We try to understand the relationship between programs and correctness, and in particular, why writing correct programs must be hard. We will review results from computability theory and complexity theory, and see that programs and complete understanding – which is required for correctness – are fundamentally at odds.

We try to understand the relationship between programs and correctness, and in particular, why writing correct programs must be hard. We will review results from computability theory and complexity theory, and see that programs and complete understanding – which is required for correctness – are

Proceedings of the National Academy of Sciences, Volume 123, Issue 42, October 2026. SignificanceTrustworthy uncertainty quantification is key for responsible data science and AI (including data-driven decision-making). Traditional statistical inference methods require specifying an underlying generative model and are often not robust to ...

We try to understand the relationship between programs and correctness, and in particular, why writing correct programs must be hard. We will review results from computability theory and complexity theory, and see that programs and complete understanding – which is required for correctness – are fundamentally at odds.

We try to understand the relationship between programs and correctness, and in particular, why writing correct programs must be hard. We will review results from computability theory and complexity theory, and see that programs and complete understanding – which is required for correctness – are fundamentally at odds.

We try to understand the relationship between programs and correctness, and in particular, why writing correct programs must be hard. We will review results from computability theory and complexity theory, and see that programs and complete understanding – which is required for correctness – are fundamentally at odds.

Combinatorics and Arithmetic for PhysicsThe meeting’s focus is about questions of discrete mathematics and number theory with an emphasis on computability. Problems are drawn mainly from theoretical p...

Quantum 10, 2257 (2026). https://doi.org/10.22331/q-2026-10-30-2257 The notion of a macroscopic quantum state must be pinned down in order to assess how well experiments probe the large-scale limits of quantum mechanics. However, the issue of quantifying so-called quantum macroscopicity is fraught with multiple approaches having varying interpretations and levels of computability and measurability. Here, we introduce two measures that capture independent macroscopic properties: i) extensive size, measuring the degree of coherence in position or momentum observables relative to atomic-scale units, and ii) entangled size, quantifying the number of entangled subsystems. These measures unify many desirable features of past proposals while having rigorously justified interpretations from quantum information theory in terms of coherence and multipartite entanglement. We demonstrate how to estimate them and obtain lower-bounds using experimental data from micro-mechanical oscillators and diffracting molecules. Notably, we find evidence for genuine multipartite entanglement of $10^5$ to $10^7$ atoms in recent mechanical superposition states.

Incompleteness and Computability is an open textbook on recursive function theory, Gödel's incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.