Hallucination is fundamental to how transformer-based language models work. In fact, it’s their greatest asset: this is the method by which language models find links between sometimes disparate ...

Mathematical logic provides a rigorous framework to analyse proofs and extract computational content, while nonexpansive mappings, which are mappings that do not increase distances between points, ...

Mathematical logic, set theory, lattices and universal algebra form an interconnected framework that underpins much of modern mathematics. At its heart, mathematical logic provides rigorous formal ...

Some scientific discoveries matter because they reveal something new — the double helical structure of DNA, for example, or the existence of black holes. However, some revelations are profound because ...

Microsoft has introduced a new set of small language models called Phi-4-reasoning, Phi-4-reasoning-plus, and Phi-4-mini-reasoning, which are described as "marking a new era for efficient AI." These ...