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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; (cintioli@campus.unicam.it)
Riccardo Silvestri
Affiliation:
Dipartimento di Scienze dell'Informazione, Università di Roma "La Sapienza" , Via Salaria 113, 00198 Roma, Italy; (silvestri@dsi.uniroma1.it)
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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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