Description Number - An Example of A Description Number

An Example of A Description Number

Say we had a Turing machine M with states q₁, ... q_R, with a tape alphabet with symbols s₁, ... s_m, with the blank denoted by s₀, and transitions giving the current state, current symbol, and actions performed (which might be to overwrite the current tape symbol and move the tape head left or right, or maybe not move it at all), and the next state. Under the original universal machine described by Alan Turing, this machine would be encoded as input to it as follows:

1. The state q_i is encoded by the letter 'D' followed by the letter 'A' repeated i times (a unary encoding)
2. The tape symbol s_j is encoded by the letter 'D' followed by the letter 'C' repeated j times
3. The transitions are encoded by giving the state, input symbol, symbol to write on the tape, direction to move (expressed by the letters 'L', 'R', or 'N', for left, right, or no movement), and the next state to enter, with states and symbols encoded as above.

The UTM's input thus consists of the transitions separated by semicolons, so its input alphabet consists of the seven symbols, 'D', 'A', 'C', 'L', 'R', 'N', and ';'. For example, for a very simple Turing machine that alternates printing 0 and 1 on its tape forever:

1. State: q₁, input symbol: blank, action: print 1, move right, next state: q₂
2. State: q₂, input symbol: blank, action: print 0, move right, next state: q₁

Letting the blank be s₀, '0' be s₁ and '1' be s₂, the machine would be encoded by the UTM as:

DADDCCRDAA;DAADDCRDA;

But then, if we replaced each of the seven symbols 'A' by 1, 'C' by 2, 'D' by 3, 'L' by 4, 'R' by 5, 'N' by 6, and ';' by 7, we would have an encoding of the Turing machine as a natural number: this is the description number of that Turing machine under Turing's universal machine. The simple Turing machine described above would thus have the description number 313322531173113325317. There is an analogous process for every other type of universal Turing machine. It is usually not necessary to actually compute a description number in this way: the point is that every natural number may be interpreted as the code for at most one Turing machine, though many natural numbers may not be the code for any Turing machine (or to put it another way, they represent Turing machines that have no states). The fact that such a number always exists for any Turing machine is generally the important thing.