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From this answer, and the comments that follow:

To clarify, this machine T′ with the fixed output for all inputs is a Turing machine? – Shashank V M

@ShashankVM, Yes, T′ is a Turing machine. – D.W.♦ (When can a deterministic finite-state-automata (DFSA) along with its input sequence be said to be a part of another DFSA?)

From what I understand, a machine with fixed output for all inputs is a Turing machine.

Consider this thought experiment, I have a "machine" which is a wall with a slit through which I can pass a tape in. The "output" of this machine is on the wall, i.e. it is painted on the wall as a fixed string $S$, which for the purpose of this experiment is "Hello". So no matter what is written on the tape, the output of this "machine" is "Hello". From what I understand this "machine" is a Turing machine.

Is my understanding correct or am I missing something?

Because I find it hard to believe that this wall with a slit can be considered a Turing machine, when a supercomputer is considered a Deterministic Finite State Automaton.

Also I cannot see how this "machine" can emulate any another Turing machine or do any useful computation.

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There are many different Turing machines. Each one runs a fixed, hardwired program. Most Turing machines do nothing useful; many of them just halt immediately.

Some are universal, meaning that they can emulate any other Turing machine given a description of its hardwired program as part of their input, but most aren't. The machine that always prints "Hello" obviously isn't universal, but it is a Turing machine.

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  • $\begingroup$ Oh, I had forgotten that a universal turing machine is different from a turing machine. $\endgroup$ Mar 1 at 7:02
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There's no notion of "can be considered to be a Turing machine". Something either is a Turing machine, or it isn't. There's a formal mathematical definition of what a Turing machine is: https://en.wikipedia.org/wiki/Turing_machine#Formal_definition

I never claimed that just because some process always produces the same output, that process must be a Turing machine.

There might be some confusion about Turing machines vs DFAs. A computer could be modelled as a DFA (if we treat its memory size as fixed), or it could be modelled as a Turing machine (if we imagine that it's possible to gain more memory; or we just don't care to make use of the fact that memory size is fixed). These two different ways to think about a computer have different advantages and disadvantages. I suggest reading Is a PDA as powerful as a CPU? and How many gates until Turing Complete and How does an automaton model a computer or something else? and How does an automaton actually "compute"?, as I suspect they get at the core issue you are struggling with.

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  • $\begingroup$ Can you suggest a good textbook to study these topics? $\endgroup$ Mar 1 at 7:00
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    $\begingroup$ @ShashankVM, Sipser is one standard textbook: math.mit.edu/~sipser/book.html $\endgroup$
    – D.W.
    Mar 1 at 7:02
  • $\begingroup$ Thanks for this. $\endgroup$ Mar 1 at 7:03

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