Perhaps the most prominent current definition of (actual) causality is due to Halpern and Pearl. It is defined using causal models (also known as structural equations models). We abstract the definition, extracting its key features, so that it can be applied to any other model where counterfactuals are defined. By abstracting the definition, we gain a number of benefits. Not only can we apply the definition in a wider range of models, including ones that allow, for example, backtracking, but we can apply the definition to determine if A is a cause of B even if A and B are formulas involving disjunctions, negations, beliefs, and nested counterfactuals (none of which can be handled by the Halpern-Pearl definition). Moreover, we can extend the ideas to getting an abstract definition of explanation that can be applied beyond causal models. Finally, we gain a deeper understanding of features of the definition even in causal models.

Causality has gained popularity in recent years. It has helped improve the performance, reliability, and interpretability of machine learning models. However, recent literature on explainable artificial intelligence (XAI) has faced criticism. The classical XAI and causality literature focuses on understanding which factors contribute to which consequences. While such knowledge is valuable for researchers and engineers, it is not what non-expert users expect as explanations. Instead, these users often await facts that cause the target consequences, i.e., actual causes. Formalizing this notion is still an open problem. Additionally, identifying actual causes is reportedly an NP-complete problem, and there are too few practical solutions to approximate formal definitions. We propose a set of algorithms to identify actual causes with a polynomial complexity and an adjustable level of precision and exhaustiveness. Our experiments indicate that the algorithms (1) identify causes for different categories of systems that are not handled by existing approaches (i.e., non-boolean, black-box, and stochastic systems), (2) can be adjusted to gain more precision and exhaustiveness with more computation time.

We address causal reasoning in multivariate time series data generated by stochastic processes. Existing approaches are largely restricted to static settings, ignoring the continuity and emission of variations across time. In contrast, we propose a learning paradigm that directly establishes causation between events in the course of time. We present two key lemmas to compute causal contributions and frame them as reinforcement learning problems. Our approach offers formal and computational tools for uncovering and quantifying causal relationships in diffusion processes, subsuming various important settings such as discrete-time Markov decision processes. Finally, in fairly intricate experiments and through sheer learning, our framework reveals and quantifies causal links, which otherwise seem inexplicable.

Miller recently proposed a definition of contrastive (counterfactual) explanations based on the well-known Halpern-Pearl (HP) definitions of causes and (non-contrastive) explanations. Crucially, the Miller definition was based on the original HP definition of explanations, but this has since been modified by Halpern; presumably because the original yields counterintuitive results in many standard examples. More recently Borner has proposed a third definition, observing that this modified HP definition may also yield counterintuitive results. In this paper we show that the Miller definition inherits issues found in the original HP definition. We address these issues by proposing two improved variants based on the more robust modified HP and Borner definitions. We analyse our new definitions and show that they retain the spirit of the Miller definition where all three variants satisfy an alternative unified definition that is modular with respect to an underlying definition of non-contrastive explanations. To the best of our knowledge this paper also provides the first explicit comparison between the original and modified HP definitions.

Causality is crucial in human reasoning and knowledge. Defining and formalizing causality has been a significant area of research in philosophy and formal methods [12, 21, 24, 11]. In recent years, with the rise of machine learning and AI, there has been growing interest in formalizing causal reasoning. One of the key areas of AI research is designing algorithms capable of comprehending causal information and performing causal reasoning [5, 29, 30]. Causal reasoning can be instrumental in formally modeling notions such as responsibility, blame, harm, and explanation, which are important aspects in designing ethical and responsible AI systems [3]. In this article we focus on the kind of causality known as "actual causality" (a.k.a.

Actual causality and a closely related concept of responsibility attribution are central to accountable decision making. Actual causality focuses on specific outcomes and aims to identify decisions (actions) that were critical in realizing an outcome of interest. Responsibility attribution is complementary and aims to identify the extent to which decision makers (agents) are responsible for this outcome. In this paper, we study these concepts under a widely used framework for multi-agent sequential decision making under uncertainty: decentralized partially observable Markov decision processes (Dec-POMDPs). Following recent works in RL that show correspondence between POMDPs and Structural Causal Models (SCMs), we first establish a connection between Dec-POMDPs and SCMs. This connection enables us to utilize a language for describing actual causality from prior work and study existing definitions of actual causality in Dec-POMDPs. Given that some of the well-known definitions may lead to counter-intuitive actual causes, we introduce a novel definition that more explicitly accounts for causal dependencies between agents' actions. We then turn to responsibility attribution based on actual causality, where we argue that in ascribing responsibility to an agent it is important to consider both the number of actual causes in which the agent participates, as well as its ability to manipulate its own degree of responsibility. Motivated by these arguments we introduce a family of responsibility attribution methods that extends prior work, while accounting for the aforementioned considerations. Finally, through a simulation-based experiment, we compare different definitions of actual causality and responsibility attribution methods. The empirical results demonstrate the qualitative difference between the considered definitions of actual causality and their impact on attributed responsibility.

