## Preliminaries and Notation

We are given a domain of entities $\mathcal{E} = {a, b, c, \ldots}$ and we are provided binary relations between these entities $\mathcal{R} = {P, Q, R, \ldots}$. 1 Also we have access to closure rules $\mathcal{C}$. Let $e_i, r_j, c_k$ be metavariables useful for indexing over the entities, relations and closure rules. The sizes of $\mathcal{E}, \mathcal{R}, \mathcal{C}$ are $\ks{E}, \ks{R}, \ks{C}$ respectively. a closure rule $c_k$ is a constraint an example of which is the following:

Now very generally we can view training examples as strings in a language of tuples where if two strings are present in a language then a third string must also be present as well. And the examples that we are presented are There are two other isomorphic ways to visualize these entities, relations and closure rules:

1. As nodes and typed edges in a graph, with constraints over the cliques in a graph. We observe edges that stay within a graph cut but not those that cross a cut. Or more generally we observe edges that stay within a connected cluster but not those that would cross across a cluster. And we may learn about the edges within a cluster independently of edges within another cluster.
2. As colored locations in a 2d automata, with constraints over the fill of the colored locations.

### An Example

Let $\mathcal{E} = {a, b, c}$, $\mathcal{R} = {P, Q, R}$ and

Our rules look rigid. If we want other rules then we would have to either express them in the specific form we have chosen or forego them.

For example, consider the following rules. Rule 1) can be expressed by unrolling it into explicit rules about the relations but that would not be possible if the number of rules is large. Rule 2) and 3) can not really be expressed easily in our framework.

Also consider the case where we have other type of rules for example

$$$e_1\ R \ e_2 \text{ If } e_2\ R\ e_1$$$

### 2d Automata Visualizations

Let us assign a color to each predicate, then our closure rules can be represented as the following:

 P Q R P P Q R R R

Our colored automata analogy breaks down if there can be more than one relation between the entities. But if the closure rules are written to ensure that there should be only type of relation between any two entities then the colored automata analogy holds true.

### Previous Work

This list is incomplete!

### Interesting Questions

This list is incomplete!

Now the interesting questions are as follows:

1. If we are given a random sample of the strings from the language then how can we best use the rules?
2. If we are given biased samples of the edges in the graph such that we don’t see edges that cross a cut boundary then how can we recover the edges?

## Footnotes

1. Throughout this article, we represent the entities in lower case and the binary relations in upper case.