# Complexity classes without machines: On complete languages for UP

## Abstract

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 *UP*^{A} has no complete languages and other relativizations for which *P*^{B} ≠ *UP*^{B} ≠ *NP*^{B} and *UP*^{B} has complete languages. Among other results we show that *P* ≠ *UP* if and only if there exists a set *S* in *P* of Boolean formulas with at most one satisfying assignment such that *S* ∩ *SAT* is not in *P*. *P* ≠ *UP* ∩ *coUP* 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 *SAT* ∩ *R* 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 *BPP*^{A} has no complete languages. If *BPP* has complete languages then it has a complete language of the form *B* ∩ *MAJORITY*, where *B* ∈ *P* and *MAJORITY* = {*f* | *f* is true for at least half of all assignments} is the canonical *PP*-complete set.

## Keywords

Boolean Formula Satisfying Assignment Relativize World Counting Class Categorical Machine## Preview

Unable to display preview. Download preview PDF.

## References

- [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 - [BG]A. Blass and Y. Gurevich. On the Unique Satisfiability Problem.
*Information and Control 55*(1982), 80–82.Google Scholar - [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 - [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 - [Co]S.A. Cook. The Complexity of Theorem-Proving Procedures.
*Proceedings ACM Symposium on Theory of Computation*(1971), 151–158.Google Scholar - [Gi]J. Gill. Computational Complexity of Probabilistic Turing Machines.
*SIAM Journal on Computing 6*(1977), 675–695.Google Scholar - [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 - [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 - [HI]J. Hartmanis and N. Immerman. On Complete Problems for
*NP*∩*CoNP*.*Automata Languages and Programming, Lecture Notes in Computer Science 194*(1985), Springer-Verlag, 250–259.Google Scholar - [HU]J.E. Hopcroft and J.D. Ullman.
*Introduction to Automata Theory, languages, and Computation*. Addison-Wesley, Reading, Massachusetts, 1979.Google Scholar - [Li]M. Li.
*Lower Bounds in Computational Complexity*. Ph.D. Dissertation, Cornell University, 1985.Google Scholar - [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 - [Va]L. Valiant. Relative Complexity of Checking and Evaluating.
*Information Processing Letters 5*(1976), 20–23.Google Scholar