Several physically inspired problems have been proven undecidable; examples are the spectral gap problem and the membership problem for quantum correlations. Most of these results rely on reductions from a handful of undecidable problems, such as the halting problem, the tiling problem, the Post correspondence problem or the matrix mortality problem. All these problems have a common property: they have an NP-hard bounded version. This work establishes a relation between undecidable unbounded problems and their bounded NP-hard versions. Specifically, we show that NP-hardness of a bounded version follows easily from the reduction of the unbounded problems. This leads to new and simpler proofs of the NP-hardness of bounded version of the Post correspondence problem, the matrix mortality problem, the positivity of matrix product operators, the reachability problem, the tiling problem, and the ground state energy problem. This work sheds light on the intractability of problems in theoretical physics and on the computational consequences of bounding a parameter.

翻译：若干源于物理思想的问题已被证明不可判定，例如谱隙问题和量子关联的成员判定问题。这些问题大多依赖于从少数几个不可判定问题（如停机问题、铺砌问题、波斯特对应问题或矩阵消亡问题）的归约。所有这些不可判定问题都有一个共同特征：它们存在NP难的有界版本。本文建立了无界不可判定问题与其有界NP难版本之间的关联。具体而言，我们证明无界问题的归约可直接导出其有界版本的NP难性。由此可得波斯特对应问题、矩阵消亡问题、矩阵乘积算符正性判定问题、可达性判定问题、铺砌问题及基态能量问题之有界版本NP难性的新颖且简化的证明。本研究揭示了理论物理中问题的计算难解性，以及参数有界化对计算复杂性的影响。