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. Furthermore, we study the satisfiability problem for probabilistic reasoning, with the additional requirement of a small model, and prove that this problem is complete for ∃ℝ^Σ.
@InProceedings{blaser_et_al:LIPIcs.ISAAC.2024.13, author = {Bl\"{a}ser, Markus and D\"{o}rfler, Julian and Li\'{s}kiewicz, Maciej and van der Zander, Benito}, title = {{The Existential Theory of the Reals with Summation Operators}}, booktitle = {35th International Symposium on Algorithms and Computation (ISAAC 2024)}, pages = {13:1--13:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-354-6}, ISSN = {1868-8969}, year = {2024}, volume = {322}, editor = {Mestre, Juli\'{a}n and Wirth, Anthony}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.13}, URN = {urn:nbn:de:0030-drops-221407}, doi = {10.4230/LIPIcs.ISAAC.2024.13}, annote = {Keywords: Existential theory of the real numbers, Computational complexity, Probabilistic logic, Models of computation, Existential second order logic} }
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