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I think it should say "if w is accepted by M" not "if w is accepted by A" in the Formal definition section. I cannot confirm this myself because what each element in the 6-tuple represents is not stated. ~hb2007 13:31, 3 May 2017 (UTC)

• @ Hb2007: Thanks for your remarks. I corrected A to M. Furthermore, I stated that the 6-tuple is exactly the same than in the page Non-deterministic Turing Machine. I hope it's more clear. Arthur MILCHIOR ( talk) 23:19, 27 May 2017 (UTC) reply

It would be good if the section on expressivity of UTMs did not assume prior knowledge of this concept by the reader. A cursory search through the complexity and TM-related articles did not reveal a formal definition of this concept (I found /info/en/?search=Expressive_power_(computer_science), but it only talks about formal languages and the Chomsky hierarchy). Perhaps we could add a short sentence or two explaining the concept? I'm posting this as a discussion because I don't feel confident making it myself. 31.222.81.237 ( talk) 23:21, 1 June 2024 (UTC) reply

I'm new to the idea of UTMs but I think the expressivity idea is:

• Each Turing machine T induces a language ${\displaystyle {\mathcal {L}}(T)\subseteq \Sigma ^{*}}$ or ${\displaystyle {\mathcal {L}}(T)\in {\mathcal {P}}(\Sigma ^{*})}$.
• The set of all possible Turing machines T is a subset of ${\displaystyle {\mathcal {P}}(\Sigma ^{*})}$, the decidable/recursive languages.
• This subset describes the expressivity of Turing machines and it's remarkably robust to changes: one tape vs. multiple tapes (including read/write only); binary vs finite alphabets; left endmarker or no left endmarker; infinite in one direction or both directions.
• Here the article claims that the expressivity of unambiguous Turing machines is the same as that of deterministic and non-deterministic Turing machines.
• However, when we apply complexity classes (e.g. polynomial time or space limitations), the expressivity changes based on whether you have a deterministic, unambiguous or non-deterministic Turing machine. The assumption is that inclusions are strict: ${\displaystyle {\text{P}}\subset {\text{UP}}\subset {\text{NP}}}$.
• This is the same idea as a lot of modifications of Turing machines (e.g. probabilistic Turing machine).

I'm also a bit hesitant to edit the article in case I've misunderstood. It would be good to add more sources to the article. I imagine any references on unambiguous Turing machines would have to talk about the concept of expressivity, whatever terminology that is expressed in. — Bilorv ( talk) 15:28, 2 June 2024 (UTC) reply

What @ Bilorv: wrote is basically right (EDIT: except the bullet about recursive/decidable, see below). I edited to clarify, but it still needs a good source. Caleb Stanford ( talk) 23:55, 2 June 2024 (UTC) reply

With one exception though. The set of all Turing machines recognizes the set RE of recursively enumerable languages, not the decidable/recursive languages. That's if we're talking about recognizability (exists a TM such that it accepts exactly on the words in the language). If we're talking about decidability (exists a TM such that it halts on words in the language and words not in the language), then depending on how things are defined UTM and NTM can recognize RE whereas DTMs recognize R. Caleb Stanford ( talk) 23:57, 2 June 2024 (UTC) reply