The existential theory of the reals and existential–universal sentences in the counting hierarchy at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The existential theory of the reals and existential–universal sentences in the counting hierarchy

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	algorithms, speed	Levels:	1
Category:	Theoretical computer science	Lean version:	not yet
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Existential–universal real sentences in the counting hierarchy. Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.

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LEVEL 1 Existential–universal real sentences in the counting hierarchy

released 2026-10-04 | 3 theorems · 12 lemmas · 23 proofs · 15,000 words | PLAY LEVEL 1 » (pdf)

We prove that the existential theory of the reals lies in the counting hierarchy. More generally, we show that the truth of existential–universal sentences over the reals can be decided in a fixed level of the counting hierarchy, even when the integer polynomials are given by arithmetic circuits.

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