Function operators spanning the arithmetical and the polynomial hierarchy
RAIRO - Theoretical Informatics and Applications (2010)
- Volume: 44, Issue: 3, page 379-418
- ISSN: 0988-3754
Access Full Article
topAbstract
topHow to cite
topHemmerling, Armin. "Function operators spanning the arithmetical and the polynomial hierarchy." RAIRO - Theoretical Informatics and Applications 44.3 (2010): 379-418. <http://eudml.org/doc/250758>.
@article{Hemmerling2010,
abstract = {
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
$\mbox\{$\Sigma^p\_\{k\}$\}$.
},
author = {Hemmerling, Armin},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Arithmetical hierarchy; polynomial hierarchy;
Boolean hierarchy; P versus NP; NP versus coNP; first
value operator; minimalization; inversion of functions; arithmetical hierarchy; Boolean hierarchy; P versus NP; first value operator},
language = {eng},
month = {10},
number = {3},
pages = {379-418},
publisher = {EDP Sciences},
title = {Function operators spanning the arithmetical and the polynomial hierarchy},
url = {http://eudml.org/doc/250758},
volume = {44},
year = {2010},
}
TY - JOUR
AU - Hemmerling, Armin
TI - Function operators spanning the arithmetical and the polynomial hierarchy
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/10//
PB - EDP Sciences
VL - 44
IS - 3
SP - 379
EP - 418
AB -
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
$\mbox{$\Sigma^p_{k}$}$.
LA - eng
KW - Arithmetical hierarchy; polynomial hierarchy;
Boolean hierarchy; P versus NP; NP versus coNP; first
value operator; minimalization; inversion of functions; arithmetical hierarchy; Boolean hierarchy; P versus NP; first value operator
UR - http://eudml.org/doc/250758
ER -
References
top- S. Buss and L. Hay, On truth-table reducibility to SAT. Inf. Comput.91 (1991) 86–102.
- J.L. Balcazar, J. Diaz and J. Gabarro, Structural complexity I and II. Springer-Verlag, Berlin (1990).
- S.J. Bellantoni and K.-H. Niggl, Ranking primitive recursions: the low Grzegorczyk classes revised. SIAM Journal on Computing29 (1999) 401–415.
- J.-Y. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sawelson, K. Wagner and G. Wechsung, The Boolean hierarchy I: structural properties. SIAM Journal on Computing17 (1988) 1232–1252.
- J.-Y. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sawelson, K. Wagner and G. Wechsung, The Boolean hierarchy II: applications. SIAM Journal on Computing18 (1989) 95–111.
- S.B. Cooper, Computability theory. Chapman & Hall/CRC, Boca Raton (2004).
- D.-Z. Du and K.-I. Ko, Theory of computational complexity. Wiley-Interscience, New York (2000).
- R.L. Epstein, R. Haas and R.L. Kramer, Hierarchies of sets and degrees below 0'. In: Logic Year (1979/80), Univ. of Connecticut, edited by M. Lerman, J.H. Schmerl, R.I. Soare. LN in Math 859. Springer Verlag, 32–48.
- Y.L. Ershov, A hierarchy of sets. I; II; III. Algebra i Logica 7 (1968) no. 1, 47–74; no. 4, 15–47; 9 (1970), no. 1, 34–51 (English translation by Plenum P.C.).
- M.R. Garey and D.S. Johnson, Computers and intractability – a guide to the theory of NP–completeness. W.H. Freeman, San Francisco (1979).
- F. Hausdorff, Grundzüge der Mengenlehre. W. de Gruyter & Co., Berlin and Leipzig (1914); Reprint: Chelsea P.C., New York (1949).
- A. Hemmerling, The Hausdorff-Ershov hierarchy in Euclidean spaces. Arch. Math. Logic45 (2006) 323–350.
- A. Hemmerling, Hierarchies of function classes defined by the first-value operator. RAIRO - Theor. Inf. Appl. 42 (2008) 253–270. Extended abstract in: Proc. of CCA'2004. Electronic Notes in Theoretical Computer Science120 (2005) 59–72.
- J. Krajicek, Bounded arithmetic, propositional logic, and complexity theory. Cambridge Univ. Press (1995).
- M.W. Krentel, The complexity of optimization problems. J. Comput. Syst. Sci.36 (1988) 490–509.
- A.I. Malcev, Algorithmen und rekursive Funktionen. Akademie-Verlag, Berlin (1974).
- P. Odifreddi, Classical recursion theory. North-Holland P.C., Amsterdam (1989).
- C.H. Papadimitriou, Computational complexity. Addison Wesley P.C., Reading (1994).
- C. Parsons, Hierarchies of primitive recursive functions. Zeitschr. f. Math. Logik u. Grundl. d. Math. 14 (1968) 357–376.
- J. Robinson, General recursive functions. Proc. Am. Math. Soc.72 (1950) 703–718.
- H. Rogers Jr, Theory of recursive functions and effective computability. McGraw-Hill, New York (1967).
- H.E. Rose, Subrecursion: functions and hierarchies. Clarendon Press, Oxford (1984).
- J. Rothe, Complexity theory and cryptology. Springer-Verlag, Berlin (2005).
- H. Schwichtenberg, Rekursionszahlen und Grzegorczyk-Hierarchie. Arch. Math. Logic12 (1969) 85–97.
- A.L. Selman, A survey of one-way functions in complexity theory. Math. Syst. Theory25 (1992) 203–221.
- A. Selman, A taxonomy of complexity classes of functions. J. Comput. Syst. Sci.48 (1994) 357–381.
- J.R. Shoenfield, On degrees of unsolvability. Ann. Math.69 (1959) 644–653.
- R.I. Soare, Recursively enumerable sets and degrees. Springer-Verlag, Berlin (1987).
- R.I. Soare, Computability and recursion. Bulletin of symbolic Logic2 (1996) 284–321.
- R.I. Soare, Computability theory and applications. Springer-Verlag, Berlin, forthcoming.
- K.W. Wagner, Bounded query classes. SIAM Journal on Computing19 (1990) 833–846.
- G. Wechsung, Vorlesungen zur Komplexitätstheorie. B.G. Teubner, Stuttgart (2000).
- K. Weihrauch, Computability. Springer-Verlag, Berlin (1987).
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.