A modified version of the classical µ-operator as well as the first value operator and the operator of inverting unary functions, applied in combination with the composition of functions and starting from the primitive recursive functions, generate all arithmetically representable functions. Moreover, the nesting levels of these operators are closely related to the stratification of the arithmetical hierarchy. The same is shown for some further function operators known from computability and complexity theory. The close relationships between nesting levels of operators and the stratification of the hierarchy also hold for suitable restrictions of the operators with respect to the polynomial hierarchy if one starts with the polynomial-time computable functions. It follows that questions around P vs. NP and NP vs. coNP can equivalently be expressed by closure properties of function classes under these operators. The polytime version of the first value operator can be used to establish hierarchies between certain consecutive levels within the polynomial hierarchy of functions, which are related to generalizations of the Boolean hierarchies over the classes .
Mots-clés : arithmetical hierarchy, polynomial hierarchy, boolean hierarchy, P versus NP, NP versus conp, first value operator, minimalization, inversion of functions
@article{ITA_2010__44_3_379_0, author = {Hemmerling, Armin}, title = {Function operators spanning the arithmetical and the polynomial hierarchy}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, pages = {379--418}, publisher = {EDP-Sciences}, volume = {44}, number = {3}, year = {2010}, doi = {10.1051/ita/2010020}, mrnumber = {2761525}, zbl = {1213.68294}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ita/2010020/} }
TY - JOUR AU - Hemmerling, Armin TI - Function operators spanning the arithmetical and the polynomial hierarchy JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2010 SP - 379 EP - 418 VL - 44 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ita/2010020/ DO - 10.1051/ita/2010020 LA - en ID - ITA_2010__44_3_379_0 ER -
%0 Journal Article %A Hemmerling, Armin %T Function operators spanning the arithmetical and the polynomial hierarchy %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2010 %P 379-418 %V 44 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ita/2010020/ %R 10.1051/ita/2010020 %G en %F ITA_2010__44_3_379_0
Hemmerling, Armin. Function operators spanning the arithmetical and the polynomial hierarchy. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 3, pp. 379-418. doi : 10.1051/ita/2010020. http://archive.numdam.org/articles/10.1051/ita/2010020/
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