Search versus Decision for $\mathsf{S}_2^\mathsf{P}$

Lance Fortnow

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arXiv · 2512.02808

Search versus Decision for $\mathsf{S}_2^\mathsf{P}$

Lance Fortnow

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Abstract

We compare the complexity of the search and decision problems for the complexity class S2P. While Cai (2007) showed that the decision problem is contained in ZPP^NP, we show that the search problem is equivalent to TFNP^NP, the class of total search problems verifiable in polynomial time with an NP oracle. This highlights a significant contrast: if search reduces to decision for S2P, then $\Sigma_2^p \cap \Pi_2^p$ is contained in ZPP^NP.

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Lance Fortnow. 2025-12-02. Search versus Decision for $\mathsf{S}_2^\mathsf{P}$. https://arxiv.org/abs/2512.02808

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