- Author:
- Sarah Easton Mocas
This thesis examines several of the most central and fundamental complexity classes. These classes are defined by polynomial and exponential time bounds both uniform and nonuniform. Showing classes to be distinct or separate has been a long standing objective in structural complexity theory. It is this objective that we address. Specifically we are interested in separating classes in the exponential time hierarchy, EXPH, from classes in the polynomial time hierarchy, PH. We show that, for any fixed integer $c, P\sp{NP\lbrack O(n\sp{c})\rbrack}\subseteq$ NEXP. This improves a previous result by Fu, Li and Zhong. Further we generalize this separation to related levels of PH and EXPH showing that, for any fixed integer c and $i\ge 1, \Delta\sbsp{i}{P\lbrack O(n\sp{c})\rbrack}\subseteq\Sigma\sbsp{i-1}{EXP}.$
There is also an interest in separating exponential time classes from classes of sets which are nonuniformly computable in polynomial time. By considering polynomial advice classes we show that $EXP \not\subseteq DTIME(2\sp{O(n\sp{c\sb1})})/n\sp{c\sb2}$for fixed integers $c\sb1$ and $c\sb2.$ This implies, for example, that $EXP \not\subseteq E/lin.$
Usually complexity theory is concerned with questions of set membership. An alternative is to allow a model which computes a partial function and outputs a value if one exists. In this way the time and space complexity of classes of partial functions is studied. We show that our results relating both uniform and nonuniform exponential and polynomial classes are true for the corresponding classes of function. Further we show that $PF\sp{NP\lbrack \log\rbrack}\subseteq EXPF\sb{PB}.$ This proof is then generalized to show that $PF\sp{\Sigma\sbsp{i}{P}\lbrack \log\rbrack}\subseteq EXPF\sb{PB}$, for $i\ge 1.$ Neither of these results is known for the corresponding classes of sets and can not be shown using proof techniques which relativize. Also we note that, for $i \ge 1, PF\sp{\Sigma\sbsp{i}{P}\lbrack \log\rbrack}\subseteq PF\sp{NP}$ unless P = NP which demonstrates that the structure of the polynomial hierarchy over function classes is very different from the polynomial hierarchy over sets unless PH collapses to $P\sp{NP}.$
Cited By
- Buhrman H, Torenvliet L, Unger F and Vereshchagin N (2019). Sparse Selfreducible Sets and Nonuniform Lower Bounds, Algorithmica, 81:1, (179-200), Online publication date: 1-Jan-2019.
- Buhrman H, Torenvliet L and Unger F Sparse selfreducible sets and polynomial size circuit lower bounds Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science, (455-468)
Index Terms
- Separating exponential time classes from polynomial time classes
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