A Universal Turing Machine is a Turing Machine with a fixed action table that can be used to simulate any other Turing Machine. Both the description of the other Turing Machine and its data are encoded onto the tape of the Universal Turing Machine. So a
Universal Turing Machine can calculate any computable function.
I have drawn from the particular machine that was created by Paul Rendell for his
Game of Life implementation, because I had similar goals - that it be reasonably simple, but also not take too many cycles to perform the simulation. Apart from reversing the tape and changing the symbol encoding, it is essentially the same machine, with 13 states and 8 symbols.
The
Turing Machine being simulated simply duplicates a string of symbols, effectively a unary multiply by 2. The data being operated on is a string of three symbols. The entire run takes over six thousand actions of the Universal Turing Machine, not bad by UTM standards, but painfully slow in this Minecraft implementation. Most of the time is spent shuttling back and forth between the program section and the data section on the tape, each time an action is simulated.
Here are the parts in more detail:
- The tape is a sequence of blocks. The colors have no significance to the operation, they just help show the motion of the tape.
- I have marked the division between the stored program and the data using red blocks. Towards the right end of the tape is the data being operated on. And to the left end is the encoded program of the Turing machine being simulated.
- The head reads the symbol from the tape in place, using redstone torches underneath the tape. These bits and their inverse are conveyed to the action table, where they can be matched by placing torches on the columns. In some cases, I have left out torches so that more than one symbol can be matched by a single action, which helps to reduce the size of the action table.
- Across from the symbol matching rows are the state matching rows. At the head of these rows are the four latches holding the current state of the machine, and again redstone torches are used to match against particular states.
- Underneath the matching rows are the wires that trigger when there is a match. At the end of the wire is a redstone torch, used to trigger the next action after the current one is finished, or left out if the machine should halt.
- Back from this there is a torch in one of two columns, which triggers the right or left tape shift at the end of the action.
- Further along is the set of four torch positions which set the next machine state for each action. I have reserved state zero for the error case when no action matches.
- And still further along is the set of three torch positions that define the change to the symbol. For simplicity, these invert bits of the symbol, rather than setting it, which also helps when combining action rows that leave the symbol unchanged.
- The outputs are all relayed underneath from the far end of the action table, so that the delay time is the same no matter which action is matched out of the array.
- The symbol change outputs feed underneath the head and in from the back. The head uses pistons to change the bit values, pulling back the top block, raising the bottom block and lowering the top block, and then pushing the new bottom block into place. A solid block at the bottom represents a bit value of 1, and a glass block a bit value of 0.
- This video features the tracks "Klik" and "Grom" by Jaap Blonk & Radbood Mens available under a Creative Commons License.
[Notes]
Symbol encodings:
'0' 000 'A' 100
'1' 001 'B' 101
'C' 010 'D' 110
'M' 011 'X' 111
Initial tape sequence :
M00C0100M1C0100M11C0101M00C1111M00C0001M11C011XBBDBABXAABABA10000000
Final tape sequence:
M00C0100M1C0100XBBDABABXAADBBBBXAADAAABXBBDABBXBBDBABXAABABABABABAB0