Automated Theory Formation in Mathematics

A program cal led " A M " is descr ibed which cai r ies on simple mathemat ics research : defining, and studying new concepts under the guidance of a large body of heuiist ic rules. The 250 h e u r K t u s communicate via an agenda mechanism, a g loba l p r i o r i t y queue of small bisk', for the program to pei f o i m and teasons why each task is plausible (e.g., "Find PENCRAHZTION. of ' p r n e s ' , because turued out to be so useful a Conccpi"). Fach concept is an active, s t ruc tured know ledge module. One bund led vei y incomplete modules are in i t ia l ly supp l ied, each one corresponding to an e l emen ta r y set theoret ic concept (e.g., union). This p r o v i d e s a de f in i te but immense space which AM begins to e x p l o r e . In one boor, AM rediscovers hundreds of common concep ts ( inc luding s ingleton sets, natural numbers, a r i thmet i c ) and theorems (e.g., unique factor izat ion). of fu l ly automatic theo ry format ion in some scientif ic f ield. "1 his inc ludes t w o act iv i t ies: (i) discovering relat ionships among k n o w n concepts (e.g., by formal manipulations, or by no t i c ing regu la r i t ies in empir ical data), and (ii) def ining new concep ts for invest igat ion. Meta-Dendral [Buchanan 75 ] p e r f o r m s on ly the f i rst of these; most domainindependent concept learn ing programs (Winston 70] per form only the la t ter of these: whi le they do create new concepts, the in i t i a t i ve is not the i rs but rather is that of a human " t e a c h e r " who al ready has the concepts in mind. What we are descr ib ing is a computet program which de f ines new concepts, investigates them, notices regu la r i t i es in the data about them, and conjectures re la t i onsh ips b e t w e e n them. This new informat ion is used by the p rog ram to evaluate the newly -de f ined concepts, concen t ra te upon the most in terest ing ones, and i terate the e n t i r e process. This paper describes such a program: AM.