Self-correcting for function fields of finite transcendental degree

Abstract

We use algebraic field extension theory to find self-correctors for a broad class of functions. Many functions whose translations are contained in a function field that is a finite degree extension of a scalar field satisfy polynomial identities that can be transformed into self-correctors. These functions can be efficiently corrected in a way that is simpler and different from how the functions are actually computed. This is an essential feature of program self-correcting. Among the functions for which we present self-correctors are many rational expressions of x,e ^x, and sin(x) (over the real and complex fields) as weD as many rational expressions of x, g ^x (g a generator) mapping the integers into a finite field and many rational expressions of x,log _h (x) (h a generator) mapping a finite field into the reals.

The new tools presented in this extended abstract will be useful to the theory of program self testing/correcting. Furthermore, they may yield new results in complexity theory. Previous work in the self-testing of polynomials had important applications in the PCP protocols that proved the hardness of approximating max-SNP problems.

Supported by NSF grant CCR-9201092.

Part of this work was done while visiting the International Computer Science Institute, Berkeley, CA 94704. Partially supported by the ESPRIT B.R.A. Project 9072 GEPPCOM.

Part of this work was performed at Sandia National Laboratories and was supported by the U.S. Department of Energy under contract DE-AC04-76DP00789; part of the work was done while at UC Berkeley supported by NSF grant number CCR-9201092.

Supported by a Fannie and John Hertz Foundation fellowship.