Automated Mathematician
Encyclopedia
The Automated Mathematician (AM) is one of the earliest successful discovery system
s. It was created by Doug Lenat in Lisp
, and in 1977 led to Lenat being awarded the IJCAI Computers and Thought Award
.
AM worked by generating and modifying short Lisp programs which were then interpreted as defining various mathematical concepts; for example, a program that tested equality between the length of two lists was considered to represent the concept of numerical equality, while a program that produced a list whose length was the product of the lengths of two other lists was interpreted as representing the concept of multiplication. The system had elaborate heuristics for choosing which programs to extend and modify, based on the experiences of working mathematicians in solving mathematical problems.
What's more, the published versions of the rules often involve vague terms that are not defined further, such as "If two expressions are structurally similar, ..." (Rule 218) or "... replace the value obtained by some other (very similar) value..." (Rule 129).
Another source of information is the user, via Rule 2: "If the user has recently referred to X, then boost the priority of any tasks involving X." Thus, it appears quite possible that much of the real discovery work is buried in unexplained procedures.
Lenat claimed that the system had rediscovered both Goldbach's conjecture
and the Unique Prime Factorization
Theorem. Later critics accused Lenat of over-interpreting the output of AM. In his paper Why AM and Eurisko appear to work, Lenat conceded that any system that generated enough short Lisp programs would generate ones that could be interpreted by an external observer as representing equally sophisticated mathematical concepts. However, he argued that this property was in itself interesting—and that a promising direction for further research would be to look for other languages in which short random strings were likely to be useful.
, which attempted to generalize the search for mathematical concepts to the search for useful heuristic
s.
Discovery system
A discovery system is an artificial intelligence system which attempts to discover new scientific concepts or laws.Notable discovery systems have included,*Autoclass*Automated Mathematician*DALTON*Eurisko*Glauber*Machine for Questions and Answers...
s. It was created by Doug Lenat in Lisp
Lisp programming language
Lisp is a family of computer programming languages with a long history and a distinctive, fully parenthesized syntax. Originally specified in 1958, Lisp is the second-oldest high-level programming language in widespread use today; only Fortran is older...
, and in 1977 led to Lenat being awarded the IJCAI Computers and Thought Award
IJCAI Computers and Thought Award
The IJCAI Computers and Thought Award is presented by the International Joint Conferences on Artificial Intelligence , recognizing outstanding young scientists in artificial intelligence. It was originally funded with royalties received from the book "Computers and Thought" , and is currently...
.
AM worked by generating and modifying short Lisp programs which were then interpreted as defining various mathematical concepts; for example, a program that tested equality between the length of two lists was considered to represent the concept of numerical equality, while a program that produced a list whose length was the product of the lengths of two other lists was interpreted as representing the concept of multiplication. The system had elaborate heuristics for choosing which programs to extend and modify, based on the experiences of working mathematicians in solving mathematical problems.
Controversy
Lenat claimed that the system was composed of hundreds of data structures called "concepts," together with hundreds of "heuristic rules" and a simple flow of control: "AM repeatedly selects the top task from the agenda and tries to carry it out. This is the whole control structure!" Yet the heuristic rules were not always represented as separate data structures; some had to be intertwined with the control flow logic. Some rules had preconditions that depended on the past history, or otherwise could not be represented in the framework of the explicit rules.What's more, the published versions of the rules often involve vague terms that are not defined further, such as "If two expressions are structurally similar, ..." (Rule 218) or "... replace the value obtained by some other (very similar) value..." (Rule 129).
Another source of information is the user, via Rule 2: "If the user has recently referred to X, then boost the priority of any tasks involving X." Thus, it appears quite possible that much of the real discovery work is buried in unexplained procedures.
Lenat claimed that the system had rediscovered both Goldbach's conjecture
Goldbach's conjecture
Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...
and the Unique Prime Factorization
Integer factorization
In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
Theorem. Later critics accused Lenat of over-interpreting the output of AM. In his paper Why AM and Eurisko appear to work, Lenat conceded that any system that generated enough short Lisp programs would generate ones that could be interpreted by an external observer as representing equally sophisticated mathematical concepts. However, he argued that this property was in itself interesting—and that a promising direction for further research would be to look for other languages in which short random strings were likely to be useful.
Successor
This intuition was the basis of AM's successor EuriskoEurisko
Eurisko is a program written by Douglas Lenat in RLL-1, a representation language itself written in the Lisp programming language. A sequel to Automated Mathematician, it consists of heuristics, i.e. rules of thumb, including heuristics describing how to use and change its own heuristics...
, which attempted to generalize the search for mathematical concepts to the search for useful heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
s.
Further reading
- Lenat, D.B., (1976), AM: An artificial intelligence approach to discovery in mathematics as heuristic search, Ph.D. Thesis, AIM-286, STAN-CS-76-570, and Heuristic Programming Project Report HPP-76-8, Stanford University, AI Lab., Stanford, CA. Published in Knowledge-based systems in artificial intelligence along with Randall Davis's Ph.D. Thesis, McGraw-Hill, 1982.
- Lenat, D. B., and Brown, J. S. (August 1984). "Why AM and EURISKO appear to work." Artificial Intelligence 23(3):269—294.
- Ritchie, G. D., and Hanna, F. K. (August 1984). "AM: A case study in AI methodology" Artificial Intelligence 23(3):249—268.
See also
- Douglas LenatDouglas LenatDouglas B. Lenat is the CEO of Cycorp, Inc. of Austin, Texas, and has been a prominent researcher in artificial intelligence, especially machine learning , knowledge representation, blackboard systems, and "ontological engineering"...
- Computer-assisted proofComputer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer.Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and...
- Automated theorem provingAutomated theorem provingAutomated theorem proving or automated deduction, currently the most well-developed subfield of automated reasoning , is the proving of mathematical theorems by a computer program.- Decidability of the problem :...
- Symbolic mathematics
- Experimental mathematicsExperimental mathematicsExperimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns...
External links
- Edmund Furse; Why did AM run out of steam?
- Ken Haase's Ph.D. Thesis; Invention and Exploration in Discovery, a rational reconstruction of Doug Lenat's seminal AM program and an analysis of the relationship between invention and exploration in discovery.