This most embarrassing "lassen" again.
I've been meditating on this sentence since early in the morning but still can figure out neither the role of "lassen" here, nor the grammar construction used, and thus, cannot get its exact meaning. Every time I come across "lassen", I'm in panic. No visible progress over time.
I've come across this sentence in today's DW news: "Erdogan hat angekündigt, seine Truppen im Nordwesten Syriens weiter in Richtung Idlib vorrücken zu lassen."
I'm also interested in the native or experienced speakers' comments here.
My (a humble fellow learner's) speculations go like this:
To move the troops forward - Truppen vorrücken lassen (Leo)
So it's something like let meaning unleash here. I've also seen this: um keine Zweifel aufkommen zu lassen - to exclude any doubt, which is equal to not to let any doubt occur.
So, in this particular piece of news I guess it as "Erdogan has announced that he will let his troops to move on in the North-West of Syria to advance towards Idlib." Or something like that.
I doubt the translation with unleash here. I think lassen simply implies that it is not a direct action. Eg. er kocht (he cooks) vs. er lässt kochen (he get's somebody else to cook). In the example above he orders the troops to move forward (maybe he tells a general to tell the troops to do it, he is not necessarily with them). But if the sentence would be Er rückt mit seinen Truppen vor it would be certain that he is with them.
Your speculations are mostly on point.
I disagree only with one small thing of your example "to exclude any doubt, which is equal to not to let any doubt occur.". I dont think these are equal and instead I think theyre only similar. The excluding is the stronger action of those two.
1 Ich mache meine Hausaufgaben um keine Zweifel aufkommen zu lassen. I do my homework to leave no doubt. I do my homework to not raise any doubt. Here the main action is doing the homework and the aim of the action is not leaving a doubt. This example is close to your construction with um; the um makes it different from the OPs construction.
2 Ich schließe aus, dass irgendein Zweifel aufkommt. I exclude that any doubt rises. Here the action itself is the excluding/preventing. You are actively (trying to make) making it impossible that any doubt rises.
3 Ich lasse keine Zweifel aufkommen. I do not let any doubts occur. Here the action itself is also the preventing. You are (trying to create) creating a situation in which no doubt occurs, but technically it would still be possible to have a doubt.
4 Ich lasse keine Zweifel zu. I do not allow any doubts. The german sentence can also describe a reflexive meaning like I am not allowing myself to have any doubts.
Last but not least, an example for 2 and 3. 2 You have seen with your own eyes how PersonA murdered PersonB. For yourself you can exclude any doubt that PersonA is a murder. 3 You got told by 500 reliable witnesses that PersonA murdered PersonB. Now you have no doubt that PersonA is a murder (but technically it could be that all those 500 people lied to hurt PersonB).