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Lemma. (Transitivity) "$\leq_p$" is a transitive relation on languages, i.e., if $L_1 \leq_p L_2$ and $L_2 \leq_p L_3$, then $L_1 \leq_p L_3$.

Proof. By definition, there are poly-time functions $f$ and $g$ such that $x \in L_1 \Leftrightarrow f(x) \in L_2$ and $y \in L_2 \Leftrightarrow g(y) \in L_3$, thus $x \in L_1 \Leftrightarrow f(x) \in L_2 \Leftrightarrow g(f(x)) \in L_3$. Obviously, $g(f(\cdot))$ is poly-time (since $|f(x)|$ is polynomial in $|x|$).

This lemma proves that "$\leq_p$" is transitive, but how would I prove that it is not antisymmetric?

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  • $\begingroup$ What makes you think it's not reflexive? $\endgroup$ Feb 18, 2018 at 18:17
  • $\begingroup$ sorry meant only for antisymmetric! it is reflexive because f(w)=w satisfies that w ∈ L <=> f(w) ∈ L $\endgroup$
    – crystyxn
    Feb 18, 2018 at 18:20

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By definition, a relation $R$ is antisymmetric if $aRb$ and $bRa$ implies $a=b$. Therefore, to show that $R$ is not antisymmetric, we have to find $a \neq b$ such that $aRb$ and $bRa$.

Specializing this to your case, you have to find two different languages $L_1 \neq L_2$ such that $L_1 \leq_p L_2$ And $L_2 \leq_p L_1$. Finding a pair of such languages isn't too challenging, so I leave it to you.

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