Computability and Decision Procedures for Number Theory and Combinatorics

点击翻译按钮获取中文摘要

Broadly speaking, my research involves two different areas and how they interact. The first area concerns mathematical models of simple computers ("automata") and their capabilities. The second area concerns the basics of pure mathematics: number theory, combinatorics, and algebra. Although these two areas superficially seem quite separated, in reality they are closely connected. How can we use insights from automata theory to contribute to pure mathematics, and vice versa? And can we use automated theorem-proving to prove our mathematical insights "purely mechanically"?To take an example, consider additive number theory: the study of how to represent numbers as the sum of members of a given set. This is a well-studied area of pure mathematics that includes such celebrated results as Waring's theorem on sums of powers of integers (proved by Hilbert), and the Goldbach conjecture about sums of primes (still unproved). Recently, Cilleruelo, Luca, and Baxter proved that, for bases b 5, every natural number is the sum of at most three numbers whose base-b representation is a palindrome (a number that reads the same forwards and backwards, like the English word radar). But they were unable to prove this for bases b = 2, 3, 4.My collaborators and I completed the additive theory of the palindromes for the remaining cases, using automata theory and a decision procedure. For example, to handle the case of base b = 2, we rephrased the assertion "every natural number is the sum of at most four binary palindromes" as a claim about the computational behavior of a particular automaton A. We then used a known decision procedure for the universality problem for this class of automata to prove that our automaton A has the specified behavior. This is just one of many similar problems that are amenable to this approach.I propose to apply these ideas to many other problems in number theory, combinatorics, and algebra. I will identify suitable problems, search for appropriate computational models that can resolve them, and apply decision procedures to prove the theorems. I will also direct the preparation of free software, so that other mathematicians and computer scientists can use this approach in their own work. Already some open-source software, called Walnut, has been created by my student Hamoon Mousavi, and is being used by other researchers.My work actively involves the training of highly-qualified personnel, ranging from undergraduate students to postdoctoral researchers. These are essential to my work, both for solving problems and for writing software.