The Existential Theory of the Reals with Summation Operators

To characterize the computational complexity of satisfiability problems for probabilistic and causal reasoning within Pearl’s Causal Hierarchy, van der Zander, Bläser, and Liśkiewicz [IJCAI 2023] introduce a new natural class, named succ-∃R. This class can be viewed as a succinct variant of the well-studied class ∃R based on the Existential Theory of the Reals (ETR). Analogously to ∃R, succ-∃R is an intermediate class between NEXP and EXPSPACE, the exponential versions of NP and PSPACE. The main contributions of this work are threefold. Firstly, we characterize the class succ-∃R in terms of nondeterministic real Random-Access Machines (RAMs) and develop structural complexity theoretic results for real RAMs, including translation and hierarchy theorems. Notably, we demonstrate the separation of ∃R and succ-∃R. Secondly, we examine the complexity of model checking and satisfiability of fragments of existential second-order logic and probabilistic independence logic. We show succ-∃R-completeness of several of these problems, for which the best-known complexity lower and upper bounds were previously NEXP-hardness and EXPSPACE, respectively. Thirdly, while succ-∃R is characterized in terms of ordinary (non-succinct) ETR instances enriched by exponential sums and a mechanism to index exponentially many variables, in this paper, we prove that when only exponential sums are added, the corresponding class ∃R^Σ is contained in PSPACE. We conjecture that this inclusion is strict, as this class is equivalent to adding a VNP-oracle to a polynomial time nondeterministic real RAM. Conversely, the addition of exponential products to ETR, yields PSPACE.