Guillermo A. Pérez

Dennis Gross, Nils Jansen, Sebastian Junges et al.

This paper presents COOL-MC, a tool that integrates state-of-the-art reinforcement learning (RL) and model checking. Specifically, the tool builds upon the OpenAI gym and the probabilistic model checker Storm. COOL-MC provides the following features: (1) a simulator to train RL policies in the OpenAI gym for Markov decision processes (MDPs) that are defined as input for Storm, (2) a new model builder for Storm, which uses callback functions to verify (neural network) RL policies, (3) formal abstractions that relate models and policies specified in OpenAI gym or Storm, and (4) algorithms to obtain bounds on the performance of so-called permissive policies. We describe the components and architecture of COOL-MC and demonstrate its features on multiple benchmark environments.

Raphael Avalos, Florent Delgrange, Ann Nowé et al.

Partially Observable Markov Decision Processes (POMDPs) are used to model environments where the full state cannot be perceived by an agent. As such the agent needs to reason taking into account the past observations and actions. However, simply remembering the full history is generally intractable due to the exponential growth in the history space. Maintaining a probability distribution that models the belief over what the true state is can be used as a sufficient statistic of the history, but its computation requires access to the model of the environment and is often intractable. While SOTA algorithms use Recurrent Neural Networks to compress the observation-action history aiming to learn a sufficient statistic, they lack guarantees of success and can lead to sub-optimal policies. To overcome this, we propose the Wasserstein Belief Updater, an RL algorithm that learns a latent model of the POMDP and an approximation of the belief update. Our approach comes with theoretical guarantees on the quality of our approximation ensuring that our outputted beliefs allow for learning the optimal value function.

Dennis Gross, Thiago D. Simao, Nils Jansen et al.

Deep Reinforcement Learning (RL) agents are susceptible to adversarial noise in their observations that can mislead their policies and decrease their performance. However, an adversary may be interested not only in decreasing the reward, but also in modifying specific temporal logic properties of the policy. This paper presents a metric that measures the exact impact of adversarial attacks against such properties. We use this metric to craft optimal adversarial attacks. Furthermore, we introduce a model checking method that allows us to verify the robustness of RL policies against adversarial attacks. Our empirical analysis confirms (1) the quality of our metric to craft adversarial attacks against temporal logic properties, and (2) that we are able to concisely assess a system's robustness against attacks.

Debraj Chakraborty, Damien Busatto-Gaston, Jean-François Raskin et al.

We study how to efficiently combine formal methods, Monte Carlo Tree Search (MCTS), and deep learning in order to produce high-quality receding horizon policies in large Markov Decision processes (MDPs). In particular, we use model-checking techniques to guide the MCTS algorithm in order to generate offline samples of high-quality decisions on a representative set of states of the MDP. Those samples can then be used to train a neural network that imitates the policy used to generate them. This neural network can either be used as a guide on a lower-latency MCTS online search, or alternatively be used as a full-fledged policy when minimal latency is required. We use statistical model checking to detect when additional samples are needed and to focus those additional samples on configurations where the learnt neural network policy differs from the (computationally-expensive) offline policy. We illustrate the use of our method on MDPs that model the Frozen Lake and Pac-Man environments -- two popular benchmarks to evaluate reinforcement-learning algorithms.

Ritam Raha, Rajarshi Roy, Nathanael Fijalkow et al.

In runtime verification, manually formalizing a specification for monitoring system executions is a tedious and error-prone process. To address this issue, we consider the problem of automatically synthesizing formal specifications from system executions. To demonstrate our approach, we consider the popular specification language Metric Temporal Logic (MTL), which is particularly tailored towards specifying temporal properties for cyber-physical systems (CPS). Most of the classical approaches for synthesizing temporal logic formulas aim at minimizing the size of the formula. However, for efficiency in monitoring, along with the size, the amount of "lookahead" required for the specification becomes relevant, especially for safety-critical applications. We formalize this notion and devise a learning algorithm that synthesizes concise formulas having bounded lookahead. To do so, our algorithm reduces the synthesis task to a series of satisfiability problems in Linear Real Arithmetic (LRA) and generates MTL formulas from their satisfying assignments. The reduction uses a novel encoding of a popular MTL monitoring procedure using LRA. Finally, we implement our algorithm in a tool called TEAL and demonstrate its ability to synthesize efficiently monitorable MTL formulas in a CPS application.

