There will at the very least be nominee directors or other human agents acting on behalf of the corporation somewhere in the chain.
The problem as I see it is that you can probably "bootstrap" some ownership by using two or more corporations owning shares in each other (you'd have problems with laws relating to controlling stakes, but they probably are not unsurmountable) and make it work in the normal case with enough automation and nominee directors with strict, limited instructions on how to act and how to seek approvals - insist the owner is a recluse who will only interact via a strictly limited app or something.
The problem is likely to be what happens if anyone starts suspecting "nobody is home". There are too many opportunities for people then to take advantage, or take actions that an algorithm will have problems countering. Something as simple as filing a basic lawsuit against the company would likely make it all come tumbling down, because even if the algorithm contains the intelligence to make the right decisions, humans have this annoying tendency to still want things like signatures or want people to appear in person barring legal rights to equal treatment for algorithms.
So such companies may be possible, whether or not technically legal, but may not end up surviving very long. For now.
To repeat myself: Even if the algorithm contains the intelligence to make the right decisions, humans have this annoying tendency to still want things like signatures or want people to appear in person barring legal rights to equal treatment for algorithms.
The algorithms might know to hire a lawfirm, but that just shifts the problem around a bit. What happens when the law firm wants a signature on engagement documents? Or needs a deposition? Trying to avoid even providing signatures will set off red flags from the outset. Banks deal with "know your customer" laws that makes them liable if they don't look into who they're dealing with. Lawyers and accounts and many others can get struck off or face legal sanctions as well.
As an example, my accountants does everything to do with my finances, no problems. I could easily script my interactions with them. Apart from one little detail: They regularly need me to sign stuff. Like my tax returns and those of my company. At which point said algorithm would need to commit fraud, or use the signature of a responsible human.
You might be able to eliminate some of these e.g. by hiring a board consisting of humans acting on behalf of a "reclusive" shareholder, but the problem is that you're then handing these people extensive power over the company with no real way of reigning them in if they uncover the truth.
The problem as I see it is that you can probably "bootstrap" some ownership by using two or more corporations owning shares in each other (you'd have problems with laws relating to controlling stakes, but they probably are not unsurmountable) and make it work in the normal case with enough automation and nominee directors with strict, limited instructions on how to act and how to seek approvals - insist the owner is a recluse who will only interact via a strictly limited app or something.
The problem is likely to be what happens if anyone starts suspecting "nobody is home". There are too many opportunities for people then to take advantage, or take actions that an algorithm will have problems countering. Something as simple as filing a basic lawsuit against the company would likely make it all come tumbling down, because even if the algorithm contains the intelligence to make the right decisions, humans have this annoying tendency to still want things like signatures or want people to appear in person barring legal rights to equal treatment for algorithms.
So such companies may be possible, whether or not technically legal, but may not end up surviving very long. For now.