what is the difference between "language" and "certificate" in complexity theory? [duplicate]

Question

Seems like they both are set of strings for which the problem returns "yes" in decision problems.

P class is consists of problems so we can only say a problem is in P. But we sometimes say that a language L is in P. why?

Answer 1

A (formal) language $L$ is a set of words over some finite alphabet. It can be an infinite set. Generally we're interested in the problem of deciding whether a particular word is in the language or not. For example $L$ can be the set of all Turing machines (encoded as binary strings for example) that halt after at most 100 steps when started with an empty tape, or $L$ can be the set of English words that don't contain the letter 'e'.

A certificate on the other hand is one particular string. When solving the decision problem I talked about above, namely checking whether some word $w$ is in some language $L$, sometimes the problem becomes much simpler if you have some additional information $c_w$. This is often called a certificate for $w \in L$ respectively $w\not \in L$.

For example if $L$ is the set of all graphs that can be colored with at most 3-colors, deciding whether a particular graph $G$ is in this set is NP-complete. But if I already give you a coloring $c_G$ it is very easy to check that it's a valid coloring. The coloring is a certificate for $G\in L$. A coloring has the nice property that it's short, just a couple of bits for each node of the graph. Note that in this case it's not easy to give a (short) certificate for some $G'$ that is not colorable. A very long certificate in this case would be an enumeration of all exponentially many possible 3-colorings of $G'$. Checking that each is not valid is again easy, and since the certificate $c_w$ is so long, the running time of the algorithm with input $G', c_w$ is polynomial as a function of input size.