2-State 3-Symbol Universal Turing Machines Do Not Exist

In this brief note, we give a simple information-theoretic proof that 2-state 3-symbol universal Turing machines cannot possibly exist, unless one loosens the definition of “universal”. Disclaimer: This article was authored by Craig Alan Feinstein in his private capacity. No official support or endorsement by the U.S. Government is intended or should be inferred. In May of 2007, Wolfram Research offered a prize to anyone who could answer the question of whether a particular 2-state 3-symbol Turing machine is universal. In October of 2007, Wolfram Research announced that Alex Smith, a student at the University of Birmingham, proved that the particular 2-state 3-symbol Turing machine is universal [1]. But not every expert in the field of theoretical computer science was convinced that Alex Smith’s proof was valid [2]. In this note, we give a simple information-theoretic proof that 2-state 3-symbol universal Turing machines cannot possibly exist, unless one loosens the definition of “universal”: A universal Turing machine must be able to perform binary operations like OR, AND, XOR, etc., between bits, and its tape-head must have the freedom to move left or right independent of the binary operations, in order to simulate other Turing machines with this property. This implies that the tape-head of a universal Turing machine must be able to keep track of at least three bits of information at a time, at least two for binary operations and at least one for the direction that the tape-head moves. The tape-head of a 2-state 3-symbol Turing machine can only keep track of log 2 (2 × 3) bits of information at a time, which is less than three bits of information; therefore, no 2-state 3-symbol universal Turing machine can possibly exist, unless one loosens the definition of “universal”.