The Helping Hierarchy

Patrizio Cintioli; Riccardo Silvestri

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 35, Issue: 4, page 367-377
  • ISSN: 0988-3754

Abstract

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Schöning [14] introduced a notion of helping and suggested the study of the class P help ( 𝒞 ) of the languages that can be helped by oracles in a given class 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 P help ( · ) 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.

How to cite

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Cintioli, Patrizio, and Silvestri, Riccardo. "The Helping Hierarchy." RAIRO - Theoretical Informatics and Applications 35.4 (2010): 367-377. <http://eudml.org/doc/222020>.

@article{Cintioli2010,
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. },
author = {Cintioli, Patrizio, Silvestri, Riccardo},
journal = {RAIRO - Theoretical Informatics and Applications},
language = {eng},
month = {3},
number = {4},
pages = {367-377},
publisher = {EDP Sciences},
title = {The Helping Hierarchy},
url = {http://eudml.org/doc/222020},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Cintioli, Patrizio
AU - Silvestri, Riccardo
TI - The Helping Hierarchy
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 4
SP - 367
EP - 377
AB - 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.
LA - eng
UR - http://eudml.org/doc/222020
ER -

References

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  9. P. Cintioli and R. Silvestri, Revisiting a Result of Ko. Inform. Process. Lett.61 (1997) 189-194.  Zbl1336.68087
  10. M. Fellows and N. Koblitz, Self-witnessing polynomial time complexity and prima factorization, in Proc. 6th Structure in Complexity Theory Conference (1992) 107-110.  Zbl0756.11042
  11. L. Hemachandra, Fault-Tolerance and Complexity, in Proc. 20th International Colloquium on Automata, Languages, and Programming. Springer-Verlag, Lecture Notes in Comput. Sci. (1993).  
  12. K. Ko, On Helping by Robust Oracle Machines. Theoret. Comput. Sci.52 (1987) 15-36.  Zbl0635.68039
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  14. U. Schöning, Robust Algorithms: A Different Approach to oracles. Theoret. Comput. Sci.40 (1985) 57-66.  Zbl0574.68041

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