Runtime monitoring of quantitative signals faces a fundamental trade-off between volatility and over-aggregation: instantaneous observations are noisy, while long-run averages obscure local structure. Localisation measures such as discounted averages offer a principled middle ground, yet remain poorly understood in runtime verification. This paper studies discounted sums from a monitoring perspective, in both deterministic and stochastic settings. We formalize the discounted monitoring problem and show that exact, sound monitoring of discounted sums cannot be achieved with finite memory. To overcome this impossibility, we introduce $\varepsilon$-approximately sound monitoring, deriving explicit bounds on memory and observation requirements. We then extend the framework to stochastic processes via expected discounted sums, defining pointwise and uniform $(\varepsilon,δ)$-soundness notions, establishing statistical optimality, and proving impossibility beyond a precision threshold. We also formalize the resource complexity of deterministic discounted monitoring via affine register machines and prove a tight worst-case lower bound. Finally, we present a specification language for arithmetic expressions over multiple discounted sums with synchronous and asynchronous semantics, and evaluate our approach on practical scenarios including algorithmic fairness.

As AI-based decision-makers increasingly influence human lives, it is a growing concern that their decisions may be unfair or biased with respect to people's protected 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 any decision sequence of fixed length. 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.

Abstract We present $$\textsf{soid}$$ soid , a tool for interrogating the decision making of autonomous agents using SMT-based automated reasoning. Relying on the Z3 SMT solver and KLEE symbolic execution engine, $$\textsf{soid}$$ soid allows investigators to receive rigorously proven answers to factual and counterfactual queries about agent behavior, enabling effective legal and engineering accountability for harmful or otherwise incorrect decisions. We evaluate $$\textsf{soid}$$ soid qualitatively and quantitatively on a pair of examples, i) a buggy implementation of a classic decision tree inference benchmark from the explainable AI (XAI) literature; and ii) a car crash in a simulated physics environment. For the latter, we also contribute the $$\textsf{soid}\hbox {-}\!\textsf{gui}$$ soid - gui , a domain-specific, web-based example interface for legal and other practitioners to specify factual and counterfactual queries without requiring sophisticated programming or formal methods expertise.

Sequential decision-making in probabilistic environments is a fundamental problem with many applications in AI and economics. In this paper, we present an algorithm for synthesizing sequential decision-making agents that optimize statistical properties such as maximum and average response times. In the general setting of sequential decision-making, the environment is modeled as a random process that generates inputs. The agent responds to each input, aiming to maximize rewards and minimize costs within a specified time horizon. The corresponding synthesis problem is known to be PSPACE-hard. We consider the special case where the input distribution, reward, and cost depend on input-output statistics specified by counter automata. For such problems, this paper presents the first PTIME synthesis algorithms. We introduce the notion of statistical abstraction, which clusters statistically indistinguishable input-output sequences into equivalence classes. This abstraction allows for a dynamic programming algorithm whose complexity grows polynomially with the considered horizon, making the statistical case exponentially more efficient than the general case. We evaluate our algorithm on three different application scenarios of a client-server protocol, where multiple clients compete via bidding to gain access to the service offered by the server. The synthesized policies optimize profit while guaranteeing that none of the server’s clients is disproportionately starved of the service.

Principled accountability for autonomous decision-making in uncertain environments requires distinguishing intentional outcomes from negligent designs from actual accidents. We propose analyzing the behavior of autonomous agents through a quantitative measure of the evidence of intentional behavior. We model an uncertain environment as a Markov Decision Process (MDP). For a given scenario, we rely on probabilistic model checking to compute the ability of the agent to influence reaching a certain event. We call this the scope of agency. We say that there is evidence of intentional behavior if the scope of agency is high and the decisions of the agent are close to being optimal for reaching the event. Our method applies counterfactual reasoning to automatically generate relevant scenarios that can be analyzed to increase the confidence of our assessment. In a case study, we show how our method can distinguish between 'intentional' and 'accidental' traffic collisions.

Evaluation of deep reinforcement learning (RL) is inherently challenging. Especially the opaqueness of learned policies and the stochastic nature of both agents and environments make testing the behavior of deep RL agents difficult. We present a search-based testing framework that enables a wide range of novel analysis capabilities for evaluating the safety and performance of deep RL agents. For safety testing, our framework utilizes a search algorithm that searches for a reference trace that solves the RL task. The backtracking states of the search, called boundary states, pose safety-critical situations. We create safety test-suites that evaluate how well the RL agent escapes safety-critical situations near these boundary states. For robust performance testing, we create a diverse set of traces via fuzz testing. These fuzz traces are used to bring the agent into a wide variety of potentially unknown states from which the average performance of the agent is compared to the average performance of the fuzz traces. We apply our search-based testing approach on RL for Nintendo's Super Mario Bros.