# Lower and upper boundsΒΆ

Oriented Bell expressions are described by a Bell expression along with a direction `<=` or `>=`. Data about an oriented Bell expression is written using a `BellExpression` along with additional data in the special properties `lower` or `upper`, describing knowledge about the `expr >= bound` and `expr <= bound` directions respectively.

Each direction `lower` and `upper` can have the following properties:

- bounds
Bounds corresponding to different sets of interest can be listed here through a mapping list. The mapping keys

`local`,`quantum`and`no-signaling`are reserved for the usual accordingly named sets. The mapping values are given by a string expression made of:- integers,
- infinity written
`inf`, - rational numbers written down using the format
`numerator/denominator`, - decimal numbers written using
`digits.digits`, - arithmetic operators
`+`,`-`,`*`,`/`, - parenthesis,
- intervals written down as
`[lb, ub]`, where`lb`,`ub`are expressions.

Here is an example of section specifying several bounds:

```
bounds:
local: 2
quantum: [-inf, 2.828427124746191]
nosignaling: 4
```

Here, the `[-inf, 2.828427124746191]` interval expression for the quantum bound interval signifies that we have obtained an approximation for the quantum bound through relaxations, so that the real upper bound is below `2.828427124746191`. The `-inf` side of the interval could be replaced by finding an explicit state and measurements that achieve a quantum value.

- keywords
Keywords can also be associated to an oriented Bell expression. For now, no keywords are reserved, nor is any keywords automatically computed for

*oriented*expressions. Keywords here respect the same syntax as the ones for Bell expressions mentioned above.Special keywords specify if a bound corresponds to a facet of some polytope. The keywords

`facet-polytope`and`not-facet-polytope`can be used for that effect, where`polytope`corresponds to the name of a bound. The keywords`(not-)facet-local`and`(not-)facet-no-signaling`are reserved. If the facet-defining property is not known, the corresponding keyword is not present.The following is a valid

`keywords`section:

```
keywords: [facet-local]
```

Two orientations of a Bell expression can be specified in a single file by incorporating both a `lower` and an `upper` section in the file. This can provide a full decription of the bounds satisfied by a given Bell expression.