Generalisation of Baker's Forcing Method to Arbitrary Prime and NP-hardness of Several $p$-adic Optimisations

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.