1 ) time on a graph of n nodes when r =1, and O ( n d ( r +1) ) time when r >1. Moreover, we show that when r =1, even the simpler membership problem has a lower bound of n 2 d based on the Strong Exponential Time Hypothesis, while reachability for d =1 has a lower bound of n 3 based on the combinatorial Boolean Matrix Multiplication Hypothesis. Thus, for r =1, our algorithm is optimal within a factor n for all levels of the hierarchy based on the dimension d (and fully optimal for d =1). We implement our MCFL reachability algorithm and evaluate it by underapproximating interleaved Dyck reachability for a standard taint analysis for Android. When combined with existing overapproximate methods, MCFL reachability discovers all tainted information on 8 out of 11 benchmarks, while it has remarkable coverage (confirming 94.3% of the reachable pairs reported by the overapproximation) on the remaining 3. To our knowledge, this is the first report of high and provable coverage for this challenging benchmark set.

In this paper a computationally efficient solution is proposed for the reachable sequence computation problem in the context of sub and superconservative discrete state Chemical Reaction Network (CRN) subclasses. It is shown that a minimal-length reachable state transition sequence and reaction vector sequence can be determined in polynomial time, assuming that the reachability relation holds. A constructive proof is provided in which a computational procedure is composed to find a reachable state transition sequence, provided a pair of initial and target states. In the studied subclasses of discrete state CRNs it is known that the reachability problem - as a decision problem - can be decided in polynomial time by solving a Linear Program formulated by means of the CRN state equation. Leveraging the LP relaxation, the constructed procedure formulates the computational problem of feasible sequence calculation by iteratively solving Linear Programs related to the decision problem of CRN reachability.

We study a family of reachability problems under waiting-time restrictions in temporal and vertex-colored temporal graphs. Given a temporal graph and a set of source vertices, we find the set of vertices that are reachable from a source via a time-respecting path, where the difference in timestamps between consecutive edges is at most a resting time. Given a vertex-colored temporal graph and a multiset query of colors, we find the set of vertices reachable from a source via a time-respecting path such that the vertex colors of the path agree with the multiset query and the difference in timestamps between consecutive edges is at most a resting time. These kinds of problems have applications in understanding the spread of a disease in a network, tracing contacts in epidemic outbreaks, finding signaling pathways in the brain network, and recommending tours for tourists, among others. We present an algebraic algorithmic framework based on constrained multilinear sieving for solving the restless reachability problems we propose. In particular, parameterized by the length k of a path sought, we show that the proposed problems can be solved in O(2^k k m Delta ) time and O(n Delta ) space, where n is the number of vertices, m the number of edges, and Delta the maximum resting time of an input temporal graph. The approach can be extended to extract paths and connected subgraphs in both static and temporal graphs, thus improving the work of Björklund et al. (in Proceedings of the European symposium on algorithms, 2014) and Thejaswi et al. (Big Data 8:335–362, 2020). In addition, we prove that our algorithms for the restless reachability problems in vertex-colored temporal graphs are optimal under plausible complexity-theoretic assumptions. Finally, with an open-source implementation, we demonstrate that our algorithm scales to large graphs with up to one billion temporal edges, despite the problems being NP-hard. Specifically, we present extensive experiments to evaluate our scalability claims both on synthetic and on real-world graphs. Our implementation is efficiently engineered and highly optimized. For instance, we can solve the restless reachability problem by restricting the path length to 9 in a real-world graph dataset with over 36 million directed edges in less than one hour on a commodity desktop with a 4-core Haswell CPU.

Legged robots face significant challenges in navigating complex environments, as they require precise real-time decisions for foothold selection and contact planning. While existing research has explored methods to select footholds based on terrain geometry or kinematics, a critical gap remains: few existing methods efficiently validate the existence of a non-collision swing trajectory. This article addresses this gap by introducing KCFRC, a novel approach for efficient foothold reachability analysis. We first formally define the foothold reachability problem and establish a sufficient condition for foothold reachability. Based on this condition, we develop the KCFRC algorithm, which enables robots to validate foothold reachability in real time. Our experimental results demonstrate that KCFRC achieves remarkable time efficiency, completing foothold reachability checks for a single leg across 900 potential footholds in an average of 2 ms. Furthermore, we show that KCFRC can accelerate trajectory optimization and is particularly beneficial for contact planning in confined spaces, enhancing the adaptability and robustness of legged robots in challenging environments.

