For a Finite State Automaton / Finite State Machine (FSM) $F$, that has an input alphabet, a set of possible states, an initial state, a set of possible final states and a state transition function, let a finite input sequence $S$ is given, such that at the end of this sequence the FSM enters a final state and stays in that state.
Can this FSM $F$ along with the input sequence $S$ be considered a separate FSM $F'$?
Analogous to this, can a Turing machine $T$ along with a finite tape $P$ be considered a separate Turing machine $T'$?
What are the conditions, if any, for this to be true assuming it is true?
Note: I expect a formal proof, or a reference/outline to a formal proof that proves that either of this can or cannot be done. Some theory related to this is also welcome.
Closely related topic:
I am aware of the principle of hardware and software equivalence, which states that a given task can be performed using hardware or software, i.e. digital hardware and software are equivalent models of computation. But I think my question is different from this one.
From this question ( Is there code below microcode? ) I think we can consider an FSM with its input sequence (microcode) to be a part of another FSM (the digital computer), but of course much more circuitry like Arithmetic and Logical Unit (ALU) and datapath is needed to make a computer. Microcode is used only for the control circuit.
This answer claims the data in the RAM of a computer along with the CPU can be considered to be a part of a bigger circuit.
The circuit is fixed (it is the gates in the processor) and part of its input is data that depends upon the program you are executing (which is stored in the RAM of the computer). However, you could consider this a larger circuit where part of it is hardcoded (i.e., the program part of the input is hardcoded); then you can view a computer running a program as a big circuit with part that is universal and identical for all programs (the gates of the processor) and part that depends on the program (the hardcoded input), and this immediately gives a mapping from programs to circuits. The mapping is implemented by a compiler.