For the sentence 'I watched a movie after I had food', are the following translations equivalent? Additionally, is the 3rd translation incorrect (is the ...了...以后... usage valid)?
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Briefly, I'd parse them as
以后 corresponds exactly to the English equivalent of 'after', while 了 has some more leeway in how it's used. All three of your sentences imply that you ate, and then you watched a movie (arguable for the first. The English equivalent would probably be "I ate food, then watched a movie"). I think 3 would be the closest in meaning to your original sentence. Although if you said that before a meal time, I'd assume you meant the upcoming meal, while if it was late at night, I'd assume that you'd already ate and watched.
When translating sentences that are in the past into Chinese it is always best to use time markers such as "yesterday", "this morning," etc. It is time markers that put sentences in a "time context," not "了" or other particles. Strictly speaking your sentences are not equivalent, because they are ambiguous when it comes to tense. Sentence 2, for example, could be in the past, present or future tense depending on context. Something similar is going on in the other two sentences. As to how to translate your sentence, I would use the pattern I would use the pattern "先...再" (or 然后) and "了" or the word "接着" and "了."
all three sentences are grammatically correct. the first one may be a little colloquial but it is definitely something i would use in a daily conversation. The real problem is: They are ambiguous. it is impossible to tell if it happened in the past, present, or future. ttt555 gives one possible, but not the only interpretation for each sentence. my observation is that, however, if no particles indicating tense is found near a verb, then it'd be normally understood as present or future tense. in your three sentences, none of them have such a particle for 看. Therefore, i'd naturally interpret ALL of them to be a prediction of the future, rather than something happened in the past.
I would recommend
to describe these events that occurred in the past. note that i replaced the first 了 with a 过 to avoid repetition. Also, i think keeping 以后 here helps making the sentence less ambiguous and getting rid of the "colloquial" flavour as found in your first sentence.
I'm not a native speaker, but I don't like the first sentence, I don't think it's correct.
To me the 2nd sentence implies a future event (I will watch a movie after I eat), whereas the 3rd sentence implies a past event (I watched a movie after I ate) although I would like to ask a native speaker whether there should be another 了 in that sentence such as 我吃了饭以后看电影了.
I would like to say something.
These 3 sentences represented a trait in Chinese, that is there is no very explicit tense in the verb of this language. So actually, they can be correctly understood as the exactly same meaning - I will watch TV after eating dinner.
Unless they provide more information like time or other adverb to tell you if it happened or is happening or finished already, either way is right.
So I think all of the answers above are right since no enough information provided. Furthermore, I don't think it is necessary to distinguish the meanings like tttt555's answer, because when the same sentence in a different environment or different paragraph, the 3 meanings can exchange each other without generating any mistake. This is Chinese, it is the same situation like news title 山东队大败河南队, you never know who actually won unless you read the details in the paragraph.