The Existential Theory of the Reals with Summation Operators
Markus Bl\"aser, Julian D\"orfler, Maciej Liskiewicz, Benito van der, Zander

TL;DR
This paper introduces the succ-$orall$R class, a succinct variant of the Existential Theory of the Reals, characterizing its complexity, structural properties, and its relation to other complexity classes, with applications to probabilistic reasoning.
Contribution
It defines and analyzes the succ-$orall$R class, establishing its complexity, structural properties, and its role in probabilistic and causal reasoning satisfiability problems.
Findings
succ-$orall$R is between NEXP and EXPSPACE in complexity.
succ-$orall$R- completeness for several logical satisfiability problems.
Adding exponential sums to ETR places the problem in PSPACE.
Abstract
To characterize the computational complexity of satisfiability problems for probabilistic and causal reasoning within the Pearl's Causal Hierarchy, arXiv:2305.09508 [cs.AI] 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 RAM machines 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…
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