Kaushik Mallik

Kyle Hsu, Rupak Majumdar, Kaushik Mallik et al.

We present a lazy version of multi-layered abstraction-based controller synthesis (ABCS) for continuous-time nonlinear dynamical systems against safety specifications. State-of-the-art multi-layered ABCS uses pre-computed finite-state abstractions of different coarseness. Our new algorithm improves this technique by computing transitions on-the-fly, and only when a particular region of the state space needs to be explored by the controller synthesis algorithm for a specific coarseness. Additionally, our algorithm improves upon existing techniques by using coarser cells on a larger subset of the state space, which leads to significant computational savings.

Thomas A. Henzinger, Konstantin Kueffner, Kaushik Mallik

As AI and machine-learned software are used increasingly for making decisions that affect humans, it is imperative that they remain fair and unbiased in their decisions. To complement design-time bias mitigation measures, runtime verification techniques have been introduced recently to monitor the algorithmic fairness of deployed systems. Previous monitoring techniques assume full observability of the states of the (unknown) monitored system. Moreover, they can monitor only fairness properties that are specified as arithmetic expressions over the probabilities of different events. In this work, we extend fairness monitoring to systems modeled as partially observed Markov chains (POMC), and to specifications containing arithmetic expressions over the expected values of numerical functions on event sequences. The only assumptions we make are that the underlying POMC is aperiodic and starts in the stationary distribution, with a bound on its mixing time being known. These assumptions enable us to estimate a given property for the entire distribution of possible executions of the monitored POMC, by observing only a single execution. Our monitors observe a long run of the system and, after each new observation, output updated PAC-estimates of how fair or biased the system is. The monitors are computationally lightweight and, using a prototype implementation, we demonstrate their effectiveness on several real-world examples.

Kaushik Mallik, Sadegh Esmaeil Zadeh Soudjani, Anne-Kathrin Schmuck et al.

Controller synthesis techniques for continuous systems with respect to temporal logic specifications typically use a finite-state symbolic abstraction of the system. Constructing this abstraction for the entire system is computationally expensive, and does not exploit natural decompositions of many systems into interacting components. We have recently introduced a new relation, called (approximate) disturbance bisimulation for compositional symbolic abstraction to help scale controller synthesis for temporal logic to larger systems. In this paper, we extend the results to stochastic control systems modeled by stochastic differential equations. Given any stochastic control system satisfying a stochastic version of the incremental input-to-state stability property and a positive error bound, we show how to construct a finite-state transition system (if there exists one) which is disturbance bisimilar to the given stochastic control system. Given a network of stochastic control systems, we give conditions on the simultaneous existence of disturbance bisimilar abstractions to every component allowing for compositional abstraction of the network system.

Kyle Hsu, Rupak Majumdar, Kaushik Mallik et al.

We present lazy abstraction-based controller synthesis (ABCS) for continuous-time nonlinear dynamical systems against reach-avoid and safety specifications. State-of-the-art multi-layered ABCS pre-computes multiple finite-state abstractions of varying granularity and applies reactive synthesis to the coarsest abstraction whenever feasible, but adaptively considers finer abstractions when necessary. Lazy ABCS improves this technique by constructing abstractions on demand. Our insight is that the abstract transition relation only needs to be locally computed for a small set of frontier states at the precision currently required by the synthesis algorithm. We show that lazy ABCS can significantly outperform previous multi-layered ABCS algorithms: on standard benchmarks, lazy ABCS is more than 4 times faster.

