Sorry to be thick, but I don’t get it. In the first Markov process, apart from the fact that you used the plural, “outputs”, when pointing to the circle with a 2 in it at the right of the diagram, how are we meant to know that this “was specified as an output twice”? And similarly, how do we know that the circle with a 2 in it at the left of the second Markov process counts as two inputs?

Whoops! I think I’ll blame Brendan for drawing these in a style that doesn’t make those distinctions clear. Later in the paper we draw the diagrams in a style that’s unambiguous… and even more importantly, we define everything mathematically, so we’re not relying on the pictures to get the idea across.

This business about states being multiple inputs or outputs is one of those nitpicky technical details that turns out to make the machine run more smoothly, but seems only weird and confusing at first. We should not have bothered to mention it in the overview section of our paper, and I should not have bothered to mention it here. In fact I should rewrite/redraw this blog post to eliminate all mention of this nuance! It’s not really what I want people to be focused on, at first.

I also have no idea what the specific implications would be of a state being an n-fold input or output.

I’ll eventually answer your questions about that, and you’ll see it’s all very nice. But right now I’m being called to dinner!

]]>(In this example, one state was specified as an output twice, which means we need to identify two input states of another Markov process with it. Get it?)

Sorry to be thick, but I don’t get it. In the first Markov process, apart from the fact that you used the plural, “outputs”, when pointing to the circle with a 2 in it at the right of the diagram, how are we meant to know that this “was specified as an output twice”? And similarly, how do we know that the circle with a 2 in it at the left of the second Markov process counts as two inputs?

I also have no idea what the specific implications would be of a state being an n-fold input or output. If a population is initially flowing into a state in process A that is nominated as a 3-fold output, how does that population flow within the composite Markov process BA in which it somehow has to be spread between 3 different states that were 3 different input states of process B? Or can a 3-fold output only be joined to a 3-fold input, rather than 3 separate 1-fold inputs?

]]>An important fact is that black boxing is ‘compositional’: if one builds a circuit from smaller pieces, the external behavior of the whole circuit can be determined from the external behaviors of the pieces. For category theorists, this means that black boxing is a functor!

Our new paper with Blake develops a similar ‘black box functor’ for detailed balanced Markov processes, and relates it to the earlier one for circuits.

]]>If the infinitesimal transformation of the potential is little, then the trajectory variation is little, and it could be possible an (infinitesimal) coordinates trasformation to obtain an invariant of the trajectory: it could be possible to obtain an invariant that connect coordinates variations with potential variation (like in the case of field theory) for a classical system. ]]>

The spectral theorem in its grown-up, infinite-dimensional form tells you how to deal with this. Needless to say, physicists *pretend* the infinite-dimensional case is just like the finite-dimensional case… but Reed and Simon give a rigorous treatment of this crucial issue.