Swen Jacobs, Guillermo A. Perez, Philipp Schlehuber-Caissier

We present an extension of the Temporal Logic Synthesis Format (TLSF). TLSF builds on standard LTL, but additionally supports high-level constructs, such as sets and functions, as well as parameters that allow a specification to define a whole a family of problems. Our extension introduces operators and a new semantics option for LTLf, i.e., LTL on finite executions.

Nathanaël Fijalkow, Arka Ghosh, Roman Kniazev et al.

Partially observable Markov decision processes (POMDPs) are a fundamental model for sequential decision-making under uncertainty. However, many verification and synthesis problems for POMDPs are undecidable or intractable. Most prominently, the seminal result of Madani et al. (2003) states that there is no algorithm that, given a POMDP and a set of target states, can compute the maximal probability of reaching the target states, or even approximate it up to a non-trivial constant. This is in stark contrast to fully observable Markov decision processes (MDPs), where the reachability value can be computed in polynomial time. In this work, we introduce posterior-deterministic POMDPs, a novel class of POMDPs. Our main technical contribution is to show that for posterior-deterministic POMDPs, the maximal probability of reaching a given set of states can be approximated up to arbitrary precision. A POMDP is posterior-deterministic if the next state can be uniquely determined by the current state, the action taken, and the observation received. While the actual state is generally uncertain in POMDPs, the posterior-deterministic property tells us that once the true state is known it remains known forever. This simple and natural definition includes all MDPs and captures classical non-trivial examples such as the Tiger POMDP (Kaelbling et al. 1998), making it one of the largest known classes of POMDPs for which the reachability value can be approximated.

Florent Delgrange, Ann Nowé, Guillermo A. Pérez

Although deep reinforcement learning (DRL) has many success stories, the large-scale deployment of policies learned through these advanced techniques in safety-critical scenarios is hindered by their lack of formal guarantees. Variational Markov Decision Processes (VAE-MDPs) are discrete latent space models that provide a reliable framework for distilling formally verifiable controllers from any RL policy. While the related guarantees address relevant practical aspects such as the satisfaction of performance and safety properties, the VAE approach suffers from several learning flaws (posterior collapse, slow learning speed, poor dynamics estimates), primarily due to the absence of abstraction and representation guarantees to support latent optimization. We introduce the Wasserstein auto-encoded MDP (WAE-MDP), a latent space model that fixes those issues by minimizing a penalized form of the optimal transport between the behaviors of the agent executing the original policy and the distilled policy, for which the formal guarantees apply. Our approach yields bisimulation guarantees while learning the distilled policy, allowing concrete optimization of the abstraction and representation model quality. Our experiments show that, besides distilling policies up to 10 times faster, the latent model quality is indeed better in general. Moreover, we present experiments from a simple time-to-failure verification algorithm on the latent space. The fact that our approach enables such simple verification techniques highlights its applicability.

Kévin Dubrulle, Véronique Bruyère, Guillermo A. Pérez et al.

As an alternative to visibly pushdown automata, we introduce visibly recursive automata (VRAs), composed of a set of classical automata that can call each other. VRAs are a strict extension of so-called systems of procedural automata, a model proposed by Frohme and Steffen. We study the complexity of standard language-theoretic operations and classical decision problems for VRAs. Since the class of deterministic VRAs forms a strict subclass in terms of expressiveness, we propose a (weaker) notion that does not restrict expressive power and which we call codeterminism. Codeterminism comes with many desirable algorithmic properties that we demonstrate by using it, e.g., as a stepping stone towards implementing complementation of VRAs.

Véronique Bruyère, Bharat Garhewal, Guillermo A. Pérez et al.

We present the first algorithm for query learning Mealy machines with timers in a black-box context. Our algorithm is an extension of the L# algorithm of Vaandrager et al. to a timed setting. We rely on symbolic queries which empower us to reason on untimed executions while learning. Similarly to the algorithm for learning timed automata of Waga, these symbolic queries can be realized using finitely many concrete queries. Experiments with a prototype implementation show that our algorithm is able to efficiently learn realistic benchmarks.

