1. Discovery Systems Author: Kenneth W. Hasse Jr. Presenter: Peter Yoon

2. Introduction Process of discovery and creativity –By constructing computer programs which Invent interesting constructions: invention Discover significant regularities: exploration

3. Prize for Math Discovery Prize of $100,000 –Edward Fredkin, 1980 –First constructed machine that made an important mathematical discovery

4. History D. Lenat, AM, 1975. D. Lenat, Eurisko, 1977. K. Haase, Cyrano, 1986. S. Fajtlowicz, Graffiti, 1986. S. Epstein, GT, 1987. R. Bagai et.al., unnamed, 1993. S. Colton et al., HR, 1998.

5. AM: Automated Mathematician AM: Automated mathematical conjecture-making AM is a computer program which defines new concepts, investigates them, notices regularities in the data about them, and conjectures relationships between them

6. Cyrano Used to experiment with interaction of exploration and invention in discovery Vocabulary: a set of data structures and combining operations together with an opaque interpreter which generates ‘examples’ from data structure Separating the exploration and invention of descriptive spaces

7. Cycle of representational extension

8. AM’s Starting Place Domain of data structures –Sets, bags(unordered collections with repeated elements), ordered sets, lists Operations –Union, difference, comparison –COALESCE: takes an operation with two inputs, define new Op taking a single input & apply self. COMPOSE -> COMPOSE-WITH-SELF

9. Inventing Cardinality

10. AM notices that the comparison operation for objects in general is rarely satisfied –Few fairs of objects pass its determination & AM decides to attempt to generalize it. Cardinality: If two sets have the same number of members, they are equivalent in SAME-SIZE’s term

11. Inventing Numbers AM’s CANONIZE operation –Takes a relation r and synthesizes a function f s.t.; I.e. the function f maps objects related by r into the same object –r(x,y) p(f(x), f(y))

12. Inventing Numbers BAG-OF-Ts –For instance, a set, bag, or list with five arbitrary elements is transformed into a bag containing five copies of the symbol T. –Equivalent to the notion of ‘number’, since there is exactly one BAF-OF-Ts for every number.

13. Prime Numbers Applying COALESCE to addition & multiplication –DOUBLE & SQUARE SQUARE-ROOT: inverting SQUARE Set of numbers with three divisors –One must be 1 and one must be n itself –Square root of n is numbers with two divisors: Primes

14. From AM to Cyrano Generality –Attempts to free itself from the domain specificity of AM –New concepts are formed by combinations of operations and categories, rather than by internal mutations Module representation: precise contract

15. From AM to Cyrano Simplicity –Definitions are considered in strict order of simplicity: Cyrano considers new definitions in order of decreasing simplicity –Search is pruned only by superficial empirical analysis Not ‘appear intelligent’ –In AM, Lenat labored under the constraint that the program ‘appear’ intelligent

16. Cyrano example [prefix][prefix[predecessor]] [prefix [predecessor] [predecessor]]

17. Numbers modulo 3 [prefix [predecessor] [prefix [predecessor] [predecessor]]]

18. Pool of definition Domain definition –Simple computer programs take particular inputs and yield particular results Fragments –Computer programs with ‘blank spaces’ where domain operations may be inserted. Cyrano construct new definition –By combining a fragmentary def & domain def.

19. Exploration New combinations of existing definitions –Order of decreasing simplicity Empirical examination by a ‘triage process’ –For definitions which appear to be reasonable to the type system –Generate examples of definition Examine for particular structure or regularities Check whether a definition is never defined

20. Invention Operates on the result of exploration Rebiasing –Arrange the constructed definition according to rules Reification –Organizes the partial evaluations of the definition into class –Add x and y : add x and 2, add x and 3, … –Ref: http://web.media.mit.edu/~haase/thesis/node23.htmlhttp://web.media.mit.edu/~haase/thesis/node23.html Construct new vocabulary

21. AM Family’s Malaise AM was not able to discover any ‘new-to- mankind’ mathematics purely on its own Eurisko, a successor to AM. Lenat did not report any conjectures it made Cyrano is a thoughtful reimplementation of Lenat’s controversial Eurisko program; Cyrano made any math conjectures

22. Conclusion First trial to mathematical automated conjecture making Does not contribute new proposition Graffiti: first program to have actually made conjectures. –Ref: C.E. Larson, “Intelligent Machinery and Mathematical Discovery”, 1999 Possibility to construct an intelligent machine