On the robustness of functional equations

SIAM Journal on Computing

Volumn 28, Issue 6, 1999, Pages 1972-1997

  Rubinfeld, Ronitt ^a  

^a Department of Computer Science and School of Operations Research and Information Engineering   (United States)

Author keywords

Functional equations; Program testing; Property testing

Indexed keywords

COMPUTATIONAL GEOMETRY; FUNCTIONS; THEOREM PROVING;

FUNCTIONAL EQUATIONS;

LINEAR EQUATIONS;

EID: 0033147353     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539796298625     Document Type: Article

References (47)

1. M. ABADI, J. FEIGENBAUM, AND J. KILIAN, On hiding information from an oracle, J. Comput. System Sei., 39 (1989), pp. 29-50.
2. L. ADLEMAN, M. HUANG, AND K. KOMPELLA, Efficient checkers for number-theoretic computations, Inform, and Comput., 121(1995), pp. 93-102.
3. J. ACZéL, Lectures on Functional Equations and Applications, Academic Press, New York, 1966.
4. J. ACZéL AND J. DIIOMBRES, Functional Equations in Several Variables, Cambridge University Press, Cambridge, UK, 1989.
5. S. AR, M. BLUM, B. CODENOTTI, AND P. GEMMELL, Checking approximate computations over the reals, in Proc. 25th Annual ACM Symp. Theory of Computing, 1993, pp. 786-795.
6. S. ARORA, C. LUND, R. MOTWANI, M. SUDAN, AND M. SZEGEDY, Proof verification and the intractability of approximation problems, in Proc. 33rd IEEE Symp. Foundations of Computer Science, 1992, pp. 14-23.
7. S. ARORA AND S. SAFRA, Probabilistic checking of proofs: A new characterization of NP, in Proc. 33rd Annual IEEE Symp. Foundations of Computer Science, 1992, pp. 2-13.
8. D. BEAVER AND J. FEIGENBAUM, Hiding instances in multioracle queries, in Proc. 7th Annual Symp. Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sei. 415, Springer-Verlag, 1990, pp. 37-48.
9. L. BABAI, L. FORTNOW, AND C. LUND, Non-deterministic exponential time has two-prover interactive protocols, Comput. Complexity, 1 (1991), pp. 3-40.
10. L. BABAI, L. FORTNOW, L. LEVIN, AND M. SZEGEDY, Checking computations in poll/logarithmic time, in Proc. 23rd Annual ACM Symposium on Theory of Computing, 1991, pp. 21-31.
11. M. BELLARE, D. COPPERSMITH, J. HASTAD, M. KIWI, AND M. SUDAN, Linearity testing in characteristic two, Proc. 3Gth Annual Symp. Foundations of Computer Science, 1995, pp. 434-441.
12. M. BELLARE, O. GOLDREICII, AND M. SUDAN, Free bits, PCPs, and nonapproximabilityToward tight results, SIAM J. Comput., 27 (1998), pp. 804-915.
13. M. BELLARE, S. GOLDWASSER, C. LUND, AND A. RUSSELL, Efficient probabilistically checkable proofs, in Proc. 25th Annual ACM Symp. Theory of Computing, 1993, pp. 294-304.
14. M. BELLARE AND M. SUDAN, Improved non-approximability results, in Proc. ACM Symp. Theory of Computing, 1994, pp. 184-193.
15. M. BLUM, Designing Programs to Check Their Work, Tech. report TR-88-009, International Computer Science Institute, University of California at Berkeley, Berkeley, CA, 1988.
16. M. BLUM, B. CODENOTTI, P. GEMMELL, AND T. SHAHOUMIAN, Self-Correcting for Function Fields of Finite Transcendental Degree, Procc. 22nd Internat. Colloquium ICALP 95, Lecture Notes in Comput. Sei. 944, 1995, pp. 547-557.
17. M. BLUM, W. EVANS, P. GEMMELL, S. KANNAN, AND M. NAOR, Checking the correctness of memories, Proc. 32nd Annual Symp. Foundations of Computer Science, 1991, pp. 90-99.
18. M. BLUM AND S. KANNAN, Program correctness checking...and the design of programs that check their work, in Proc. 21st Annual ACM Symp. on Theory of Computing, 1989, pp. 86-97.
19. M. BLUM, M. LUBY, AND R. RUBINFELD, Self-testing/correcting with applications to numerical problems, J. Comput. System Sei., 47 (1993), pp. 549-595.
20. M. BLUM AND S. MICALI, How to generate cryptographically strong sequences of pseudo-random bits, SIAM J. Comput., 13 (1984), pp. 850-864.