Florent Delgrange, Guy Avni, Anna Lukina et al.

We propose a novel framework to controller design in environments with a two-level structure: a known high-level graph ("map") in which each vertex is populated by a Markov decision process, called a "room". The framework "separates concerns" by using different design techniques for low- and high-level tasks. We apply reactive synthesis for high-level tasks: given a specification as a logical formula over the high-level graph and a collection of low-level policies obtained together with "concise" latent structures, we construct a "planner" that selects which low-level policy to apply in each room. We develop a reinforcement learning procedure to train low-level policies on latent structures, which unlike previous approaches, circumvents a model distillation step. We pair the policy with probably approximately correct guarantees on its performance and on the abstraction quality, and lift these guarantees to the high-level task. These formal guarantees are the main advantage of the framework. Other advantages include scalability (rooms are large and their dynamics are unknown) and reusability of low-level policies. We demonstrate feasibility in challenging case studies where an agent navigates environments with moving obstacles and visual inputs.

Kasper Engelen, Guillermo A. Pérez, Marnix Suilen

Safe policy improvement (SPI) is an offline reinforcement learning problem in which a new policy that reliably outperforms the behavior policy with high confidence needs to be computed using only a dataset and the behavior policy. Markov decision processes (MDPs) are the standard formalism for modeling environments in SPI. In many applications, additional information in the form of parametric dependencies between distributions in the transition dynamics is available. We make SPI more data-efficient by leveraging these dependencies through three contributions: (1) a parametric SPI algorithm that exploits known correlations between distributions to more accurately estimate the transition dynamics using the same amount of data; (2) a preprocessing technique that prunes redundant actions from the environment through a game-based abstraction; and (3) a more advanced preprocessing technique, based on satisfiability modulo theory (SMT) solving, that can identify more actions to prune. Empirical results and an ablation study show that our techniques increase the data efficiency of SPI by multiple orders of magnitude while maintaining the same reliability guarantees.

Kasper Engelen, Guillermo A. Pérez, Shrisha Rao

Parametric Markov chains (pMC) are used to model probabilistic systems with unknown or partially known probabilities. Although (universal) pMC verification for reachability properties is known to be coETR-complete, there have been efforts to approach it using potentially easier-to-check properties such as asking whether the pMC is monotonic in certain parameters. In this paper, we first reduce monotonicity to asking whether the reachability probability from a given state is never less than that of another given state. Recent results for the latter property imply an efficient algorithm to collapse same-value equivalence classes, which in turn preserves verification results and monotonicity. We implement our algorithm to collapse "trivial" equivalence classes in the pMC and show empirical evidence for the following: First, the collapse gives reductions in size for some existing benchmarks and significant reductions on some custom benchmarks; Second, the collapse speeds up existing algorithms to check monotonicity and parameter lifting, and hence can be used as a fast pre-processing step in practice.

Kasper Engelen, Guillermo A. Pérez, Shrisha Rao

We study the complexity of reductions for weighted reachability in parametric Markov decision processes. That is, we say a state p is never worse than q if for all valuations of the polynomial indeterminates it is the case that the maximal expected weight that can be reached from p is greater than the same value from q. In terms of computational complexity, we establish that determining whether p is never worse than q is coETR-complete. On the positive side, we give a polynomial-time algorithm to compute the equivalence classes of the order we study for Markov chains. Additionally, we describe and implement two inference rules to under-approximate the never-worse relation and empirically show that it can be used as an efficient preprocessing step for the analysis of large Markov decision processes.

Florent Delgrange, Ann Nowé, Guillermo A. Pérez

We consider the challenge of policy simplification and verification in the context of policies learned through reinforcement learning (RL) in continuous environments. In well-behaved settings, RL algorithms have convergence guarantees in the limit. While these guarantees are valuable, they are insufficient for safety-critical applications. Furthermore, they are lost when applying advanced techniques such as deep-RL. To recover guarantees when applying advanced RL algorithms to more complex environments with (i) reachability, (ii) safety-constrained reachability, or (iii) discounted-reward objectives, we build upon the DeepMDP framework introduced by Gelada et al. to derive new bisimulation bounds between the unknown environment and a learned discrete latent model of it. Our bisimulation bounds enable the application of formal methods for Markov decision processes. Finally, we show how one can use a policy obtained via state-of-the-art RL to efficiently train a variational autoencoder that yields a discrete latent model with provably approximately correct bisimulation guarantees. Additionally, we obtain a distilled version of the policy for the latent model.

