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The universal Turing machine $U_{TM}$ is a TM that takes in as input an encoding of a TM and a string, then runs the TM on the string and does whatever the simulated TM does. The language of the universal TM is the set of all encodings of a TM/string pair as a string for which the TM accepts the string.

I have seen several different names for the language of the universal Turing machine. Michael Sipser refers to it as $A_{TM}$ (the acceptance language for Turing machines), while Hopcroft, Ullman, and Motwani call it $L_u$ (the universal language).

Is the a standardized term for the language of the universal Turing machine? I would understand if there might be many "universal Turing machines" that vary in their encoding schemes, so if the answer is "no, there is no general term for this" that would be good to know. I'm mostly asking because I'm teaching an introductory course in the theory of computation and have been using the term $A_{TM}$ for this without knowing if there is a better term to use.

Thanks!

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No, there is no general term for this.

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