Counter Machine - Two-counter Machines Are Turing Equivalent (with A Caveat) - Step 3: Four Counters Can Be Simulated By Two Counters.

Step 3: Four Counters Can Be Simulated By Two Counters.

As before, one of the counters is used as scratchpad. The other, real counter holds an integer whose prime factorization is 2a3b5c7d. The exponents a, b, c, and d can be thought of as four virtual counters that are being simulated. If the real counter is set to zero then incremented once, that is equivalent to setting all the virtual counters to zero. If the real counter is doubled, that is equivalent to incrementing a, and if it's halved, that's equivalent to decrementing a. By a similar procedure, it can be multiplied or divided by 3, which is equivalent to incrementing or decrementing b. Similarly, c and d can be incremented or decremented. To check if a virtual counter such as c is equal to zero, just divide the real counter by 5, see what the remainder is, then multiply by 5 and add back the remainder. That leaves the real counter unchanged. The remainder will have been nonzero if and only if c was zero.

As a result, an FSM with two counters can simulate four counters, which are in turn simulating two stacks, which are simulating a Turing machine. Therefore, an FSM plus two counters is at least as powerful as a Turing machine. A Turing machine can easily simulate an FSM with two counters, therefore the two machines have equivalent power.