Counter Machine - Two-counter Machines Are Turing Equivalent (with A Caveat)

Two-counter Machines Are Turing Equivalent (with A Caveat)

For every Turing machine, there is a 2CM that simulates it, given that the 2CM's input and output are properly encoded. This is proved in Minsky's book (Computation, 1967, p.255-258), and an alternative proof is sketched below in three steps. First, a Turing machine can be simulated by a finite-state machine (FSM) equipped with two stacks. Then, two stacks can be simulated by four counters. Finally, four counters can be simulated by two counters.