Weakly Hard Problems

• Authors: Jack H. Lutz
• Venue: SIAM Journal on Computing
• Year: 1995

A weak completeness phenomenon is investigated in the complexity class ${\rm E}= {\rm DTIME}(2^{\rm linear})$. According to standard terminology, a language $H$ is $\leq^{\rm P}_{m}$-hard for E if the set ${\rm P}_{m}(H)$, consisting of all languages $A \leq^{\rm P}_{m}H$, contains the entire class E. A language $C$ is $\leq^{\rm P}_{m}$-complete for E if it is $\leq^{\rm P}_{m}$-hard for E and is also an element of E. Generalizing this, a language $H$ is weakly $\leq^{\rm P}_{m}$-hard for E if the set ${\rm P}_{m}(H)$ does not have measure 0 in E. A language $C$ is weakly $\leq^{\rm P}_{m}$-complete for E if it is weakly $\leq^{\rm P}_{m}$-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 $\leq^{\rm P}_{m}$-complete, but not $\leq^{\rm P}_{m}$-complete, for E. The existence of such languages implies that previously known strong lower bounds on the complexity of weakly $\leq^{\rm P}_{m}$-hard problems for E (given by work of Lutz, Mayordomo, and Juedes) are indeed more general than the corresponding bounds for $\leq^{\rm P}_{m}$-hard problems for E. The proof of this result introduces a new diagonalization method, called martingale diagonalization. Using this method, one simultaneously develops an infinite 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.