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Generalized structural equations models (GSEMs) are, as the name suggests, a generalization of structural equations models (SEMs). They can deal with (among other things) infinitely many variables with infinite ranges, which is critical for capturing dynamical systems. We provide a sound and complete axiomatization of causal reasoning in GSEMs that is an extension of the sound and complete axiomatization provided by Halpern for SEMs. Considering GSEMs helps clarify what properties Halpern's axioms capture. 

Generalized structural equations models (GSEMs) are, as the name suggests, a generalization of structural equations models (SEMs). They can deal with (among other things) infinitely many variables with infinite ranges, which is critical for capturing dynamical systems. We provide a sound and complete axiomatization of causal reasoning in GSEMs that is an extension of the sound and complete axiomatization provided by Halpern for SEMs. Considering GSEMs helps clarify what properties Halpern's axioms capture. 

Consider a bank that uses an AI system to decide which loan applications to approve. We want to ensure that the system is fair, that is, it does not discriminate against applicants based on a predefined list of sensitive attributes, such as gender and ethnicity. We expect there to be a regulator whose job it is to certify the bank's system as fair or unfair. We consider issues that the regulator will have to confront when making such a decision, including the precise definition of fairness, dealing with proxy variables, and dealing with what we call allowed variables, that is, variables such as salary on which the decision is allowed to depend, despite being correlated with sensitive variables. We show (among other things) that the problem of deciding fairness as we have defined it is co-NP-complete, but then argue that, despite that, in practice the problem should be manageable. 

We investigate how to model the beliefs of an agent who becomes more aware. We use the framework of Halpern and Rego (2013) by adding probability, and define a notion of a model transition that describes constraints on how, if an agent becomes aware of a new formula φ in state s of a model M, she transitions to state s* in a model M*. We then discuss how such a model can be applied to information disclosure. 

Scientific models describe natural phenomena at different levels of abstraction. Abstract descriptions can provide the basis for interventions on the system and explanation of observed phenomena at a level of granularity that is coarser than the most fundamental account of the system. Beckers and Halpern [2019], building on work of Rubenstein et al.[2017], developed an account of \emph{abstraction} for causal models that is exact. Here we extend this account to the more realistic case where an abstract causal model offers only an approximation of the underlying system. We show how the resulting account handles the discrepancy that can arise between low- and high-level causal models of the same system, and in the process provide an account of how one causal model approximates another, a topic of independent interest. Finally, we extend the account of approximate abstractions to probabilistic causal models, indicating how and where uncertainty can enter into an approximate abstraction. 

Abstract Sufficient conditions are given under which ratifiable acts exist. 

We consider a sequence of successively more restrictive definitions of abstraction for causal models, starting with a notion introduced by Rubenstein et al. called exact transformation that applies to probabilistic causal models, moving to a notion of uniform transformation that applies to deterministic causal models and does not allow differences to be hidden by the ``right'' choice of distribution, and then to abstraction, where the interventions of interest are determined by the map from low-level states to high-level states, and strong abstraction, which takes more seriously all potential interventions in a model, not just the allowed interventions. We show that procedures for combining micro-variables into macro-variables are instances of our notion of strong abstraction, as are all the examples considered by Rubenstein et al.