CPSC 681 Graduate Seminar, Fall 1999

Monday, October 18, 1999
4:10-5:25 pm
Room 124, H. R. Bright Building

Proving Elementary Calculus Theorems in ACL2

Dr. Ruben A. Gamboa

Dr. Gamboa received a B.S. degree from Angelo State University in 1984, an M.C.S. from Texas A&M University in 1986, and a Ph.D. from the University of Texas at Austin in 1999. All degrees are in computer science. As part of his dissertation he modified the ACL2 theorem prover to support the irrational numbers, using the principles of non-standard analysis. His interests range over many issues in computation, encompassing deductive databases, computability, artificial intelligence, and automated theorem proving. He has been with Logical Information Machines (LIM) since 1990, where he is the chief architect for its financial database server. Before joining LIM, he was a member of the technical staff at MCC, a research consortium located in Austin, TX.

This talk presents an overview of the architecture of the Boyer-Moore theorem prover, Nqthm, and its "industrial-strength" cousin, ACL2. Nqthm and ACL2 are theorem provers over a first-order logic with no quantifiers. Users of Nqthm and ACL2 provide function definitions and theorems in the syntax of LISP. Typical proof efforts involve the definition of several recursive functions and the use of induction to prove lemmas about these functions. Nqthm was a pioneer in the automatic discovery of induction schemes to prove theorems.

Non-standard analysis -- invented by Robinson, also the inventor of resolution theorem proving -- which formalizes the intuitive reasoning of infinitesimal numbers used by Leibnitz when he co-invented calculus, by Newton in the Principia, and by countless scientists and engineers since will also be discussed. A feature of non-standard analysis is that induction takes the place of many of the compactness and limit arguments used in traditional analysis. Since ACL2 has strong support for induction, it is a perfect theorem prover for automating non-standard analysis, even though it lacks the expressiveness usually associated with calculus, such as quantifiers and sets. Proofs will be presented in ACL2 of a few theorems from elementary calculus, such as the intermediate-value and mean-value theorems.

Everyone is invited and welcome to attend the seminars in this series.