Guillermo A. Perez

In these notes we will tackle the problem of finding optimal policies for Markov decision processes (MDPs) which are not fully known to us. Our intention is to slowly transition from an offline setting to an online (learning) setting. Namely, we are moving towards reinforcement learning.

Raphaël Berthon, Adrien Boiret, Guillermo A. Perez et al.

Active learning is a setting in which a student queries a teacher, through membership and equivalence queries, in order to learn a language. Performance on these algorithms is often measured in the number of queries required to learn a target, with an emphasis on costly equivalence queries. In graybox learning, the learning process is accelerated by foreknowledge of some information on the target. Here, we consider graybox active learning of subsequential string transducers, where a regular overapproximation of the domain is known by the student. We show that there exists an algorithm using string equation solvers that uses this knowledge to learn subsequential string transducers with a better guarantee on the required number of equivalence queries than classical active learning.

Damien Busatto-Gaston, Debraj Chakraborty, Shibashis Guha et al.

In this paper, we investigate the combination of synthesis, model-based learning, and online sampling techniques to obtain safe and near-optimal schedulers for a preemptible task scheduling problem. Our algorithms can handle Markov decision processes (MDPs) that have 1020 states and beyond which cannot be handled with state-of-the art probabilistic model-checkers. We provide probably approximately correct (PAC) guarantees for learning the model. Additionally, we extend Monte-Carlo tree search with advice, computed using safety games or obtained using the earliest-deadline-first scheduler, to safely explore the learned model online. Finally, we implemented and compared our algorithms empirically against shielded deep Q-learning on large task systems.

Dennis Gross, Nils Jansen, Guillermo A. Pérez et al.

We give a formal verification procedure that decides whether a classifier ensemble is robust against arbitrary randomized attacks. Such attacks consist of a set of deterministic attacks and a distribution over this set. The robustness-checking problem consists of assessing, given a set of classifiers and a labelled data set, whether there exists a randomized attack that induces a certain expected loss against all classifiers. We show the NP-hardness of the problem and provide an upper bound on the number of attacks that is sufficient to form an optimal randomized attack. These results provide an effective way to reason about the robustness of a classifier ensemble. We provide SMT and MILP encodings to compute optimal randomized attacks or prove that there is no attack inducing a certain expected loss. In the latter case, the classifier ensemble is provably robust. Our prototype implementation verifies multiple neural-network ensembles trained for image-classification tasks. The experimental results using the MILP encoding are promising both in terms of scalability and the general applicability of our verification procedure.

Floris Geerts, Filip Mazowiecki, Guillermo A. Pérez

In this paper we cast neural networks defined on graphs as message-passing neural networks (MPNNs) in order to study the distinguishing power of different classes of such models. We are interested in whether certain architectures are able to tell vertices apart based on the feature labels given as input with the graph. We consider two variants of MPNNS: anonymous MPNNs whose message functions depend only on the labels of vertices involved; and degree-aware MPNNs in which message functions can additionally use information regarding the degree of vertices. The former class covers a popular formalisms for computing functions on graphs: graph neural networks (GNN). The latter covers the so-called graph convolutional networks (GCNs), a recently introduced variant of GNNs by Kipf and Welling. We obtain lower and upper bounds on the distinguishing power of MPNNs in terms of the distinguishing power of the Weisfeiler-Lehman (WL) algorithm. Our results imply that (i) the distinguishing power of GCNs is bounded by the WL algorithm, but that they are one step ahead; (ii) the WL algorithm cannot be simulated by "plain vanilla" GCNs but the addition of a trade-off parameter between features of the vertex and those of its neighbours (as proposed by Kipf and Welling themselves) resolves this problem.

