Notes

Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations


[Methods for] proof searching

To find a proof of some statement LongEqual[p,q] in a multiway system one can always in principle just start from p, evolve the system until it first generates q, then pick out the sequence of strings on the path from p to q. But doing this will usually involve building up a vast network of strings. And although at some level computational irreducibility and NP completeness (see page 766) imply that in general only a limited amount of this computational work can be saved, there are in practice quite often important optimizations that can be made. For finding a proof of LongEqual[p,q] is like searching for a path satisfying the constraint of going from p to q. And just like in the systems based on constraints in Chapter 5 one can usually do at least somewhat better than just to look at every possible path in turn.

For a start, in generating the network of paths one only ever need keep a single path that leads to any particular string; just like in many of my pictures of multiway systems one can in effect always drop any duplicate strings that occur. One might at first imagine that if p and q are both short strings then one could also drop any very long strings that are produced. But as we have seen, it is perfectly possible for long intermediate strings to be needed to get from p to q. Still, it is often reasonable to weight things so that at least at first one looks at paths that involve only shorter strings.

In the most direct approach, one takes a string and at each step just applies the underlying rules or axioms of the multiway system. But as soon as one knows that there is a path from a string u to a string v, one can also imagine applying the rule u->v to any string—in effect like a lemma. And one can choose which lemmas to try first by looking for example at which involve the shortest or commonest strings.

It is often important to minimize the number of lemmas one has to keep. Sometimes one can do this by reducing every lemma—and possibly every string—to some at least partially canonical form. One can also use the fact that in a multiway system if u->v and r->s then u<>r -> v<>s.

If one wants to get from p to q the most efficient thing is to use properties of q to avoid taking wrong turns. But except in systems with rather simple structure this is usually difficult to achieve. Nevertheless, one can for example always in effect work forwards from p, and backwards from q, seeing whether there is any overlap in the sets of strings one gets.


From Stephen Wolfram: A New Kind of Science [citation]