Exotic Quantifiers, Complexity Classes, and Complete Problems

Abstract

We define new complexity classes in the Blum-Shub-Smale theory of computation over the reals, in the spirit of the polynomial hierarchy, with the help of infinitesimal and generic quantifiers. Basic topological properties of semialgebraic sets like boundedness, closedness, compactness, as well as the continuity of semialgebraic functions are shown to be complete in these new classes. All attempts to classify the complexity of these problems in terms of the previously studied complexity classes have failed. We also obtain completeness results in the Turing model for the corresponding discrete problems. In this setting, it turns out that infinitesimal and generic quantifiers can be eliminated, so that the relevant complexity classes can be described in terms of the usual quantifiers only.

Computational Complexity of Multi-Player Evolutionarily Stable Strategies

“…These results stand in contrast to the two-player setting, where the same decision problems are NP-complete [GZ89;CS08]. The class ∃R is the complexity class that captures the decision problem for the existential the-ory of the reals [SŠ17], or alternatively, is the constant-free Boolean part of the real analogue NP R in the Blum-Shub-Smale model of computation [BC09]. Clearly we have NP ⊆ ∃R, and from the decision procedure for the existential theory of the reals by Canny [Can88] it follows that ∃R ⊆ PSPACE.…”

Computational Complexity of Multi-player Evolutionarily Stable Strategies

“…Most prominently is the class BP(NP 0 R ) which also captures the complexity of the existential theory of the reals. It has been named ∃R by Schaefer and Štefankovič [SŠ17] as well as NPR by Bürgisser and Cucker [BC09]; we shall use the former notation ∃R. We further let ∀R = BP(coNP 0 R ) as well as…”

“…These results stand in contrast to the two-player setting, where the same decision problems are NP-complete [GZ89;CS08]. The class ∃R is the complexity class that captures the decision problem for the existential theory of the reals [SŠ17], or alternatively, is the constant free Boolean part of the real analogue NP R in the Blum-Shub-Smale model of computation [BC09]. Clearly we have NP ⊆ ∃R, and from the decision procedure for the existential theory of the reals by Canny [Can88] it follows that ∃R ⊆ PSPACE.…”

Existential Theory of the Reals Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games

“…Schaefer and Štefankovič [27] defined the complexity class ∃R as the closure of ETR under polynomial time many-one reductions. Alternatively, ∃R is equal to the constant-free Boolean part of the class NP R [8], which is the analogue class to NP in the Blum-Shub-Smale model of computation [4]. Clearly NP ⊆ ∃R and from the decision procedure by Canny [9] we have that ∃R ⊆ PSPACE.…”