Short answers, pulled from the story.
Automated reasoning is a subfield of artificial intelligence and computer science dedicated to building computer programs that can reason completely, or nearly completely, without human input. It draws on formal logic, theoretical computer science, and philosophy, and its most developed areas include automated theorem proving, interactive theorem proving, and automated proof checking.
The Logic Theorist, built by Allen Newell, Cliff Shaw, and Herbert A. Simon in 1956, proved thirty-eight of fifty-two theorems drawn from chapter two of Principia Mathematica. For one theorem, the program produced a proof more elegant and efficient than the one written by Whitehead and Russell, requiring fewer steps than the original.
The Boyer-Moore Theorem Prover, also called NQTHM, was started in 1971 at Edinburgh, Scotland. It was a fully automatic theorem prover built in Pure Lisp, designed with influence from John McCarthy and Woody Bledsoe, and it used rewriting, symbolic evaluation, and an induction heuristic to prove mathematical theorems.
Rocq, formerly known as Coq, is an automated proof assistant developed in France. It can automatically extract executable programs from specifications, producing either Objective CAML or Haskell source code. Properties, programs, and proofs are all expressed in a single language called the Calculus of Inductive Constructions.
The TPTP library, created by Sutcliffe and Suttner in 1998, is a regularly updated collection of formal logic problems used to benchmark automated theorem provers. Problems from the library are selected for a regular competition among theorem provers held at the CADE conference.
In the 2020s, AI researchers developed two main approaches to improve large language model reasoning: reasoning language models that spend additional time on a problem before generating an answer, and neuro-symbolic architectures that combine neural networks with symbolic reasoning systems to prevent hallucinations.