Thinking in terms of causality helps us structure how different parts of a system depend on each other, and how interventions on one part of a system may result in changes to other parts. Therefore, formal models of causality are an attractive tool for reasoning about security, which concerns itself with safeguarding properties of a system against interventions that may be malicious. As we show, many security properties are naturally expressed as nested causal statements: not only do we consider what caused a particular undesirable effect, but we also consider what caused this causal relationship itself to hold. We present a natural way to extend the Halpern-Pearl (HP) framework for causality to capture such nested causal statements. This extension adds expressivity, enabling the HP framework to distinguish between causal scenarios that it could not previously naturally tell apart. We moreover revisit some design decisions of the HP framework that were made with non-nested causal statements in mind, such as the choice to treat specific values of causal variables as opposed to the variables themselves as causes, and may no longer be appropriate for nested ones.

Pearl opened the door to formally defining actual causation using causal models. His approach rests on two strategies: first, capturing the widespread intuition that X=x causes Y=y iff X=x is a Necessary Element of a Sufficient Set for Y=y, and second, showing that his definition gives intuitive answers on a wide set of problem cases. This inspired dozens of variations of his definition of actual causation, the most prominent of which are due to Halpern & Pearl. Yet all of them ignore Pearl's first strategy, and the second strategy taken by itself is unable to deliver a consensus. This paper offers a way out by going back to the first strategy: it offers six formal definitions of causal sufficiency and two interpretations of necessity. Combining the two gives twelve new definitions of actual causation. Several interesting results about these definitions and their relation to the various Halpern & Pearl definitions are presented. Afterwards the second strategy is evaluated as well. In order to maximize neutrality, the paper relies mostly on the examples and intuitions of Halpern & Pearl. One definition comes out as being superior to all others, and is therefore suggested as a new definition of actual causation.

Halpern and Pearl introduced a definition of actual causality; Eiter and Lukasiewicz showed that computing whether X = x is a cause of Y = y is NP-complete in binary models (where all variables can take on only two values) and Σ^P_2 -complete in general models. In the final version of their paper, Halpern and Pearl slightly modified the definition of actual cause, in order to deal with problems pointed out by Hopkins and Pearl. As we show, this modification has a nontrivial impact on the complexity of computing whether {X} = {x} is a cause of Y = y. To characterize the complexity, a new family D_k^P , k = 1, 2, 3, . . ., of complexity classes is introduced, which generalises the class DP introduced by Papadimitriou and Yannakakis (DP is just D_1^P). We show that the complexity of computing causality under the updated definition is D_2^P -complete. Chockler and Halpern extended the definition of causality by introducing notions of responsibility and blame, and characterized the complexity of determining the degree of responsibility and blame using the original definition of causality. Here, we completely characterize the complexity using the updated definition of causality. In contrast to the results on causality, we show that moving to the updated definition does not result in a difference in the complexity of computing responsibility and blame.

Causal models defined in terms of structural equations have proved to be quite a powerful way of representing knowledge regarding causality. However, a number of authors have given examples that seem to show that the Halpern-Pearl (HP) definition of causality gives intuitively unreasonable answers. Here it is shown that, for each of these examples, we can give two stories consistent with the description in the example, such that intuitions regarding causality are quite different for each story. By adding additional variables, we can disambiguate the stories. Moreover, in the resulting causal models, the HP definition of causality gives the intuitively correct answer. It is also shown that, by adding extra variables, a modification to the original HP definition made to deal with an example of Hopkins and Pearl may not be necessary. Given how much can be done by adding extra variables, there might be a concern that the notion of causality is somewhat unstable. Can adding extra variables in a "conservative" way (i.e., maintaining all the relations between the variables in the original model) cause the answer to the question "Is X=x a cause of Y=y" to alternate between "yes" and "no"? It is shown that we can have such alternation infinitely often, but if we take normality into consideration, we cannot. Indeed, under appropriate normality assumptions. adding an extra variable can change the answer from "yes" to "no", but after that, it cannot cannot change back to "yes".