Abstract of Paper |

*Measure Theoretic Completeness Notions for the Exponential Time Classes*

by Klaus Ambos-Spies

**Abstract:**

The resourcebounded measure theory of Lutz leads to weakenings of the classical hardness and completeness notions. While a set $A$ is hard (under polynomialtime manyone 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$(2^{\mbox{lin}(n)})$ and EXP${}={}$DTIME$(2^{\mbox{poly}(n)})$ 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. In the first part of this paper 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. In the second part of the paper we give structural separations of completeness and almost completeness in terms of immunity, and of almost completeness and weak completeness in terms of incompressibility. Moreover, we look at almost completeness for the exponentialtime classes under boundedtruthtable and Turing reductions of fixed norm: We establish the nontriviality of these concepts and prove some hierachy theorems.