21. M. BLUM AND H. WASSERMAN, Program result-checking: A theory of testing meets a test of theory, in Proc. 35th FOCS, 1994, pp. 382-392.
22. E. CASTILLO AND M. R. RuiZ-COBO, Functional Equations and Modelling in Science and Engineering, Marcel Dekker, New York, 1992.
23. R. CLEVE AND M. LUBY, A Note on Self-Testing/Correcting Methods for Trigonometric Functions, Tech. report TR-90-032, International Computer Science Institute, University of California at Berkeley, Berkeley, CA, 1990.
24. W. J. CODY, Performance evaluation of programs related to the real gamma function, ACM Trans. Math. Software, 17 (1991), pp. 46-54.
25. W. J. CODY AND L. STOLTZ, The use of Taylor series to test accuracy of function programs, ACM Trans. Math. Software, 17 (1991), pp. 55-63.
26. D. COPPERSMITH, untitled manuscript, December 1989 (result described in [19]).
27. F. ERGüN, S. R. KUMAR, AND R. RUBINFELD, Approximate checking of polynomials and functional equations, Proc. 37th IEEE Symp. Foundations of Computer Science, 1996, pp. 292-303.
28. F. ERGÜN, S. R. KUMAR, AND D. SIVAKUMAR, Self-testing without the generator bottleneck, SIAM J. Comput., to appear.
29. U. FEIGE, S. GOLDWASSER, L. LOVASZ, S. SAFRA, AND M. SZEGEDY, Approximating clique is almost NP-complete, in Proc. 32nd IEEE Symp. on Foundations of Computer Science, 1991, pp. 2-12.
30. J. FEIGENBAUM AND L. FORTNOW, Random-self-reducibility of complete sets, SIAM J. Cornput., 22 (1993), pp. 994-1005.
31. P. GEMMELL, R. LIPTON, R. RUBINFELD, M. SUDAN, AND A. WIGDERSON, Selftesting/correcting for polynomials and for approximate functions, in Proc. 23rd Annual ACM Symposium on Theory of Computing, 1991, pp. 32-42.
32. M. KLAWE, Limitations on explicit constructions of expanding graphs, SIAM J. Comput., 13 (1984), pp. 156-1C6.
33. S. RAVI KUMAR AND D. SIVAKUMAR, Efficient self-testing/self-correction of linear recurrences, Proc. 37th IEEE Foundations of Computer Science, 1996, pp. 602-611.
34. R. LlPTON, New directions in testing, in Distributed Computing and Cryptography, DIMACS Ser. Discrete Math. Theoret. Comput. Sei. 2, AMS, Providence, RI, 1991, pp. 191-202.
35. S. MlCALI, Computationally-sound proofs, Proc. 35th Annual Symp. Foundations of Computer Science, 1994, pp. 436-453.
36. A. POLISCIIUK AND D. SPIELMAN, Nearly linear size holographic proofs, Proc. 26th ACM Symp. on Theory of Computing, 1994, pp. 194-203.
37. R. RUBINFELD AND M. SUDAN, Testing polynomial functions efficiently and over rational domains, in Proc. 3rd Annual ACM-SIAM Symp. on Discrete Algorithms, SIAM, Philadelphia, 1992, pp. 23-43.
38. R. RUBINFELD AND M. SUDAN, Robust characterizations of polynomials with applications to program testing, SIAM J. Comput., 25 (1996), pp. 252-271.
39. N. SCHRIVER, personal communication, February 1990.
40. M. SUDAN, Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems, Ph.D. thesis, University of California, Berkeley, CA, 1992.
41. [4l] M. SUDAN, personal communication, 1994.
42. F. VAINSTEIN, Error detection and correction in numerical computations by algebraic methods, Proc. 9th Internat. Symp. AAECC-9, Lecture Notes in Comput. Sei. 539, SpringerVerlag, New York, 1991.
43. F. VAINSTEIN, Algebraic Methods in Hardware/Software Testing, Ph.D. thesis, Boston University, Boston, MA, 1993.
44. F. VAINSTEIN, Low redundancy polynomial checks for numerical computation, Appl. Algebra Engrg., Comm. Comput., 7 (1996), pp. 439-447.
45. F. VAINSTEIN, Self checking design technique for numerical computations, J. VLSI Design, 5 (1995), pp. 385-392.
46. R. ZlPPEL, Zero testing of algebraic functions, Inform. Process. Lett., 61 (1997), pp. 63-67.
47. R. ZIPPEL, Effective Polynomial Computation, Kluwer Academic Publishers, Norwell, MA, 1993.