# Effectively universal Turing machines and Turing-completeness?

An effectively universal Turing machine $$T$$ is a Turing machine for which there exists a recursive reduction $$f$$ such that $$\forall A:U(A)=T(f(A))$$, where $$A, f(A)$$ are finite sequences of symbols (that are given as inputs) and $$U$$ is a universal Turing machine.

Is there any connection between effectively universal Turing machines and Turing-completeness?

Also, given a Turing machine $$T$$ that is not effectively universal, is it possible to decide if it halts on a specific input? As far as I am aware, the diagonalization argument would not apply, since the TM used in the construction would be effectively universal.

My attempt so far was to use an argument from contradiction:

If this variant of halting problem is undecidable, then there exists a recursive reduction from the (general) halting problem to it. However, if such a reduction exists, it contradicts the premise that $$T$$ is not effectively universal.

I am not sure, however, if this is correct or if it misses anything.

• You seem to be missing some "for all $A,B$" somewhere. Please edit the question to clarify the first sentence and place the quantifiers in the correct place. Also, what are $A,B$? Also, what does "$T$ behave on $B$" mean? Can you make the definition more precise?
– D.W.
Commented Feb 16 at 22:42