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The Helping Hierarchy

Published online by Cambridge University Press:  15 April 2002

Patrizio Cintioli  and
Riccardo Silvestri
Patrizio Cintioli
Affiliation:
Dipartimento di Matematica e Fisica, Università di Camerino, Via Madonna delle Carceri, 62032 Camerino (MC), Italy; ([email protected])
Riccardo Silvestri
Affiliation:
Dipartimento di Scienze dell'Informazione, Università di Roma "La Sapienza" , Via Salaria 113, 00198 Roma, Italy; ([email protected])
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Abstract

Schöning [14] introduced a notion of helping and suggested the study of the class ${\rm P}_{\rm help}({\cal C})$ of the languages that can be helped by oracles in a given class ${\cal C}$. Later, Ko [12], in order to study the connections between helping and "witness searching" , introduced the notion of self-helping for languages. We extend this notion to classes of languages and show that there exists a self-helping class that we call SH which contains all the self-helping classes. We introduce the Helping hierarchy whose levels are obtained applying a constant number of times the operator ${\rm P}_{\rm help}(\cdot)$ to the set of all the languages. We show that the Helping hierarchy collapses to the k-th level if and only if SH is equal to the k-th level. We give characterizations of all the levels and use these to construct a relativized world in which the Helping hierarchy is infinite.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 4 , July 2001 , pp. 367 - 377
Copyright
© EDP Sciences, 2001

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References

V. Arvind, A Note on the Self-Witnessing Property of Computational Problems, Proc. 2nd Annual International Conference on Computing and Combinatorics (COCOON'96). Springer-Verlag, Lecture Notes in Comput. Sci. 1090 (1996) 241-249.
J.L. Balcázar, Self-reducibility, Proc. 4th Symposium on Theoretical Aspects of Computer Science. Springer-Verlag, Lecture Notes in Comput. Sci. 247 (1987) 136-147.
Balcázar, J.L., Self-reducibility structures and solutions of NP problems. Rev. Mat. Complut. 2 (1989) 175-184. CrossRef
D.P. Bovet and P. Crescenzi, Introduction to the Theory of Complexity. Prentice-Hall (1994).
J.L. Balcázar, J. Díaz and J. Gabarró, Structural Complexity I, Vol. 1. Springer-Verlag (1988).
J.L. Balcázar, J. Díaz and J. Gabarró, Structural Complexity II, Vol. 2. Springer-Verlag (1990).
J. Cai, L. Hemachandra and J. Viskoc, Promises Problems and Access to Unambiguous Computation, Proc. 17th Symposium on Mathematical Foundations of Computer Science. Springer-Verlag, Lecture Notes in Comput. Sci. 629 (1992) 162-171.
Cintioli, P. and Silvestri, R., Helping by Unambiguous Computation and Probabilistic Computation. Theory Comput. Syst. 30 (1997) 165-180. CrossRef
Cintioli, P. and Silvestri, R., Revisiting a Result of Ko. Inform. Process. Lett. 61 (1997) 189-194. CrossRef
M. Fellows and N. Koblitz, Self-witnessing polynomial time complexity and prima factorization, in Proc. 6th Structure in Complexity Theory Conference (1992) 107-110.
L. Hemachandra, Fault-Tolerance and Complexity, in Proc. 20th International Colloquium on Automata, Languages, and Programming. Springer-Verlag, Lecture Notes in Comput. Sci. (1993).
On Helping, K. Ko by Robust Oracle Machines. Theoret. Comput. Sci. 52 (1987) 15-36.
Ogihara, M., Helping, On by Parity-like Languages. Inform. Process. Lett. 54 (1995) 41-43. CrossRef
Schöning, U., Robust Algorithms: A Different Approach to oracles. Theoret. Comput. Sci. 40 (1985) 57-66. CrossRef