The Centrality of Heuristics in the Mathematical Discoveries of AM

   1) The Centrality of Heuristics in the Mathematical Discoveries of AM ( Automatic Mathematician)

   [A]t one point AM [Automatic Mathematician] had some notions of sets, set-operations, numbers, and simple arithmetic. One heuristic rule it knew said " If F is an interesting relation, then look at its inverse". This rule fired after AM had studied "multiplication" for a while. The r.h.s. of the rule then directed AM to define and study the relation "divisors-of" (e.g. divisors-of (12) {1,2,3,4,6,12}. Another heuristic rule that later fired said " If f is a relation from A into B, then it's worth examining those members of A which map into extremal members of B." In this case, f was matched to "divisors-of", A was "numbers", B was "sets of numbers", and an extremal member of B might be, e.g., a very small set of numbers. Thus this heuristic rule caused AM to define the set of numbers with no divisors, the set of numbers with only 1 divisor, with only 2 divisors, etc. One of these sets (the last [ sic] mentioned) turned out subsequently to be quite important; these numbers are of course the primes. (Lenat & Harris, 1978, p. 30)

   2) The Power of Heuristics in Problem Solving

   Extraordinarily rapid progress during the early stages of an attack on a new problem area is a rather common occurrence in AI research; it merely signifies that the test cases with which the system has been challenged are below the level of difficulty where combinatorial explosion of the number of pathways in the problem space sets in.... It is the goal of AI research to move that threshold higher and higher on the scale of problem complexity through the introduction of heuristics-heuristics to reduce the rate of growth of the solution tree, heuristics to guide the development of the tree so that it will be rich in pathways leading to satisfactory problem solutions, and heuristics to direct the search to the "best" of these pathways. (Gelernter, quoted in Barr & Feigenbaum, 1982, pp. 139-140)