Measure Theoretic Completeness Notions for the Exponential Time Classes
The resource-bounded measure theory of Lutz leads to variants of the classical hardness and completeness notions. While a set A is hard (under polynomial time many-one reducibility) for a complexity class C if every set in C can be reduced to A, a set A is almost hard if the class of reducible sets has measure 1 in C, and a set A is weakly hard if the class of reducible sets does not have measure 0 in C. If, in addition, A is a member of C then A is almost complete and weakly complete for C, respectively. Weak hardness for the exponential time classes E = DTIME(2lin(n)) and EXP = DTIME(2poly(n)) has been extensively studied in the literature, whereas the nontriviality of the concept of almost completeness has been established only recently.
Here we continue the investigation of these measure theoretic hardness notions for the exponential time classes and we establish the relations among these notions which had been left open. In particular, we show that almost hardness for E and EXP are independent. Moreover, there is a set in E which is almost complete for EXP but not weakly complete for E. These results exhibit a surprising degree of independence of the measure concepts for E and EXP.
Finally, we give structural separations for some of these concepts and we show the nontriviality of almost hardness for the bounded query reducibilities of fixed norm.
KeywordsIEEE Computer Society Complexity Class Exponential Time Structural Separation Measure Concept
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