Guy Avni, Kaushik Mallik, Suman Sadhukhan

Many sequential decision-making tasks require satisfaction of multiple, partially contradictory objectives. Existing approaches are monolithic, namely all objectives are fulfilled using a single policy, which is a function that selects a sequence of actions. We present auction-based scheduling, a modular framework for multi-objective decision-making problems. Each objective is fulfilled using a separate policy, and the policies can be independently created, modified, and replaced. Understandably, different policies with conflicting goals may choose conflicting actions at a given time. In order to resolve conflicts, and compose policies, we employ a novel auction-based mechanism. We allocate a bounded budget to each policy, and at each step, the policies simultaneously bid from their available budgets for the privilege of being scheduled and choosing an action. Policies express their scheduling urgency using their bids and the bounded budgets ensure long-run scheduling fairness. We lay the foundations of auction-based scheduling using path planning problems on finite graphs with two temporal objectives. We present decentralized algorithms to synthesize a pair of policies, their initially allocated budgets, and bidding strategies. We consider three categories of decentralized synthesis problems, parameterized by the assumptions that the policies make on each other: (a) strong synthesis, with no assumptions and strongest guarantees, (b) assume-admissible synthesis, with weakest rationality assumptions, and (c) assume-guarantee synthesis, with explicit contract-based assumptions. For reachability objectives, we show that, surprisingly, decentralized assume-admissible synthesis is always possible when the out-degrees of all vertices are at most two.

Guy Avni, Thomas A. Henzinger, Kaushik Mallik et al.

We study the problem of generating paths on a graph that satisfy a collection of ω-regular objectives. We propose a decoupled framework in which each objective is assigned to an independent agent that selects a local policy, while a scheduler -- oblivious to the graph and objective -- dynamically composes these policies into a single path. We ask when such a composition satisfies all objectives, assuming their conjunction is realizable. The framework enables modular policy design but raises fundamental compositional challenges. We show that even extremely fair deterministic schedulers do not ensure correctness, and that stochastic schedulers, while necessary, are insufficient without coordination. For safety objectives, we demonstrate that fully decentralized implementations are impossible, and we introduce a protocol for synchronizing on maximal safe actions. For non-safety objectives, we introduce conventions -- simple, a priori restrictions agreed upon before the graph or objectives are revealed -- that guarantee satisfaction of all objectives when followed by all agents. We characterize minimally restrictive conventions for major subclasses of ω-regular objectives. In particular, Büchi objectives admit universal composition of finite-memory policies without scheduler communication; co-Büchi objectives require only knowledge of whether the agent was scheduled; and parity objectives additionally require knowledge of which agent was scheduled.

Guruprerana Shabadi, Kaushik Mallik

We consider multi-objective reinforcement learning problems where objectives come from an identical family -- such as the class of reachability objectives -- and may appear or disappear at runtime. Our goal is to design adaptive policies that can efficiently adjust their behaviors as the set of active objectives changes. To solve this problem, we propose a modular framework where each objective is supported by a selfish local policy, and coordination is achieved through a novel auction-based mechanism: policies bid for the right to execute their actions, with bids reflecting the urgency of the current state. The highest bidder selects the action, enabling a dynamic and interpretable trade-off among objectives. Going back to the original adaptation problem, when objectives change, the system adapts by simply adding or removing the corresponding policies. Moreover, as objectives arise from the same family, identical copies of a parameterized policy can be deployed, facilitating immediate adaptation at runtime. We show how the selfish local policies can be computed by turning the problem into a general-sum game, where the policies compete against each other to fulfill their own objectives. To succeed, each policy must not only optimize its own objective, but also reason about the presence of other goals and learn to produce calibrated bids that reflect relative priority. In our implementation, the policies are trained concurrently using proximal policy optimization (PPO). We evaluate on Atari Assault and a gridworld-based path-planning task with dynamic targets. Our method achieves substantially better performance than monolithic policies trained with PPO.

Ali Asadi, Thomas A. Henzinger, Ehsan Kafshdar Goharshady et al.

Two-player games on graphs are a classical framework for analyzing strategic decision making. In turn-based games, two players move a token along the edges of the graph, and the right to move the token is determined by the current vertex. In pure bidding games the right to move the token is determined at each step through bidding; here we consider Richman bidding, where the winning player of a bid pays the losing player. The winner is decided based on a temporal or quantitative specification evaluated over the resulting infinite play. We combine turn-based games and pure bidding games into generalized bidding games, with player-1 vertices, player-2 vertices, and bidding vertices. This natural and simple generalization of bidding games has far-reaching consequences. We show that, as a model, generalized bidding games are more expressive than pure bidding games, and we provide several applications. We also show that generalized Richman bidding games are structurally equivalent to simple stochastic games: they are linearly interreducible to each other. As was previously known, the special case of pure Richman bidding games corresponds to random-turn games. In other words, generalized bidding games extend pure bidding games in the same way that simple stochastic games extend random-turn games. We use this connection to solve generalized Richman bidding games for temporal and quantitativ specifications. We establish that generalized bidding games with parity and mean-payoff specifications retain the best known upper bounds for turn-based games and pure bidding games, namely $NP\cap coNP$. We study a repair problem that asks whether bidding vertices can be assigned owners so as to bring the threshold budget required to win the game below a given target. This problem has direct applications in compositional policy synthesis for multi-objective settings, and we show it to be NP-complete.

