Automated conjecturing III : property-relations conjectures

Discovery in mathematics is a prototypical intelligent behavior, and an early and continuing goal of artificial intelligence research. We present a heuristic for producing mathematical conjectures of a certain typical form and demonstrate its utility. Our program conjectures relations that hold between properties of objects (property-relation conjectures). These objects can be of a wide variety of types. The statements are true for all objects known to the program, and are the simplest statements which are true of all these objects. The examples here include new conjectures for the hamiltonicity of a graph, a well-studied property of graphs. While our motivation and experiments have been to produce mathematical conjectures-and to contribute to mathematical research-other kinds of interesting property-relation conjectures can be imagined, and this research may be more generally applicable to the development of intelligent machinery

Automated conjecturing II : chomp and reasoned game play

We demonstrate the use of a program that generates conjectures about positions of the combinatorial game Chomp—explanations of why certain moves are bad. These could be used in the design of a Chomp-playing program that gives reasons for its moves. We prove one of these Chomp conjectures—demonstrating that our conjecturing program can produce genuine Chomp knowledge. The conjectures are generated by a general purpose conjecturing program that was previously and successfully used to generate mathematical conjectures. Our program is initialized with Chomp invariants and example game boards—the conjectures take the form of invariant-relation statements interpreted to be true for all board positions of a certain kind. The conjectures describe a theory of Chomp positions. The program uses limited, natural input and suggests how theories generated on-the-fly might be used in a variety of situations where decisions—based on reasons—are required

Top-down Automated Theorem Proving (Notes for Sir Timothy)

We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code all relevant domain concepts. These concepts encode a large portion of the knowledge in any domain. Furthermore, researchers should write programs that produce proofs of the kind that human mathematicians write (and publish); this means proofs that might sometimes have mistakes; and this means making inferences that are sometimes invalid. This approach is meant to contrast with the historically dominant "bottom up" approach: coding fundamental types (typically sets), axioms and rules for (valid) inference, and building up from this foundation to the theorems of mathematical practice and to their outstanding questions. It is an important fact that the actual proofs that mathematicians publish in math journals do not look like the formalized proofs of Russell & Whitehead's Principia Mathematica (or modern computer systems like Lean that automate some of this formalization). We believe some "lack of rigor" (in mathematical practice) is human-like, and can and should be leveraged for ATP.Comment: Cross list with cs.A

Conjecturing-Based Computational Discovery of Patterns in Data

Modern machine learning methods are designed to exploit complex patterns in data regardless of their form, while not necessarily revealing them to the investigator. Here we demonstrate situations where modern machine learning methods are ill-equipped to reveal feature interaction effects and other nonlinear relationships. We propose the use of a conjecturing machine that generates feature relationships in the form of bounds for numerical features and boolean expressions for nominal features that are ignored by machine learning algorithms. The proposed framework is demonstrated for a classification problem with an interaction effect and a nonlinear regression problem. In both settings, true underlying relationships are revealed and generalization performance improves. The framework is then applied to patient-level data regarding COVID-19 outcomes to suggest possible risk factors.Comment: 25 pages, 6 figure