Inheritable Properties and Computer Assisted Proofs in Dynamics

1 0 Introduction. Computer assisted proofs are becoming an important factor of present day mathematics. Since there exist short theorems having arbitrarily long proofs (a consequence of GG odel's incompletness theorems, see 4, 6, 23]) and the power of computers is growing exponentially, this shoudbe expected. The general scheme for constructing computer assisted proofs is actually very simple and is a direct consequence of the fact that any computer may deal only with nite sets. A proof recipe 1. Take a suitable theory which reduces the problem to a question concerning a given nite set. 2. Apply an appropriate algorithm to answer the question. If the rst step of the scheme is performed succesfully then the recipe is guaranteed to work, at least theoretically, because the case analysis algorithm may be always applied. In practice, the availability of a fast algorithm is often crucial. The scheme also shows that computers will never replace mathematitians, since both developing the reduction in Step 1 and constructing the algorithm in Step 2 are jobs for a mathematician. Actually, the job is often quite challenging as in the case of the proof of the four color hypothesis by Appel and Haken 1, 2], probably the most famous computer assisted proof. Computer assisted proofs in discrete mathematics and algebra are relatively common. There are theories in which one can rigorously prove or refute any conjectured statement in a given class by reducing the problem to a question which may be answered by running an appropriate algorithm. An example is the algorithmic proof theory of hypergeometric identities (see 27]).