Filip Cano, Thomas A. Henzinger, Bettina Könighofer et al.

As AI-based decision-makers increasingly influence human lives, it is a growing concern that their decisions are often unfair or biased with respect to people's sensitive attributes, such as gender and race. Most existing bias prevention measures provide probabilistic fairness guarantees in the long run, and it is possible that the decisions are biased on specific instances of short decision sequences. We introduce fairness shielding, where a symbolic decision-maker -- the fairness shield -- continuously monitors the sequence of decisions of another deployed black-box decision-maker, and makes interventions so that a given fairness criterion is met while the total intervention costs are minimized. We present four different algorithms for computing fairness shields, among which one guarantees fairness over fixed horizons, and three guarantee fairness periodically after fixed intervals. Given a distribution over future decisions and their intervention costs, our algorithms solve different instances of bounded-horizon optimal control problems with different levels of computational costs and optimality guarantees. Our empirical evaluation demonstrates the effectiveness of these shields in ensuring fairness while maintaining cost efficiency across various scenarios.

Ashutosh Gupta, Thomas A. Henzinger, Konstantin Kueffner et al.

Input-output robustness appears in various different forms in the literature, such as robustness of AI models to adversarial or semantic perturbations and individual fairness of AI models that make decisions about humans. We propose runtime monitoring of input-output robustness of deployed, black-box AI models, where the goal is to design monitors that would observe one long execution sequence of the model, and would raise an alarm whenever it is detected that two similar inputs from the past led to dissimilar outputs. This way, monitoring will complement existing offline ``robustification'' approaches to increase the trustworthiness of AI decision-makers. We show that the monitoring problem can be cast as the fixed-radius nearest neighbor (FRNN) search problem, which, despite being well-studied, lacks suitable online solutions. We present our tool Clemont, which offers a number of lightweight monitors, some of which use upgraded online variants of existing FRNN algorithms, and one uses a novel algorithm based on binary decision diagrams -- a data-structure commonly used in software and hardware verification. We have also developed an efficient parallelization technique that can substantially cut down the computation time of monitors for which the distance between input-output pairs is measured using the $L_\infty$ norm. Using standard benchmarks from the literature of adversarial and semantic robustness and individual fairness, we perform a comparative study of different monitors in \tool, and demonstrate their effectiveness in correctly detecting robustness violations at runtime.

Davide Corsi, Kaushik Mallik, Andoni Rodriguez et al.

Shielding has emerged as a promising approach for ensuring safety of AI-controlled autonomous systems. The algorithmic goal is to compute a shield, which is a runtime safety enforcement tool that needs to monitor and intervene the AI controller's actions if safety could be compromised otherwise. Traditional shields are designed statically for a specific safety requirement. Therefore, if the safety requirement changes at runtime due to changing operating conditions, the shield needs to be recomputed from scratch, causing delays that could be fatal. We introduce dynamic shields for parametric safety specifications, which are succinctly represented sets of all possible safety specifications that may be encountered at runtime. Our dynamic shields are statically designed for a given safety parameter set, and are able to dynamically adapt as the true safety specification (permissible by the parameters) is revealed at runtime. The main algorithmic novelty lies in the dynamic adaptation procedure, which is a simple and fast algorithm that utilizes known features of standard safety shields, like maximal permissiveness. We report experimental results for a robot navigation problem in unknown territories, where the safety specification evolves as new obstacles are discovered at runtime. In our experiments, the dynamic shields took a few minutes for their offline design, and took between a fraction of a second and a few seconds for online adaptation at each step, whereas the brute-force online recomputation approach was up to 5 times slower.

Guy Avni, Martin Kurečka, Kaushik Mallik et al.

