Weakly Hard Problems

Aweak completeness phenomenon is investigated in the complexity class E = DTIME(2linear). According to standard terminology, a language H is m-hard for E if the set Pm(H), consisting of all languages A m H , contains the entire class E. A language C is P m-complete for E if it is m-hard for E and is also an element of E. Generalizing this, a language H is weakly m-hard for E if the set Pm(H) does not have measure 0 in E. A language C is weakly m-complete for E if it is weakly m-hard for E and is also an element of E. The main result of this paper is the construction of a language that is weakly m-complete, but not P m-complete, for E. The existence of such languages implies that previously known strong lower bounds on the complexity of weakly m-hard problems for E (given by work of Lutz, Mayordomo, and Juedes) are indeed more general than the corresponding bounds for m-hard problems for E. The proof of this result introduces a new diagonalization method, called martingale diagonalization. Using this method, one simultaneously develops an in nite family of polynomial time computable martingales (betting strategies) and a corresponding family of languages that defeat these martingales (prevent them from winning too much money) while also pursuing another agenda. Martingale diagonalization may be useful for a variety of applications. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International and Microware Systems Corporation.