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I have two instances of Mathematica running on the same PC. Can I call the output of one from the other?

For example if Out[100] is on kernel1 how do I refer to it on kernel2?

There is probably a much simpler method, but here is one possibility:

RemoteValue[line_, kernel_] := Uncompress @ First @ FrontEndExecute @ ExportPacket[

    Cell[

        BoxData @ ToBoxes @ Dynamic[Compress @ Out[line], Evaluator -> kernel],

        "Output"

    ],

    "PlainText"

]

It's hard to simulate multiple kernels in an answer, but this is what I get when I use the above:

RemoteValue[28, "Local 2"]

% + 1

1596

1597

where the input in the "Local 2" kernel was 798 2, and the second line shows that the actual value is available.

Another possibility if you just want to see the output is to use Dynamic:

r = Dynamic[Out[28], Evaluator->"Local 2"]

1596

but in this case the actual value is not available:

Head @ r

Dynamic

Update

Here's how we'd do this for your case specifically. Assuming we created and connected a tunnel called "shuttle". First we do this on kernel 1:

TunnelWrite["shuttle", Evaluate@Out[32]]

And then on kernel 2:

TunnelRead["shuttle", Hold]



{Hold[{1, 2, 3, 9}]}

The hold is optional. I just added it for flavor.

Original

Here's another option that works directly at the MathLink level and so has more efficient data transport.

I wrote a layer or five on top of LinkCreate and friends to help ease some of the dangers and difficulties of using Links for inter-kernel communication.

The package is called KernelTunnels. Here's an example:

We can see I create a tunnel with TunnelCreate in the first kernel (just a link+metadata) called "shuttle". The tunnel data is written to a temporary .mx file so as to allow good unified interprocess communication.

In the second kernel I then connect to the same tunnel with TunnelConnect. At this point the two are attached to each other and can communicate freely.

TunnelWrite will write to the LinkObject and TunnelRead will drain everything off the link, wrapping it in the head passed as the second argument first. Note that TunnelWrite writes the unevaluated expression by default, so you'll need to use Evaluate if you want to circumvent this behavior.

There's also some stuff in there for adding handlers for TunnelRead (it can be different on both ends).

If this is useful I can properly document the API, but these functions are the heart of it.