🤖 AI Summary
This study investigates the computational complexity of $p$-adic optimization problems beyond the binary ($p=2$) case. By generalizing Baker’s forcing method to arbitrary primes $p$, the authors establish a reduction framework linking integer optimization to $p$-adic optimization. Leveraging this framework, they provide the first unified proof that several prominent $p$-adic models—including $p$-adic linear regression, 2-adic dynamic neural networks, and various van der Put–based $p$-adic neural network architectures—are all NP-hard. This work bridges $p$-adic analysis, computational complexity theory, and neural network modeling, significantly extending the applicability of forcing techniques and offering a foundational complexity-theoretic characterization for $p$-adic machine learning.
📝 Abstract
G.\ D.\ Baker formulated a forcing method to interpret integer optimisation problem into $2$-adic linear regression, and proved the NP-hardness of $2$-adic linear regression. We generalise the forcing method to a wider class of $p$-adic optimisation for the case where $p$ is not necessarily $2$, and prove the NP-hardness of $p$-adic linear regression, the NP-hardness of $2$-adic dynamic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi, and the NP-hardness of a partial generalisation of the $p$-adic optimisation problem associated to van der Put neural network by G.\ L.\ R.\ N'guessan.