Halting problem in neural networks: Some AI Systems are Impossible to Compute (spectrum.ieee.org)

Researchers have shown that there are fundamental computability limits for deep neural networks: even when a stable, accurate network is known to exist in theory, there may be no algorithm that a digital computer can actually construct it. The paper (PNAS, 16 March) formalizes a gap between existence proofs for neural nets and what can be computed in practice, proving that for some problems no algorithm—regardless of how much or how precise the data—is guaranteed to compute a network that is both stable (robust to small input changes) and accurate. The work frames these limits in computability terms related to Turing and Gödel: some desirable networks are like recipes without realizable mixers, and small numerical or model perturbations can send computation to a qualitatively different solution. For practitioners this changes how we interpret universal approximation and stability claims: theoretical representability doesn’t imply constructibility or safe deployability. The study also documents a practical tradeoff between stability and accuracy in many settings and introduces “fast iterative restarted networks” (FIRENETs) as a design that can improve robustness in tasks like medical imaging. The authors call for a classification theory to identify which stable, accurate networks are algorithmically attainable and quantify how close one can get when exact computation is impossible — a direction with direct implications for safety‑critical AI, verification, and the limits of automated model design.