Complexity classes without machines: On complete languages for UP

  • Juris Hartmanis
  • Lane Hemachandra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 226)


This paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines. Often, counting classes, probabilistic classes, and intersection classes lack such enumerations. Concentrating on the counting class UP, we show that there are relativizations for which UPA has no complete languages and other relativizations for which PBUPBNPB and UPB has complete languages. Among other results we show that PUP if and only if there exists a set S in P of Boolean formulas with at most one satisfying assignment such that SSAT is not in P. PUPcoUP if and only if there exists a set S in P of uniquely satisfiable Boolean formulas such that no polynomial-time machine can compute the solutions for the formulas in S. If UP has complete languages then there exists a set R in P of Boolean formulas with at most one satisfying assignment so that SATR is complete for UP. Finally, we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class BPP. There is a relativized world where BPPA has no complete languages. If BPP has complete languages then it has a complete language of the form BMAJORITY, where BP and MAJORITY = {f | f is true for at least half of all assignments} is the canonical PP-complete set.


Boolean Formula Satisfying Assignment Relativize World Counting Class Categorical Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BD]
    A. Borodin and A. Demers. Some Comments on Functional Self-Reducibility and the NP Hierarchy. Department of Computer Science Technical Report TR76-284, July 1976. Cornell University, Ithaca, New York.Google Scholar
  2. [BG]
    A. Blass and Y. Gurevich. On the Unique Satisfiability Problem. Information and Control 55 (1982), 80–82.Google Scholar
  3. [Be]
    P. Berman. Relations Between Density and Deterministic Complexity of NP-Complete Languages. Proceedings Symposium on Mathematical Foundations of Computer Science, 1978, Springer-Verlag, 63–71.Google Scholar
  4. [CH]
    J. Cai and L. Hemachandra. The Boolean Hierarchy: Hardware over NP. To appear in Proceedings of the Structure in Complexity Theory Conference, Lecture Notes in Computer Science (1986), Springer-Verlag.Google Scholar
  5. [Co]
    S.A. Cook. The Complexity of Theorem-Proving Procedures. Proceedings ACM Symposium on Theory of Computation (1971), 151–158.Google Scholar
  6. [Gi]
    J. Gill. Computational Complexity of Probabilistic Turing Machines. SIAM Journal on Computing 6 (1977), 675–695.Google Scholar
  7. [GJ]
    M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co., 1979.Google Scholar
  8. [GS]
    J. Grollmann and A.L. Selman. Complexity Measures for Public-Key Cryptosystems. Proceedings IEEE Symposium on Foundations of Computer Science (1984), 495–503.Google Scholar
  9. [HI]
    J. Hartmanis and N. Immerman. On Complete Problems for NPCoNP. Automata Languages and Programming, Lecture Notes in Computer Science 194 (1985), Springer-Verlag, 250–259.Google Scholar
  10. [HU]
    J.E. Hopcroft and J.D. Ullman. Introduction to Automata Theory, languages, and Computation. Addison-Wesley, Reading, Massachusetts, 1979.Google Scholar
  11. [Li]
    M. Li. Lower Bounds in Computational Complexity. Ph.D. Dissertation, Cornell University, 1985.Google Scholar
  12. [Si]
    M. Sipser. On Relativization and the Existence of Complete Sets. Automata, Languages and Programming, Lecture Notes in Computer Science 140 (1982), Springer-Verlag, 523–531.Google Scholar
  13. [Va]
    L. Valiant. Relative Complexity of Checking and Evaluating. Information Processing Letters 5 (1976), 20–23.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Juris Hartmanis
    • 1
  • Lane Hemachandra
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthaca

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