First, a rather broad question: has there been any work on what, given a model $M$ of set theory, we can say about those models of set theory $N$ and posets $\mathbb{P}$ such that $\mathbb{P}\in N$ and $M=N[G]$ for some $G$ $\mathbb{P}$-generic over $N$?

Second, a more specific question. Let a poset $\mathbb{P}$ be $detectable$ if there is some sentence $\phi_\mathbb{P}$ in the language of set theory + a constant denoting $\mathbb{P}$ such that for all $M$ with $\mathbb{P}\in M$, we have $(M, \mathbb{P})\models\phi\iff M=N[G]$ for some model $N$ with $\mathbb{P}\in N$ and $G$ $\mathbb{P}$-generic over $N$. What posets are detectable? [Answered in the comments by Amit Kumar Gupta.]

Finally, an incredibly general question. Let $M$ be a model of $ZFC$, $\mathcal{C}$ a class of posets in $M$. Say $\mathcal{C}$ is $consistent$ if there is some elementary extension $N$ of $M$ such that, for all $\mathbb{P}\in \mathcal{C}$, there is some $N_\mathbb{P}\models ZFC$ with $\mathbb{P}\in N_\mathbb{P}$ and some $G$ $\mathbb{P}$-generic over $N_\mathbb{P}$ such that $N=N_\mathbb{P}[G]$. What are the consistent classes like? Can we say anything interesting about them?

I'm not sure if these questions are meaningful, or - even assuming they are - if they are interesting. Basically, what I'm interested in is the notion of inverse forcing - similar in an aesthetic sense, at least to me, to inverse Galois theory - and I haven't run into anything along these lines yet.

nota forcing extension of any inner model $W$ by a nontrivial forcing $\mathbb{P} \in W$. In your second paragraph you seem to define a notion ofdetectabilitybut it doesn't appear that you ask an actual question. A result of Laver's is that if $V=M[G]$ is a forcing extension of $M$ by a set forcing $\mathbb{P} \in M$ then $M$ is definable in $V$ from parameters in $M$. So in a sense this says that if $V$ is a forcing extension, then it can detect the ground model. – Amit Kumar Gupta Nov 8 '10 at 6:36