Graph games are fundamental in strategic reasoning of multi-agent systems and their environments. We study a new family of graph games which combine stochastic environmental uncertainties and auction-based interactions among the agents, formalized as bidding games on (finite) Markov decision processes (MDP). Normally, on MDPs, a single decision-maker chooses a sequence of actions, producing a probability distribution over infinite paths. In bidding games on MDPs, two players -- called the reachability and safety players -- bid for the privilege of choosing the next action at each step. The reachability player's goal is to maximize the probability of reaching a target vertex, whereas the safety player's goal is to minimize it. These games generalize traditional bidding games on graphs, and the existing analysis techniques do not extend. For instance, the central property of traditional bidding games is the existence of a threshold budget, which is a necessary and sufficient budget to guarantee winning for the reachability player. For MDPs, the threshold becomes a relation between the budgets and probabilities of reaching the target. We devise value-iteration algorithms that approximate thresholds and optimal policies for general MDPs, and compute the exact solutions for acyclic MDPs, and show that finding thresholds is at least as hard as solving simple-stochastic games.

Thomas A. Henzinger, Fabian Kresse, Kaushik Mallik et al.

We study the problem of predictive runtime monitoring of black-box dynamical systems with quantitative safety properties. The black-box setting stipulates that the exact semantics of the dynamical system and the controller are unknown, and that we are only able to observe the state of the controlled (aka, closed-loop) system at finitely many time points. We present a novel framework for predicting future states of the system based on the states observed in the past. The numbers of past states and of predicted future states are parameters provided by the user. Our method is based on a combination of Taylor's expansion and the backward difference operator for numerical differentiation. We also derive an upper bound on the prediction error under the assumption that the system dynamics and the controller are smooth. The predicted states are then used to predict safety violations ahead in time. Our experiments demonstrate practical applicability of our method for complex black-box systems, showing that it is computationally lightweight and yet significantly more accurate than the state-of-the-art predictive safety monitoring techniques.

Satyankar Chandra, Ashutosh Gupta, Kaushik Mallik et al.

Many critical decision-making tasks are now delegated to machine-learned models, and it is imperative that their decisions are trustworthy and reliable, and their outputs are consistent across similar inputs. We identify a new source of unreliable behaviors-called glitches-which may significantly impair the reliability of AI models having steep decision boundaries. Roughly speaking, glitches are small neighborhoods in the input space where the model's output abruptly oscillates with respect to small changes in the input. We provide a formal definition of glitches, and use well-known models and datasets from the literature to demonstrate that they have widespread existence and argue they usually indicate potential model inconsistencies in the neighborhood of where they are found. We proceed to the algorithmic search of glitches for widely used gradient-boosted decision tree (GBDT) models. We prove that the problem of detecting glitches is NP-complete for tree ensembles, already for trees of depth 4. Our glitch-search algorithm for GBDT models uses an MILP encoding of the problem, and its effectiveness and computational feasibility are demonstrated on a set of widely used GBDT benchmarks taken from the literature.

Thomas A. Henzinger, Mahyar Karimi, Konstantin Kueffner et al.

Machine-learned systems are in widespread use for making decisions about humans, and it is important that they are fair, i.e., not biased against individuals based on sensitive attributes. We present a general framework of runtime verification of algorithmic fairness for systems whose models are unknown, but are assumed to have a Markov chain structure, with or without full observation of the state space. We introduce a specification language that can model many common algorithmic fairness properties, such as demographic parity, equal opportunity, and social burden. We build monitors that observe a long sequence of events as generated by a given system, and output, after each observation, a quantitative estimate of how fair or biased the system was on that run until that point in time. The estimate is proven to be correct modulo a variable error bound and a given confidence level, where the error bound gets tighter as the observed sequence gets longer. We present two categories of monitoring algorithms, namely ones with a uniform error bound across all time points, and ones with weaker non-uniform, pointwise error bounds at different time points. Our monitoring algorithms use statistical tools that are adapted to suit the dynamic requirements of monitoring and the special needs of the fairness specifications. Using a prototype implementation, we show how we can monitor if a bank is fair in giving loans to applicants from different social backgrounds, and if a college is fair in admitting students while maintaining a reasonable financial burden on the society. In these experiments, our monitors took less than a millisecond to update their verdicts after each observation.

