On ≤1-ttp-Sparseness and Nondeterministic Complexity Classes (Extended Abstract)

For any reduction r, a set is called “≤ r p -sparse” if it is ≤ r p -reducible to a sparse set. The difficulty of sets in nondeterministic complexity classes is investigated in terms of non-≤ 1−tt p -sparseness, i.e., not being ≤ 1−tt p -reducible to any sparse set. In particular, nondeterministic complexity classes used to specify various types of one-way functions are mainly considered, i.e., UP, N(poly), UBPP, and UP. For each such class, we prove that it contains a non-≤ 1−tt p -sparse set unless it is included in P. Since these classes are included in more general nondeterministic complexity classes such as NP, easy consequences of our observations show the nonsparseness of ≤ 1−tt p -hard sets for such classes.

[1]  Ker-I Ko,et al.  Studies in complexity theory , 1986 .

[2]  Ker-I Ko,et al.  On Sets Truth-Table Reducible to Sparse Sets , 1988, SIAM J. Comput..

[3]  Ker-I Ko,et al.  On sets reducible to sparse sets , 1987, Computational Complexity Conference.

[4]  Osamu Watanabe,et al.  Polynomial time reducibility to a set of small density , 1987, SCT.

[5]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[6]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[7]  Steven Fortune,et al.  A Note on Sparse Complete Sets , 1978, SIAM J. Comput..

[8]  Alan L. Selman,et al.  Complexity Measures for Public-Key Cryptosystems , 1988, SIAM J. Comput..

[9]  Alan L. Selman,et al.  Promise Problems Complete for Complexity Classes , 1988, Inf. Comput..

[10]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[11]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[12]  Stephen R. Mahaney.  Sparse Complete Sets of NP: Solution of a Conjecture of Berman and Hartmanis , 1982, J. Comput. Syst. Sci..

[13]  Yacov Yacobi,et al.  The Complexity of Promise Problems with Applications to Public-Key Cryptography , 1984, Inf. Control..

[14]  Leslie G. Valiant,et al.  Relative Complexity of Checking and Evaluating , 1976, Inf. Process. Lett..

[15]  Eric Allender,et al.  The Complexity of Sparse Sets in P , 1986, SCT.

[16]  Juris Hartmanis.  Feasible computations and provable complexity properties , 1989, CBMS-NSF regional conference series in applied mathematics.

[17]  Ding-Zhu Du,et al.  On One-Way Functions and Polynomial-Time Isomorphisms , 1986, Theor. Comput. Sci..

[18]  Mihalis Yannakakis,et al.  The complexity of facets (and some facets of complexity) , 1982, STOC '82.

[19]  守屋 悦朗,et al.  J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979 , 1980 .

[20]  Yaacov Yesha,et al.  On Certain Polynomial-Time Truth-Table Reducibilities of Complete Sets to Sparse Sets , 1983, SIAM J. Comput..

[21]  Esko Ukkonen.  Two Results on Polynomial Time Truth-Table Reductions to Sparse Sets , 1983, SIAM J. Comput..

[22]  Deborah Joseph,et al.  Some Remarks on Witness Functions for Nonpolynomial and Noncomplete Sets in NP , 1985, Theor. Comput. Sci..

[23]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[24]  Osamu Watanabe.  On One-One Polynomial Time Equivalence Relations , 1985, Theor. Comput. Sci..

[25]  R.E. Ladner,et al.  A Comparison of Polynomial Time Reducibilities , 1975, Theor. Comput. Sci..

[26]  Leonard M. Adleman,et al.  Two theorems on random polynomial time , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).