Can you ask/instruct a robot on shabbos to perform melacha on your behalf? Would this fall under the same category as amirah l'akum?
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The consensus of authorities permits setting timers before Shabbat. It is similarly permissible to allow a website to sell goods on one's behalf on Shabbat.2
Apparently the consensus is that causing machines to work for us on Shabbat is not under the prohibition of amirah l'akum (instructing Gentiles). However, the use of a robot on Shabbat may still be forbidden for other reasons. For example, if it employs a microphone to listen for instructions, there are cases when use of a microphone is allowed, such as in hearing aids (Shemirat Shabbat Kehilchato, vol. 1, 34), and other cases in which it is forbidden, such as for recording or amplifying a performance (ibid. vol. 1, 28).
Robots exist now that can act based on voice instructions. You can't tell such a robot to do melacha for you just like you can't indirectly cause any melacha. Additionally, we normally assume using electric devices is assur, and you are directly activating to one.
Ignoring the electric issue, what if the robots developed more advanced AI's so it was a separate agent which made your act of speaking less directly connected to the action it performed?
The gemara states one can acquire an animal by calling it to you (so it walks 4 amos), but not an eved, since it comes on it own free will. Assumably, a robot will be more like an animal since it is unlikely to have human-like free will. If the case by kinyan is parallel to shabbos, it should be assur to cause (through speech) animals or robots to do melacha on shabbos.
Also, it would definitely go against the spirit of shabbos, but probably is not included in amirah l'akum.
Robots have no free will (just like animals) and I remember seeing a Teshuva which says you can't tell animals to do Melacha for you on Shabbat.