Reduction over intersection of languages

Question

Given two languages $L1$ and $L2$, such that $L2$ is NP-Hard under polytime (many-one or Turing) reduction. Let $L=L1\cap L2$.

1- Is it true that if $L2$ is polytime (many-one or Turing) reducible to a third language $L3$, then $L$ is polytime reducible to $L1\cap L3$ ?

2- If $L \in$ P, what can it be concluded about the complexity of $L1$ ?

Thank you for your answers.

Answer 1

1: No. For example, let X be an EXP-hard language, and let L=L₁=L₂={0x: x∈X} and L₃={1x: x∈X}. Then,

• Both L and L₂ are EXP-hard. In particular, L does not belong to P and L₂ is NP-hard.
• L₂ is polynomial-time many-one reducible to L₃.
• L₁∩L₃ is the empty set. Because L∉P, this implies that L is not polynomial-time Turing reducible to L₁∩L₃.

2: Nothing. Let X be an arbitrary language and Y be an NP-hard language. Let L₁={0x: x∈X} and L₂={1y: y∈Y}. Although L₁∩L₂ is the empty set and therefore trivially belongs to P, and L₂ is indeed NP-hard, this does not tell anything about the complexity of L₁.