The Automated Mathematician (AM) is one of the earliest successful discovery systems in artificial intelligence. Created by Douglas Lenat in 1977, it was designed to autonomously explore and generate mathematical concepts by manipulating short Lisp programs. AM was awarded the IJCAI Computers and Thought Award in 1977, recognizing its significance in the field of artificial intelligence and automated reasoning.
AM operated 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 employed elaborate heuristics for selecting which programs to extend and modify, based on the experiences of working mathematicians in solving problems.
Architecture and Control Flow
Lenat described AM as composed of hundreds of data structures called "concepts," together with hundreds of "heuristic rules" and a simple flow of control. He stated: "AM repeatedly selects the top task from the agenda and tries to carry it out. This is the whole control structure!" The system maintained an agenda of tasks, each associated with a priority, and iteratively selected the highest-priority task to execute. This approach allowed AM to focus its computational resources on the most promising avenues of exploration.
The heuristic rules guided the system in deciding which concepts to extend, merge, or modify. They encoded domain knowledge about mathematical practice, such as recognizing analogies between concepts or identifying interesting patterns. The agenda-based control structure was a precursor to later planning and search techniques in machine learning and automated reasoning.
Discovery Claims and Controversy
Lenat claimed that AM had rediscovered both Goldbach's conjecture and the fundamental theorem of arithmetic. However, later critics accused Lenat of over-interpreting the output of AM. 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 history, or otherwise could not be represented in the framework of the explicit rules.
Published versions of the rules often involved vague terms that were 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 was the user, via Rule 2: "If the user has recently referred to X, then boost the priority of any tasks involving X." This suggested that much of the real discovery work might have been buried in unexplained procedures or user interactions.
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: Eurisko
This intuition was the basis of AM's successor Eurisko, which attempted to generalize the search for mathematical concepts to the search for useful heuristics. Eurisko extended AM's approach by not only discovering mathematical concepts but also discovering and refining the heuristics themselves, allowing the system to improve its own search strategies over time. This represented an early step toward meta-learning and self-improving AI systems.
Legacy and Influence
AM's legacy extends beyond its immediate results. It influenced subsequent work in automated discovery, including systems like HR and Graffiti, which are related math discovery programs. The questions AM raised about the nature of creativity, the role of heuristics, and the interpretation of AI output remain relevant in modern generative AI research. AM's approach of representing knowledge as executable programs anticipated later developments in neural networks and program synthesis, where models generate code or mathematical expressions.
Despite its controversies, AM demonstrated that computers could explore mathematical spaces in ways that surprised their creators, and it highlighted the importance of careful interpretation in evaluating AI discoveries. The system's focus on heuristics and agenda-based control influenced later work in automated planning and search, and its emphasis on user interaction foreshadowed interactive machine learning approaches.
See Also
- Automated theorem proving
- Symbolic mathematics
- Experimental mathematics