Herbrandss Theorem :: essays research papers

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 much experience and insight. Brief History In 1930 Kurt Godel and Jaques Herbrand proved the first version of what is now the completeness of predicate calculus. Godel and Herbrand both demonstrated that the proof machinery of the predicate calculus can provide a formal proof for every logically true proposition, while also giving a constructive method for finding the proof, given the proposition. In 1936 Alonzo Church and Alain Turing independently discovered a fundamental negative property of the predicate calculus. â€Å"Until then, there had been an intense search for a positive solution to what was called the decision problem – which was to create an algorithm for the predicate calculus which would correctly determine, for any formal sentence B and any set A of formal sentences, whether or not B is a logical consequence of A. Church and Turing found that despite the existence of the proof procedure, which correctly recognizes (by constructing the proof of B from A) all cases where B is in fact a logical consequence of A, there is not and cannot be an algorithm which can similarly correctly recognize all cases in which B is not a logical consequence of A. "It means that it is pointless to try to program a computer to answer 'yes' or 'no' correctly to every question of the form 'is this a logically true sentence ?'" Church and Turing proved that it was impossible to find a general decision to verify the inconsistency of a formula.