Mathematical discoveries from program search with large language models.

Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements1,2. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches3. Applying FunSearch to a central problem in extremal combinatorics-the cap set problem-we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.

Divorce in Turkish and Moroccan Communities in Belgium.

This paper focuses on divorce amongst Turkish and Moroccan Belgians, with a specific focus on the effect of partner-choice patterns. Divorce patterns of marriages established between 01 January 2001 and either 31 December 2003 (descriptive part), or 31 December 2005 (event-history analyses) are analysed and compared to marriages established between 01 January 1988 and 31 December 1990. We distinguish three marriage types: transnational marriages (i.e. marrying a partner from Morocco or Turkey), local intra-ethnic marriages (marrying another Moroccan of Turkish Belgian) and mixed marriages (i.e. marrying someone with a Belgian or other Western-European citizenship). To research divorce rates, we analysed population data from the Belgian national register, using piecewise constant log-rate event-history analyses with effect coding on all marriages taking place between 01 January 2001 and 31 December 2005 (N Turkish = 9631, N Moroccan = 17,786). First, the results reveal that in the past 15 years, divorce rates have doubled within Turkish and Moroccan migrant groups. Second, divorce rates are much higher amongst the Moroccan group. Third, there are clear differences between marriage types. Local intra-ethnic marriages have the lowest divorce levels, mixed marriages the highest, and transnational marriages take up a middle position.