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.



When can we detect forcing?
Amit Gupta's comment gives the answer to your first question, which I will expand on slightly:


Every poset is detectable, and even more, the class of all posets is uniformly detectable. That is, there is a single formula $\phi(x)$ that takes as input a poset $P$, such that $\phi(P)$ is true in a model $M$ exactly when $M$ is a forcing extension of an inner model $N$ by $P$ (let us call $N$ a ground model of $M$ by $P$). It follows that the same is true for any set of posets (or definable class $C$ of posets) in $M$: we can detect when $M$ is a $C$-forcing extension.


Regarding your second question ("consistent classes"), there are many examples of consistent classes. For example, let $C$ be the class of posets that add a subset to a regular cardinal using the poset as defined in $L$ (that is, $C$ is the class of posets of the form $ Add(\kappa,1)^L $ ). If we start in $V$ and force with the (class) product of $C$, then in the resulting model $V[G]$, for every member $P$ of $C$ there is a ground model of $V[G]$ by $P$. This is because, for a fixed $\kappa$, we can use the commutativity of the product to rewrite the extension as class forcing (adding a subset to each cardinal except $\kappa$), followed by set forcing adding a subset to $\kappa$. This allows us to rewrite $V[G]=V[G^\prime][g]$, and $V[G]$ is an extension of $V[G^\prime]$ by $Add(\kappa,1)^L$ as required. 


We can modify this argument in various ways to handle other classes. It will work for any class of posets from $L$ whose product preserves $ZFC$. We can also take our posets from another absolute inner model, such as $L[mu]$, instead of from $L$. Dropping the requirement that the posets be from a designated inner model (for example, taking $C$ to be the class of posets $Add(\kappa,1)$) may still possible but would require a more delicate argument. The issue here is that the definition of the poset $Add(\kappa,1)$ is not absolute, and so the poset defined in $V$ may not exist in every ground model. Nonetheless I suspect that the same model defined in the previous example will work.


There are also some classes that are clearly not consistent, for example the the class $C$ of partial orders that collapse $\kappa$ to $\omega$ (for every $\kappa$). Any model containing generics for all such posets will fail to satisfy $ZFC$, as every cardinal will be collapsed. 


For the more general case, one avenue of attack might be to use the Maximality Principle (introduced by Joel Hamkins). This principle says that any statement that can be forced in such a way that further forcing cannot make it false, is already true, and existence of a ground model for a particular poset is just such a statement.


