On the Limitations of Locally Robust Positive Reductions

Polynomial-time positive reductions, as introduced by Selman, are by definition globally robust — they are positive with respect to all oracles. This paper studies the extent to which the theory of positive reductions remains intact when their global robustness assumption is removed. We note that two-sided locally robust positive reductions — reductions that are positive with respect to the oracle to which the reduction is made — are sufficient to retain all crucial properties of globally robust positive reductions. In contrast, we prove absolute and relativized results showing that one-sided local robustness fails to preserve fundamental properties of positive reductions, such as the downward closure of NP.

[1]  Ker-I Ko,et al.  On Helping by Robust Oracle Machines , 1987, Theor. Comput. Sci..

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

[3]  Ker-I Ko On Self-Reducibility and Weak P-Selectivity , 1983, J. Comput. Syst. Sci..

[4]  Nancy A. Lynch,et al.  Comparison of polynomial-time reducibilities , 1974, STOC '74.

[5]  Juris Hartmanis,et al.  Robust Machines Accept Easy Sets , 1990, Theor. Comput. Sci..

[6]  Neil Immerman,et al.  Sparse sets in NP-P: Exptime versus nexptime , 1983, Inf. Control..

[7]  Alan L. Selman,et al.  P-Selective Sets, Tally Languages, and the Behavior of Polynomial Time Reducibilities on NP , 1979, ICALP.

[8]  Uwe Schöning,et al.  Robust Algorithms: A Different Approach to Oracles , 1985, Theor. Comput. Sci..

[9]  Alan L. Selman,et al.  Reductions on NP and P-Selective Sets , 1982, Theor. Comput. Sci..

[10]  Gábor Tardos,et al.  Query complexity, or why is it difficult to separateNPA∩coNPA fromPA by random oraclesA? , 1989, Comb..

[11]  Timothy J. Long,et al.  Quantitative Relativizations of Complexity Classes , 1984, SIAM J. Comput..

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

[13]  Alan L. Selman,et al.  Qualitative Relativizations of Complexity Classes , 1985, J. Comput. Syst. Sci..

[14]  Ronald V. Book,et al.  Tally Languages and Complexity Classes , 1974, Inf. Control..

[15]  Manuel Blum,et al.  Generic oracles and oracle classes , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).