"Definition of NP via relations and quantifiers; not via NTMs"

Question

I have the following question on an assignment, and despite asking my prof, I can't get a grasp on it..

Let $L_1, L_2$ be languages in ${\sf NP}$. Using the definition of ${\sf NP}$ via relations and quantifiers (not the non-deterministic Turing machines) prove that the following language is in ${\sf NP}$: $L=\{ x | x \in L_1 \text{ or } xx \in L_2 \}$, where $xx$ is two concatenated copies of $x$.

Her notes say the following on this:

Let $L\subseteq\Sigma^{*}$. We say $L\in{\sf NP}$ if there is a two-place predicate $R\subseteq\Sigma^{*}\times\Sigma^{*}$ such that $R$ is computable in polynomial time, and such that for some $c,d\in\mathbb{N}$ we have $\forall x\in\Sigma^{*},x\in L\iff\exists y\in\Sigma^{*},|y|\leq c|x|^d$ and $R(x,y)$.

I would imagine that this is a well known definition (though possibly worded differently)... but I don't know where else I can get details on it. My textbook has a somewhat similar definition, but it doesn't talk about this $c,d\in\mathbb{N}$ stuff..

A verifier for a language $A$ is an algorithm $V$, where $$A=\{w\mid V\text{ accepts }\langle w,c\rangle\text{ for some string c}\}.$$ We measure the time of a verifier only in terms of the length of $w$, so a polynomial time verifier runs in polynomial time in the length of $w$. A language $A$ is polynomially verifiable if it has a polynomial time verifier.

The $y$ in the first definition is supposed to be analogous to $c$ in the second; it is a certificate or witness used by the verifier. But that's all I can understand between the two.

Now... what I think I know about this question is that $R(x,y)$ is supposed to be a verifier for $L$, while $R_1(x,y_1)$ is a verifier for $L_1$ and $R_2(x,y_2)$ is a verifier for $L_2$. She said something about $y=\langle y_1, y_2\rangle$ being an encoding of the two certificates for the verifiers $R_1$ and $R_2$. But from there I'm lost. I have no idea how to answer this question even after asking for help two or three times.

Anybody here able to help? Thanks in advance.

Answer 1

I think I just realized, after finding a similar problem online, that this is asking to simply show that ${\sf NP}$ is closed under union. Can anybody give any insight into whether or not this is a proper solution, or if I'm missing anything? I mostly understand my answer, but I don't really know what I'm doing with the $c$ and $d$ constants... I put them in there because I feel like I'm going to need to.

This question is asking to prove that ${\sf NP}$ is closed under union. It is the same as asking if $L=L_{1}\cup L_{2}$ is in ${\sf NP}$.

Since $L_{1}$ and $L_{2}$ are both in ${\sf NP}$, it follows that they both have polynomial time verifiers $R_i(x_i,y_i)$ that evaluate to true when $y_{i}$ is a valid certificate/witness for $x_i\in L_i$, where $i\in \{1,2\}$.

To show that $L=L_1\cup L_2$ is also in ${\sf NP}$, we need to construct a polynomial-time verifier $R(x,y)$. The certificate $y=\langle y_1,y_2 \rangle$ for $L$ will be an encoding of the two certificates $y_1$ and $y_2$ in such a way that $R(x,y)$ evaluates to true when $\displaystyle\bigvee_{1\leq i\leq 2} R_i(x,y_i)$ evaluates to true.

Given that $R_{i}(x_i,y_i)$ for $i\in\{1,2\}$ are polynomial time verifiers with running time $O(c_i|x_i|^{d_i})$ for some $c,d\in\mathbb{N}$, $R(x,y)$ will have a running time of at most $O(2\cdot max(c_i)|x|^{max(d_i)})$, which is still polynomial. Therefore, $L\in {\sf NP}$.