Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

72 the mathematical infinitymachine is not a good model of a computer with respect to execution time. At anyrate, it turns out that one can actually produce relatively simple examples of universalTuring machines with very few states and symbols. Minsky gave an example of sucha machine U with seven states and four symbols and showed that any Turing machinecan be represented by a machine with only two symbols. It is not known what is thesmallest number of states for a machine U with two symbols, although there are somecandidates for it requiring only four states. 7The third busy beaverS = ABCHI = A0 → 1RB, A1 → 1LC, B0 → 1LA, B1 → 1RB, C0 → 1LB, C1 → HThe program runs as follows, where the highlighted box shows the current position ofthe tape head before the next reading step:A ...0 0 0 0 0 0 0 0 0 ...B ...0 0 0 0 0 1 0 0 0 ...A ...0 0 0 0 0 1 1 0 0 ...C ...0 0 0 0 0 1 1 0 0 ...B ...0 0 0 0 1 1 1 0 0 ...A ...0 0 0 1 1 1 1 0 0 ...B ...0 0 1 1 1 1 1 0 0 ...B ...0 0 1 1 1 1 1 0 0 ...B ...0 0 1 1 1 1 1 0 0 ...B ...0 0 1 1 1 1 1 0 0 ...B ...0 0 1 1 1 1 1 0 0 ...A ...0 0 1 1 1 1 1 1 0 ...C ...0 0 1 1 1 1 1 1 0 ...H HALTgiving a string of six 1’s, taking thirteen steps to end, and using six cells of the tape(two to the right and three to the left). Note also that this busy beaver is not theslowest, because S(3) = 21; this number of steps is attained by another busy beaver,thus showing that there may be more than one champion beaver.Notwithstanding these results, it was proved by Tibor Radó in 1962 (Radó 1962;Lin and Radó 1965) that the function (N) is not computable by a Turing machine,because it grows faster than any function computable by a Turing machine. In otherwords, there is no Turing machine that for all N computes (N) on input N. Theexistence of uncomputable functions (i.e., not representable by any computer program)gives us a hint of the unapproachability of infinity by finite means. The halting problemfor Turing machines is undecidable, so there is no finite algorithm that, for every Turingmachine, decides whether a Turing machine will halt or not.7 A recent claim (2007) for a machine U with two states and three symbols, for which a $25,000 prize has beenawarded, has given rise to controversy because it allows the use of special infinite input data, which is excludedin classical models of Turing machines. See the criticism in http://cs.nyu.edu/pipermail/fom/2007-October/012156.html and the response in http://forum.wolframescience.com/showthread.php?s=&theadid=1472.