We state it here for clarity and completeness. The data generating mechanism for ( X, A,Z, W, U) is summarized in Table 1, and the setups of varying parameters in each scenario are summarized in Table 2. Table 1: Data generating mechanism and setup for fixed parameters across scenarios.21 X)null + ωW, (20) where the first equality is due to Assumption 1. Furthermore, note that E[h( W, 1, X)|X, Z,U ] = E[h( W, 1, X)| X,U ] = E[Y | X,A = 1, U] = E[Y | X,A = 1, Z,U ] = b X)null, 22 where the first and third equality is due to Assumption 1, the second equality follows from Theorem 1 of Miao et al. (2018a) under Assumptions 2 and 3, and the last equality is by (19). X) null + ωW, where the second equality is due to Assumption 1, and the third equality is due to Theorem 2.2 of Cui et al. (2023) under Assumptions 4 and 5, and the last equality is due to (20). Step (i) The method we adopt is neural maximum moment restriction (NMMR), which employs multilayer perceptron (MLP) to estimate the confounding bridges (Kompa et al., 2022).

Query answering under existential rules — implications with existential quantifiers in the head — is known to be decidable when imposing restrictions on the rule bodies such as frontier-guardedness [Baget et al., 2010; Baget et al., 2011a]. Query answering is also decidable for description logics [Baader, 2003], which further allow disjunction and functionality constraints (assert that certain relations are functions); however, they are focused on ER-type schemas, where relations have arity two. This work investigates how to get the best of both worlds: having decidable existential rules on arbitrary arity relations, while allowing rich description logics, including functionality constraints, on arity-two relations. We first show negative results on combining such decidable languages. Second, we introduce an expressive set of existential rules (frontier-one rules with a certain restriction) which can be combined with powerful constraints on arity-two relations (e.g. GC2, ALCQIb) while retaining decidable query answering. Further, we provide conditions to add functionality constraints on the higher-arity relations.

Let ϕ be a sentence (i.e. a formula with no free variables) in some logical fragment, ψ(ȳ) a formula with free variables ȳ, a set of ground, function-free literals, and ā a tuple of individual constants with the same arity as ȳ. We are to think of as being a body of data, ϕ a background theory, and ψ(ā) a query which we wish to answer. That answer should be positive just in case {ϕ} entails ψ(ā). What is the computational complexity of our task? A fair reply depends on what, precisely, we take the inputs to our problem to be. For, in practice, the background theory ϕ is static, and the query ψ(ȳ) small: only the database, which is devoid of logical complexity, is large and indefinitely extensible.