Approximate testing with error relative to input size

We formalize the notion and initiate the investigation of approximate testing for arbitrary forms of the error term. Until now only the case of absolute error had been addressed ignoring the fact that often only the most significant figures of a numerical calculation are valid. This work considers approximation errors whose magnitude grows with the size of the input to the program. We demonstrate the viability of this new concept by addressing the basic and benchmark problem of self-testing for the class of linear and polynomial functions. We obtain stronger versions of results of Ergun et al. (Proceedings of the 37th FOCS, 1996, pp. 592-601) by exploiting elegant techniques from Hyers-Ulam stability theory.

[1]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[2]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[3]  Manuel Blum,et al.  Checking approximate computations over the reals , 1993, STOC.

[4]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[5]  Donald E. Knuth.  The Art of Computer Programming 2 / Seminumerical Algorithms , 1971 .

[6]  Ronitt Rubinfeld,et al.  On the robustness of functional equations , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[7]  Frédéric Magniez.  Multi-linearity Self-Testing with Relative Error , 2000, STACS.

[8]  Ronitt Rubinfeld,et al.  Self-testing/correcting for polynomials and for approximate functions , 1991, STOC '91.

[9]  D. Djoković,et al.  A representation theorem for $(X_1-1)(X_2-1)...(X_n-1)$ and its applications , 1969 .

[10]  Ronitt Rubinfeld,et al.  Self-testing polynomial functions efficiently and over rational domains , 1992, SODA '92.

[11]  Zbigniew Gajda.  Local stability of the functional equation characterizing polynomial functions , 1990 .

[12]  R. Rubinfeld.  A mathematical theory of self-checking, self-testing and self-correcting programs , 1991 .

[13]  Themistocles M. Rassias,et al.  On the behavior of mappings which do not satisfy Hyers-Ulam stability , 1992 .

[14]  Joan Feigenbaum,et al.  Hiding Instances in Multioracle Queries , 1990, STACS.

[15]  G. Forti.  Hyers-Ulam stability of functional equations in several variables , 1995 .

[16]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[17]  T. Rassias.  On the stability of the linear mapping in Banach spaces , 1978 .

[18]  Richard J. Lipton,et al.  New Directions In Testing , 1989, Distributed Computing And Cryptography.

[19]  D. H. Hyers.  On the Stability of the Linear Functional Equation. , 1941, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[21]  Manuel Blum,et al.  Designing programs that check their work , 1989, STOC '89.

[22]  Ronitt Rubinfeld,et al.  Approximate checking of polynomials and functional equations , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[23]  Manuel Blum,et al.  Program result-checking: a theory of testing meets a test of theory , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[24]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[25]  Ronitt Rubinfeld,et al.  Approximate Checking of Polynomials and Functional Equations (extended abstract). , 1996, IEEE Annual Symposium on Foundations of Computer Science.

[26]  Themistocles M. Rassias,et al.  Approximate homomorphisms , 1992 .

[27]  Michael H. Albert,et al.  Functions with bounded nth differences , 1983 .