Many bounded versions of undecidable problems are NP-hard

TL;DR

This paper demonstrates that many problems in physics and computation, which are undecidable in their unbounded form, become NP-hard when bounded, providing simpler proofs and insights into their intractability.

Contribution

It establishes a general relation showing NP-hardness of bounded versions of undecidable problems derived from classical reductions, simplifying proofs and understanding of their computational complexity.

Findings

Bounded versions of undecidable problems are NP-hard.

Simplified proofs for NP-hardness of several problems.

Insights into the complexity of physically inspired problems.

Abstract

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.