## CPSC 681 Graduate Seminar, Fall 1999

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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.*