Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem

Journal of Computer and System Sciences

Volumn 57, Issue 2, 1998, Pages 127-143

  Agrawal, Manindra ^a   Allender, Eric ^b   Rudich, Steven ^c  

^a INDIAN INSTITUTE OF TECHNOLOGY KANPUR   (India)

^b RUTGERS UNIVERSITY   (United States)

^c Carnegie Mellon University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

PROBLEM SOLVING; THEOREM PROVING;

GAP THEOREM; ISOMORPHISM THEOREM;

COMPUTATIONAL COMPLEXITY;

EID: 0032181939     PISSN: 00220000     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcss.1998.1583     Document Type: Article

References (37)

1. M. Agrawal, On the isomorphism problem for weak reductibilities, J. Comput. System Sci. 53 (1996), 267-282.
2. M. Agrawal, DSPACE(n) (questioned equal to) NSPACE(n): A degree theoretic characterization, J. Comput. System Sci. 54 (1997), 383-392.
3. M. Agrawal, For completeness, sublogarithmic space is no space, manuscript.
4. M. Agrawal and E. Allender, An isomorphism theorem for circuit complexity, in "Proc. 11th Annual IEEE Conference on Computational Complexity, 1996," pp. 2-11.
5. M. Agrawal, E. Allender, R. Impagliazzo, T. Pitassi, and S. Rudich, Reducing the complexity of reductions, in "Proc. 29th ACM Symposium on Theory of Computing, 1997," pp. 730-738.
6. 1 formulae on finite structures, Ann. Pure Appl. Logic 24 (1983), 1-48.
7. E. Allender, P-uniform circuit complexity, J. Assoc. Comput. Mach. 36 (1989), 912-928.
8. E. Allender, J. Balcázar, and N. Immerman, A first-order isomorphism theorem, SIAM J. Comput. 26 (1997), 557-567.
9. E. Allender and V. Gore, Rudimentary reductions revisited, Inform. Process. Lett. 40 (1991), 89-95.
10. N. Alon and J. Spencer, "The Probabilistic Method," Wiley, New York, 1992.
11. 0-reductions cannot prove the PCP theorem, manuscript, 1995.
12. J. Balcázar, J. Díaz, and J. Gabarró, "Structural Complexity, I, II," Springer-Verlag, New York/Berlin, 1988, 1990.
13. 1, J. Comput. System Sci. 41 (1990), 274-306.
14. P. Beame, A switching lemma primer, manuscript, http:// www.cs.washington.edu/homes/beame/papers.html.
15. L. Berman and J. Hartmanis, On isomorphism and density of NP and other complete sets, SIAM J. Comput. 6 (1977), 305-322.
16. R. Boppana and M. Sipser, The complexity of finite functions, in "Handbook of Theoretical Computer Science, Vol. A: Algorithms and Complexity" (J. van Leeuwen, Ed.), pp. 757-804, Elsevier, Amsterdam, 1990.
17. A. Borodin, Computational complexity and the existence of complexity gaps, J. Assoc. Comput. Mach. 19 (1972), 158-174.
18. P. Erdo″s and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960), 85-90.
19. L. Fortnow and S. Laplante, Circuit lower bounds à la Kolmogorov, Inform. Comput. 123 (1995), 121-126.
20. M. Furst, J. Saxe, and M. Sipser, Parity, circuits, and the polynomial-time hierarchy, Math. Systems Theory 17 (1984), 13-27.
21. 0, Inform. Process. Lett. 26 (1987), 153-155.
22. J. Håstad, R. Impagliazzo, L. Levin, and M. Luby, "Construction of a Pseudo-Random Generator from Any One-Way Function," Technical Report, 91-068 ICSI, 1990.
23. N. Jones, Space-bounded reducibility among combinatorial problems, J. Comput. System Sci. 11 (1975), 68-85.
24. D. Joseph and P. Young, Some remarks on witness functions for non-polynomial and non-complete sets in NP, Theor. Comput. Sci. 39 (1985), 225-237.
25. D. Joseph, R. Pruim, and P. Young, Collapsing degrees in subexponential time, in "Proc. 9th Structure in Complexity Theory Conference, 1994" pp. 367-382.
26. K.-I. Ko, T. J. Long, and D.-Z. Du, On one-way functions and polynomial-time isomorphisms, Theor. Comput. Sci. 47 (1986), 263-276.
27. S. Kurtz, S. Mahaney, and J. Royer, Collapsing degrees, J. Comput. System Sci. 37 (1988), 247-268.
28. S. Kurtz, S. Mahaney, and J. Royer, The structure of complete degrees, in "Complexity Theory Retrospective" (A. Selman, Eds.), pp. 108-146, Springer-Verlag, New York/Berlin, 1990.
29. S. Kurtz, S. Mahaney, and J. Royer, The isomorphism conjecture fails relative to a random oracle, J. Assoc. Comput. Mach. 42 (1995), 401-420.
30. M. Li and P. Vitányi, "An Introduction to Kolmogorov Complexity and its Applications," Springer-Verlag, New York/Berlin, 1993.
31. S. Lindell, How to define exponentiation from addition and multiplication in first-order logic on finite structures, manuscript. This improves an earlier characterization: S. Lindell, A purely logical characterization of circuit uniformity, in "Proc. 7th Structure in Complexity Theory Conference, 1992," pp. 185-192.
32. M. Luby, "Pseudorandomness and Cryptographic Applications," Princeton Univ. Press, Princeton, NJ, 1996.
33. N Nisan, "Using Hard Problems to Create Pseudorandom Generators," MIT Press, Cambridge, MA, 1992.
34. W. J. Paul, N. Pippenger, E. Szemerédi, and W. Trotter, On determinism versus non-determinism and related problems, in "Proc. Foundations of Computer Science, 1983," pp. 429-438.
35. J. Rogers, The isomorphism conjecture holds and one-way functions exist relative to an oracle, J. Comput. System Sci. 54 (1997), 412-423.
36. A. Selman, A survey of one way functions in complexity theory, Math. Systems Theory 25 (1992), 203-221.
37. B. A. Trakhtenbrot, Turing computations with logarithmic delay, Algebra i Logika 3 (1964), 33-48.