Do you mean writing systems? Mycenaean Greek wasn’t *that* different from later forms of Greek. The language wasn’t the issue – and in fact the Greeks themselves tried to decipher Mycenaean texts without success.
In this case, Linear B was deciphered in a few steps:
* [Alice Kober](https://en.wikipedia.org/wiki/Alice_Kober) reasoned that the number of characters – about 200 – meant it couldn’t be an alphabet (no language has that many sounds) and couldn’t be a logographic system like Chinese (because 200 is too few characters for such a system; modern logographic systems have ~a thousand or more even for basic use). She therefore reasoned – it turned out correctly – that Linear B was a syllabary: a system that writes words in terms of syllables. (Few modern writing systems work this way; Korean [hangul](https://en.wikipedia.org/wiki/Hangul) and Japanese [kana](https://en.wikipedia.org/wiki/Kana) are the main two, though others exist, mostly in East Asia and for Native American languages.)
* Kober was able to figure out, with that hypothesis in mind, that many words carried common endings. This wasn’t obvious at first because the inflectional endings (like -ing in English) got combined into larger syllables, kind of like how (in my dialect, anyway) the word “doing” has -ing on its own but the word “running” attaches the doubled “n” to the suffix (that is, it ends in -ning, not -ing, and if I’m enunciating I’ll say *run-ning*, not *run-ing*). By doing so, she was able to work out that it represented an inflected language – and to confirm that her hypothesis that it was at least somewhat syllabic was correct.
* The decipherment was completed by two guys named Michael Ventris and John Chadwick. They built on Kober’s idea that Linear B must be syllabic and noticed that different sequences of characters only appeared on tablets in certain locations. They guessed – and could confirm – that these were place-names. Using those place names as guesses for pronunciations, they could start figuring out what each symbol represented, and could decipher enough to reveal that the written language was a form of Greek. From there, the rest was downhill.
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