Thomas A. Henzinger, Mahyar Karimi, Konstantin Kueffner et al.

Machine-learned systems are in widespread use for making decisions about humans, and it is important that they are fair, i.e., not biased against individuals based on sensitive attributes. We present runtime verification of algorithmic fairness for systems whose models are unknown, but are assumed to have a Markov chain structure. We introduce a specification language that can model many common algorithmic fairness properties, such as demographic parity, equal opportunity, and social burden. We build monitors that observe a long sequence of events as generated by a given system, and output, after each observation, a quantitative estimate of how fair or biased the system was on that run until that point in time. The estimate is proven to be correct modulo a variable error bound and a given confidence level, where the error bound gets tighter as the observed sequence gets longer. Our monitors are of two types, and use, respectively, frequentist and Bayesian statistical inference techniques. While the frequentist monitors compute estimates that are objectively correct with respect to the ground truth, the Bayesian monitors compute estimates that are correct subject to a given prior belief about the system's model. Using a prototype implementation, we show how we can monitor if a bank is fair in giving loans to applicants from different social backgrounds, and if a college is fair in admitting students while maintaining a reasonable financial burden on the society. Although they exhibit different theoretical complexities in certain cases, in our experiments, both frequentist and Bayesian monitors took less than a millisecond to update their verdicts after each observation.

Thomas A. Henzinger, Mahyar Karimi, Konstantin Kueffner et al.

A machine-learned system that is fair in static decision-making tasks may have biased societal impacts in the long-run. This may happen when the system interacts with humans and feedback patterns emerge, reinforcing old biases in the system and creating new biases. While existing works try to identify and mitigate long-run biases through smart system design, we introduce techniques for monitoring fairness in real time. Our goal is to build and deploy a monitor that will continuously observe a long sequence of events generated by the system in the wild, and will output, with each event, a verdict on how fair the system is at the current point in time. The advantages of monitoring are two-fold. Firstly, fairness is evaluated at run-time, which is important because unfair behaviors may not be eliminated a priori, at design-time, due to partial knowledge about the system and the environment, as well as uncertainties and dynamic changes in the system and the environment, such as the unpredictability of human behavior. Secondly, monitors are by design oblivious to how the monitored system is constructed, which makes them suitable to be used as trusted third-party fairness watchdogs. They function as computationally lightweight statistical estimators, and their correctness proofs rely on the rigorous analysis of the stochastic process that models the assumptions about the underlying dynamics of the system. We show, both in theory and experiments, how monitors can warn us (1) if a bank's credit policy over time has created an unfair distribution of credit scores among the population, and (2) if a resource allocator's allocation policy over time has made unfair allocations. Our experiments demonstrate that the monitors introduce very low overhead. We believe that runtime monitoring is an important and mathematically rigorous new addition to the fairness toolbox.

Kaushik Mallik, Anne-Kathrin Schmuck, Sadegh Soudjani et al.

Controller synthesis techniques for continuous systems with respect to temporal logic specifications typically use a finite-state symbolic abstraction of the system model. Constructing this abstraction for the entire system is computationally expensive, and does not exploit natural decompositions of many systems into interacting components. We describe a methodology for compositional symbolic abstraction to help scale controller synthesis for temporal logic to larger systems. We introduce a new relation, called (approximate) disturbance bisimulation, as the basis for compositional symbolic abstractions. Disturbance bisimulation strengthens the standard approximate alternating bisimulation relation used in control. It extends naturally to systems which are composed of weakly interconnected sub-components possibly connected in feedback, and models the coupling signals as disturbances. After proving this composability of disturbance bisimulation for metric systems we apply this result to the compositional abstraction of networks of input-to-state stable deterministic non-linear control systems. We give conditions that allow to construct finite-state abstractions compositionally for each component in such a network, so that the abstractions are simultaneously disturbance bisimilar to their continuous counterparts. Combining these two results, we show conditions under which one can compositionally abstract a network of non-linear control systems in a modular way while ensuring that the final composed abstraction is disturbance bisimilar to the original system. We discuss how we get a compositional abstraction-based controller synthesis methodology for networks of such systems against local temporal specifications as a by-product of our construction.