Abstract
Given a setA, we investigate the lattice structure formed by those of its subsets that belong to the complexity classP, taken modulo finite variations and ordered by inclusion. We show that up to isomorphism, only three structures are possible for this lattice. IfA isP-immune, itsP-subset structure degenerates to the trivial one-element lattice. IfA is “almostP-immune” but notP-immune (for instance, ifA is inP), itsP-subset structure is isomorphic to the countable atomless Boolean latticeℬ. In all other cases theP-subset structure is isomorphic toℬ (ω), the weak countable power ofℬ. All natural intractable sets appear to fall in the third category. The results generalize to many other complexity classes, and similar characterizations hold for, e.g., the structures formed by recursive complexity cores.
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B. Aspvall, M. F. Plass, and R. E. Tarjan. A linear time algorithm for testing the truth of certain quantified Boolean formulas.Inform. Process. Lett.,8 (1979), 121–123.
J. L. Balcázar. Separating, strongly separating, and collapsing complexity classes.Proc. 11th Symp. on Mathematical Foundations of Computer Science (1984), Lecture Notes in Computer Science, Vol. 176. Springer-Verlag, Berlin, 1984, pp. 1–16.
J. L. Balcázar and U. Schöning. Bi-immune sets for complexity classes.Math. Systems Theory,18 (1985), 1–10.
C. H. Bennett and J. Gill. Relative to a random oracleA, P A ≠NP A ≠ co-NP A with probability 1.SIAM J. Comput.,10 (1981), 96–113.
L. Berman. On the structure of complete sets: almost everywhere complexity and infinitely often speedup.Proc. 17th Ann. IEEE Symp. on Foundations of Computer Science (1976), 76–80.
L. Berman and J. Hartmanis. On isomorphism and density ofNP and other complete sets.SIAM J. Comput.,6 (1977), 305–322.
S. Breidbart. On splitting recursive sets.J. Comput. System Sci.,17 (1978), 56–64.
S. A. Cook. The complexity of theorem-proving procedures.Proc. 3rd Ann. ACM Symp. on Theory of Computing (1971), 151–158.
P. Flajolet and J. M. Steyaert. On sets having only hard subsets.Proc. 2nd Internat. Colloq. on Automata, Languages, and Programming (1974), Lecture Notes in Computer Science, Vol. 14. Springer-Verlag, Berlin, 1974, pp. 446–457.
J. G. Geske, D. T. Huynh, and A. L. Selman. A hierarchy theorem for almost everywhere complex sets with application to polynomial complexity degrees.Proc. 4th Symp. on Theoretical Aspects of Computer Science (1987), Lecture Notes in Computer Science, Vol. 247. Springer-Verlag, Berlin, 1987, pp. 125–135.
G. Grätzer.Lattice Theory. Freeman, San Francisco, 1971.
L. Henschen and L. Wos. Unit refutations and Horn sets.J. Assoc. Comput. Mach.,21 (1974), 590–605.
S. Homer and W. Maass. Oracle dependent properties of the lattice ofNP sets.Theoret. Comput. Sci.,24 (1983), 279–289.
J. E. Hopcroft and J. D. Ullman.Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading, Mass., 1979.
K. Ko and D. Moore. Completeness, approximation and density.SIAM J. Comput.,10 (1981), 787–796.
N. Lynch. On reducibility to complex or sparse sets.J. Assoc. Comput. Mach.,22 (1975), 341–345.
A. R. Meyer and M. S. Paterson. With what frequency are apparently intractable problems difficult? Report LCS/TM-126, Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Mass., 1979.
P. Orponen. A classification of complexity core lattices.Theoret. Comput. Sci.,47 (1986), 121–130.
P. Orponen, D. A. Russo, and U. Schöning. Optimal approximations and polynomially levelable sets.SIAM J. Comput.,15 (1986), 399–408.
P. Orponen and U. Schöning. The density and complexity of polynomial cores for intractable sets.Inform. and Control,70 (1986), 54–68.
H. Rogers, Jr.Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967.
U. Schöning. OnNP-decomposable sets.SIGACT News,14(1) (1982), 18–20.
U. Schöning and R. V. Book. Immunity, relativizations and nondeterminism.SIAM J. Comput.,13 (1984), 329–337.
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This research was supported in part by the Emil Aaltonen Foundation, the Academy of Finland, and the National Science Foundation under Grant No. MCS83-14272. Part of the work was carried out while the second author was visiting the Department of Mathematics, University of California at Santa Barbara.
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Russo, D.A., Orponen, P. OnP-subset structures. Math. Systems Theory 20, 129–136 (1987). https://doi.org/10.1007/BF01692061
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DOI: https://doi.org/10.1007/BF01692061