2-State 3-Color Universal Turing Machines Don't Exist

In this brief note, we give a simple information-theoretic proof that 2-state 3-color universal Turing machines don’t exist. Let us suppose that there exists a 2-state 3-color universal Turing machine U (without a halting state). Then it must be possible to program U to emulate any Turing machine T (without a halting state) by initializing the tape of U with a string of colors that specify both the machine structure of T and also the input tape of T . And in order to emulate Turing machine T , universal Turing machine U must emulate the following infinite loop: (1) Read the status of the tape-head of Turing machine T and determine the rule of T corresponding to the status of its tape-head. (2) Apply the rule corresponding to the status of the tape-head of T . Goto (1). At every stage in the evolution of universal Turing machine U ’s emulation of Turing machine T , U must keep track of the step that it is carrying out in the above infinite loop, either step (1) or step (2). So the information content of the tape-head of U must be at least one bit. Also, in order for U to emulate T carrying out either step (1) or step (2), it is necessary for U to keep track of either the status of the emulated tape-head of T (for step (1)) or the rule corresponding to the status of the emulated tape-head of T (for step (2)). So the information content of the tape-head of U must be at least two bits. Furthermore, it is necessary for U to be able to determine when it is time to switch from step (1) to step (2) and from step (2) to step (1). So the information content of the tape-head of U must be at least three bits. Therefore, at any given time in the evolution of universal Turing machine U ’s emulation of Turing machine T , the information content of both the state of the tapehead of U and the color of the cell where the tape-head of U rests must be at least three bits. But U is a 2-state 3color machine, so since 2×3 = 6, the information content of both the state of its tape-head and the color of the cell where its tape-head rests is actually log 2 6 bits, which is less than three bits. Contradiction! Hence, no 2-state 3-color universal Turing machine can possibly exist.