Theory Driven Discovery - AM (1976)

AM is a program that discovers concepts in elementary mathematics and set theory.

AM has 2 inputs:

• A description of some concepts of set theory (in LISP form). E.g. set union, intersection, the empty set.
• Information on how to perform mathematics. E.g. functions.

Given the above information AM discovered:

Integers

-- it is possible to count the elements of this set and this is an the image of this counting function -- the integers -- interesting set in its own right.

Addition

-- The union of two disjoint sets and their counting function.

Multiplication

-- Having discovered addition and multiplication as laborious set-theoretic operations more effective descriptions were supplied by hand.

Prime Numbers

-- factorisation of numbers and numbers with only one factor were discovered.

Golbach's Conjecture

-- Even numbers can be written as the sum of 2 primes. E.g. 28 = 17 + 11.

Maximally Divisible Numbers

-- numbers with as many factors as possible. A number k is maximally divisible is k has more factors than any integer less than k. E.g. 12 has six divisors 1,2,3,4,6,12.

How does AM work?

AM employs many general-purpose AI techniques:

• A frame based representation of mathematical concepts.
  • AM can create new concepts (slots) and fill in their values.
• Heuristic search employed
  • 250 heuristics represent hints about activities that might lead to interesting discoveries.
  • How to employ functions, create new concepts, generalisation etc.
• Hypothesis and test based search.
• Agenda control of discovery process.