> You could even implement a linear bounded Turing machine as an FSM it would just take an ungodly number of states.

To be clear, these things have formal definitions, and this statement is not correct.

1. A linear bounded automaton[0] is a Turing machine that can only overwrite symbols presented on its input tape as input. However, the definition of the automaton is still required to be finite, but it is required to operate on unboundedly large input tapes.

2. A finite state machine[1] is a model of computation where the sequence of input symbols are observed once and the machine is at all times in one of a finite number of states. It is equivalent to a TM that can only move right (and consequently cannot read anything it writes to the tape).

They are different, formally[2]. There are languages that a LBA can accept that a FSM cannot accept (famously, the strings of balanced parentheses cannot be recognized by a FSM).

[0]: https://en.wikipedia.org/wiki/Linear_bounded_automaton

[1]: https://en.wikipedia.org/wiki/Finite-state_machine

[2]: https://en.wikipedia.org/wiki/Pumping_lemma_for_regular_lang...