# Measure Theoretic Completeness Notions for the Exponential Time Classes

## Abstract

The resource-bounded measure theory of Lutz leads to variants of the classical hardness and completeness notions. While a set *A* is hard (under polynomial time many-one reducibility) for a complexity class **C** if every set in **C** can be reduced to *A*, a set *A* is *almost hard* if the class of reducible sets has measure 1 in **C**, and a set *A* is *weakly hard* if the class of reducible sets does not have measure 0 in **C.** If, in addition, *A* is a member of **C** then *A* is *almost complete* and *weakly complete* for *C*, respectively. Weak hardness for the exponential time classes **E** = **DTIME**(2^{lin}(*n*)) and **EXP** = **DTIME**(2^{poly}(*n*)) has been extensively studied in the literature, whereas the nontriviality of the concept of almost completeness has been established only recently.

Here we continue the investigation of these measure theoretic hardness notions for the exponential time classes and we establish the relations among these notions which had been left open. In particular, we show that almost hardness for **E** and **EXP** are independent. Moreover, there is a set in **E** which is almost complete for **EXP** but not weakly complete for **E**. These results exhibit a surprising degree of independence of the measure concepts for **E** and **EXP**.

Finally, we give structural separations for some of these concepts and we show the nontriviality of almost hardness for the bounded query reducibilities of fixed norm.

## Keywords

IEEE Computer Society Complexity Class Exponential Time Structural Separation Measure Concept## Preview

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## References

- 1.K. Ambos-Spies and E. Mayordomo. Resource-bounded measure and randomness. In: A. Sorbi (ed.),
*Complexity, logic, and recursion theory*, p. 1–47, Dekker, 1997.Google Scholar - 2.K. Ambos-Spies, E. Mayordomo and X. Zheng. A comparison of weak completeness notions. In:
*Proceedings of the 11th Ann. IEEE Conference on Computational Complexity*, p. 171–178, IEEE Computer Society Press, 1996.Google Scholar - 3.K. Ambos-Spies, W. Merkle, J. Reimann, and S.A. Terwijn. Almost complete sets. In:
*STACS 2000*, LNCS 1770, p. 419–430, Springer, 2000.CrossRefGoogle Scholar - 4.K. Ambos-Spies, H.-C. Neis and S.A. Terwijn. Genericity and measure for exponential time.
*Theoretical Computer Science*, 168:3–19, 1996.zbMATHCrossRefMathSciNetGoogle Scholar - 5.K. Ambos-Spies, S.A. Terwijn and X. Zheng. Resource bounded randomness and weakly complete problems.
*Theoretical Computer Science*, 172:195–207, 1997.zbMATHCrossRefMathSciNetGoogle Scholar - 6.J.L. Balcázar, J. Díaz, and J. Gabarró.
*Structural Complexity*, volume I. Springer, 1995.Google Scholar - 7.L. Berman.
*Polynomial reducibilities and complete sets*. Ph.D. thesis, Cornell University, 1977.Google Scholar - 8.H. Buhrmann and L. Torenvliet. On the structure of complete sets. In:
*Proceedings of the 9th Ann. Structure in Complexity Conference*, p. 118–133, IEEE Computer Society Press, 1994.Google Scholar - 9.S. Homer. Structural properties for complete problems for exponential time. In:
*Complexity Theory Retrospective*II (Hemaspaandra, L.A. et al., eds.), p. 135–153, Springer, 1997.Google Scholar - 10.D.W. Juedes and J.H. Lutz. The complexity and distribution of hard problems.
*SIAM Journal on Computing*, 24:279–295, 1995.zbMATHCrossRefMathSciNetGoogle Scholar - 11.D.W. Juedes and J.H. Lutz. Weak completeness in E and E
_{2}.*Theoretical Computer Science*, 143:149–158, 1995.zbMATHCrossRefMathSciNetGoogle Scholar - 12.J.H. Lutz. Almost everywhere high nonuniform complexity.
*Journal of Computer and System Sciences*, 44:220–258, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - 13.J.H. Lutz. Weakly hard problems.
*SIAM Journal on Computing*24:1170–1189, 1995.zbMATHCrossRefMathSciNetGoogle Scholar - 14.J.H. Lutz. The quantitative structure of exponential time. In:
*Complexity Theory Retrospective*II (Hemaspaandra, L.A. et al., eds.), p. 225–260, Springer, 1997.Google Scholar - 15.E. Mayordomo. Almost every set in exponential time is P-bi-immune.
*Theoretical Computer Science*, 136:487–506, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - 16.K. Regan, D. Sivakumar and J.-Y. Cai. Pseudorandom generators, measure theory and natural proofs. In:
*Proceedings of the 36th Ann. IEEE Symposium an Foundations of Computer Science*, p. 171–178, IEEE Computer Society Press, 1995.Google Scholar