ABSTRACTThis paper presents a comprehensive approach for achieving multi‐task safe optimal adaptive tracking (MSOAT) for a class of nonlinear discrete‐time systems, particularly those in strict‐feedback form, utilizing a multi‐layer neural network (MNN)‐based framework. To begin, a cost function with a novel Barrier function (BF) term is introduced for each subsystem to address the weak safely reachable problem, serving as a crucial tool for guiding the system's trajectory toward the safe set while avoiding unwanted sets. To deal with the tracking problem, the Hamilton‐Jacobi‐Bellman (HJB) framework is used through the actor‐critic MNN‐based backstepping technique to estimate the solution of the value functions and obtain both virtual and actual optimal control policies for each subsystem, effectively circumventing non‐causality issues. Further, to mitigate catastrophic forgetting in multi‐tasking scenarios, a regularizer term, which is derived from the online version of the Elastic Weight Consolidation (EWC) method, is included in the critic and actor MNN update laws without directly computing the Fisher information matrix. To enhance the convergence rate, the critic MNN is tuned with a hybrid learning technique involving weight adjustments both at specific sampling instants and iteratively within those intervals. A control barrier function (CBF) with a time‐varying BF is also integrated into the actor update law, collaborating with the BF to keep the trajectory in the safe set with a smaller trade‐off factor, simultaneously validating the safety condition in real‐time. Finally, the overall stability is established. An example of a 6‐DOF autonomous underwater vehicle (AUV) is used to assess the effectiveness of the proposed approach.

A set of configurations $H$ is a home-space for a set of configurations $X$ of aPetri net if every configuration reachable from (any configuration in) $X$ can reach (some configuration in) $H$. The semilinear home-space problem for Petri nets asks, given a Petri net and semilinear sets of configurations $X$, $H$, if $H$ is a home-space for $X$. In 1989, David de Frutos Escrig and Colette Johnen proved that the problem is decidable when $X$ is a singleton and $H$ is a finite union of linear sets with the same periods. In this paper, we show that the general (semilinear) problem is decidable. This result is obtained by proving a duality between the reachability problem and the non-home-space problem. In particular, we prove that for any Petri net and any semilinear set of configurations $H$ we can effectively compute a semilinear set $C$ of configurations, called a non-reachability core for $H$, such that for every set $X$ the set $H$ is not a home-space for $X$ if, and only if, $C$ is reachable from $X$. We show that the established relation to the reachability problem yields the Ackermann-completeness of the (semilinear) home-space problem. For this we also show that, given a Petri net with an initial marking, the set of minimal reachable markings can be constructed in Ackermannian time.

Questions related to the extension of reachability problems and aimed at the construction of attraction sets, which are asymptotic analogs of reachable sets in the situation of successive relaxation of the constraint system, are studied. Finitely additive measures with the property of weak absolute continuity with respect to a fixed measure are used as generalized elements; the measure (in the case of control problems) is usually defined as the restriction of the Lebesgue measure to some family of measurable sets. The properties of relaxed reachability problems and the connection of their extensions with attraction sets in the class of ordinary solutions (controls), as well as the properties of these sets that have the sense of stability when the constraints are relaxed and asymptotic insensitivity when some “part” of the constraints is relaxed, are studied.

Multi-agent reinforcement learning with synchronized and decomposed reward automaton synthesized from reactive temporal logic

This article introduces a new method for modelling and verifying the execution of timed security protocols (TSPs) and their time-dependent security properties. The method, which is novel and reliable, uses an extension of interpreted systems, accessible semantics in multi-agent systems, and timed interpreted systems (TISs) with dense time semantics to model TSP executions. We enhance the models of TSPs by incorporating delays and varying lifetimes to capture real-life aspects of protocol executions. To illustrate the method, we model a timed version of the Needham–Schroeder Public Key Authentication Protocol. We have also developed a new bounded model checking reachability algorithm for the proposed structures, based on Satisfiability Modulo Theories (SMTs), and implemented it within the tool. The method comprises a new procedure for modelling TSP executions, translating TSPs into TISs, and translating TISs’ reachability problem into the SMT problem. The paper also includes thorough experimental results for nine protocols modelled by TISs and discusses the findings in detail.

We propose an automated procedure to prove polyhedral abstractions (also known as polyhedral reductions) for Petri nets. Polyhedral abstraction is a new type of state space equivalence, between Petri nets, based on the use of linear integer constraints between the marking of places. In addition to defining an automated proof method, this paper aims to better characterize polyhedral reductions, and to give an overview of their application to reachability problems. Our approach relies on encoding the equivalence problem into a set of SMT formulas whose satisfaction implies that the equivalence holds. The difficulty, in this context, arises from the fact that we need to handle infinite-state systems. For completeness, we exploit a connection with a class of Petri nets, called flat nets, that have Presburger-definable reachability sets. We have implemented our procedure, and we illustrate its use on several examples.

Leader-follower control and APF for Multi-USV coordination and obstacle avoidance