Michaël Cadilhac, Guillermo A. Pérez, Marie van den Bogaard

Discounted-sum games provide a formal model for the study of reinforcement learning, where the agent is enticed to get rewards early since later rewards are discounted. When the agent interacts with the environment, she may regret her actions, realizing that a previous choice was suboptimal given the behavior of the environment. The main contribution of this paper is a PSPACE algorithm for computing the minimum possible regret of a given game. To this end, several results of independent interest are shown. (1) We identify a class of regret-minimizing and admissible strategies that first assume that the environment is collaborating, then assume it is adversarial---the precise timing of the switch is key here. (2) Disregarding the computational cost of numerical analysis, we provide an NP algorithm that checks that the regret entailed by a given time-switching strategy exceeds a given value. (3) We show that determining whether a strategy minimizes regret is decidable in PSPACE.

Nikhil Balaji, Stefan Kiefer, Petr Novotný et al.

Value iteration is a fundamental algorithm for solving Markov Decision Processes (MDPs). It computes the maximal $n$-step payoff by iterating $n$ times a recurrence equation which is naturally associated to the MDP. At the same time, value iteration provides a policy for the MDP that is optimal on a given finite horizon $n$. In this paper, we settle the computational complexity of value iteration. We show that, given a horizon $n$ in binary and an MDP, computing an optimal policy is EXP-complete, thus resolving an open problem that goes back to the seminal 1987 paper on the complexity of MDPs by Papadimitriou and Tsitsiklis. As a stepping stone, we show that it is EXP-complete to compute the $n$-fold iteration (with $n$ in binary) of a function given by a straight-line program over the integers with $\max$ and $+$ as operators.

Jan Křetínský, Guillermo A. Pérez, Jean-François Raskin

We formalize the problem of maximizing the mean-payoff value with high probability while satisfying a parity objective in a Markov decision process (MDP) with unknown probabilistic transition function and unknown reward function. Assuming the support of the unknown transition function and a lower bound on the minimal transition probability are known in advance, we show that in MDPs consisting of a single end component, two combinations of guarantees on the parity and mean-payoff objectives can be achieved depending on how much memory one is willing to use. (i) For all $ε$ and $γ$ we can construct an online-learning finite-memory strategy that almost-surely satisfies the parity objective and which achieves an $ε$-optimal mean payoff with probability at least $1 - γ$. (ii) Alternatively, for all $ε$ and $γ$ there exists an online-learning infinite-memory strategy that satisfies the parity objective surely and which achieves an $ε$-optimal mean payoff with probability at least $1 - γ$. We extend the above results to MDPs consisting of more than one end component in a natural way. Finally, we show that the aforementioned guarantees are tight, i.e. there are MDPs for which stronger combinations of the guarantees cannot be ensured.

Stephane Le Roux, Guillermo A. Perez

We study the never-worse relation (NWR) for Markov decision processes with an infinite-horizon reachability objective. A state q is never worse than a state p if the maximal probability of reaching the target set of states from p is at most the same value from q, regard- less of the probabilities labelling the transitions. Extremal-probability states, end components, and essential states are all special cases of the equivalence relation induced by the NWR. Using the NWR, states in the same equivalence class can be collapsed. Then, actions leading to sub- optimal states can be removed. We show the natural decision problem associated to computing the NWR is coNP-complete. Finally, we ex- tend a previously known incomplete polynomial-time iterative algorithm to under-approximate the NWR.

Krishnendu Chatterjee, Petr Novotný, Guillermo A. Pérez et al.

A standard objective in partially-observable Markov decision processes (POMDPs) is to find a policy that maximizes the expected discounted-sum payoff. However, such policies may still permit unlikely but highly undesirable outcomes, which is problematic especially in safety-critical applications. Recently, there has been a surge of interest in POMDPs where the goal is to maximize the probability to ensure that the payoff is at least a given threshold, but these approaches do not consider any optimization beyond satisfying this threshold constraint. In this work we go beyond both the "expectation" and "threshold" approaches and consider a "guaranteed payoff optimization (GPO)" problem for POMDPs, where we are given a threshold $t$ and the objective is to find a policy $σ$ such that a) each possible outcome of $σ$ yields a discounted-sum payoff of at least $t$, and b) the expected discounted-sum payoff of $σ$ is optimal (or near-optimal) among all policies satisfying a). We present a practical approach to tackle the GPO problem and evaluate it on standard POMDP benchmarks.