Herbrands’s Theorem

Herbrand’s Theorem

Automated theorem proving has two goals: (1) to prove theorems and (2) to do it automatically. Fully automated theorem provers for first-order logic have been developed, starting in the 1960’s, but as theorems get more complicated, the time that theorem provers spend tends to grow exponentially. As a result, no really interesting theorems of mathematics can be proved this way- the human life span is not long enough.

Therefore a major problem is to prove interesting theorems and the solution is to give the theorem provers heuristics, rules of thumb for knowledge and wisdom. Some heuristics are fairly general, for example, in a proof that is about t break into several cases do as much as possible that will be of broad applicability before the division into cases occurs. But many heuristics are area-specific; for instance, heuristics appropriate for plane geometry will probably not be appropriate for group theory. The development of good heuristics is a major area of research and requires