Embedding causal team languages into predicate logic

Causal team semantics ([2]) supports causal-observational languages, which enrich the languages for deterministic causation ([11], [18]) with dependencies and other team-specific operators. Handling the causal aspects of these languages requires a richer semantics than propositional team semantics; nonetheless, in this paper we show that the causal-observational languages considered in [2] can be embedded into first-order dependence logic by means of a translation and a careful choice of models. We show that, in some significant cases, the translation can be refined to an embedding into the Bernays-Schönfinkel-Ramsey fragment of dependence logic or, in the restricted case of recursive causal models, into the existential fragment. As an application, we use the embeddings to show the decidability of a satisfiability problem for the causal-observational languages. Along the way, we question the correctness of the semantics for interventionist counterfactuals proposed by Halpern ([18]) and propose an alternative one which behaves as usual in the uncontroversial recursive case.