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What I mean is, is there an algorithm that given any finitely presented group with soluble word problem can solve the word problem on that group?

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No. This is the Boone-Rogers theorem on unsolvability of the uniform word problem. It's actually mentioned in the Wikipedia article on the word problem. https://en.wikipedia.org/wiki/Word_problem_for_groups

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  • $\begingroup$ Thanks! If you don't mind me asking, it says that there exists no "uniform partial algorithm" that solves the word problem for all finitely presented groups with soluble word problem, I can grok what "partial algorithm" means from the link, but what does the word "uniform" signify? $\endgroup$ – user3201708 May 6 '18 at 23:52
  • $\begingroup$ @user3 Honestly I'm not an expert on this, I found your question interesting and looked up the answer on Wikipedia. I would guess uniform just means it works for everything. $\endgroup$ – Matt Samuel May 6 '18 at 23:59

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