Elliott Battle
12/07/12
The Turing Machine
Tankurn Kotravivatana’s presentation on the Turing machine sought to demonstrate the
limitations of computability. He started –off by first asking the question, “what is computable?”.
The definition given by Alan Turing was that a number was computable so long as a machine
could write-out its decimal numbers using instructions in binary. For example, the number
0.01010101… in binary produces an infinite series that converges at 1/3, thereby allowing the
computer to print 1/3 and making 1/3 a computable number.
Using scanning methods and algorithms Turing described, one could even write irrational
numbers on a computer. A computer capable of doing this, a “turning machine”, would be able
to do this task competently. In a specific case, a universal Turing Machine, U, can take an input
string w such that U(M,w) is accepted iff M(w) is accepted.
This, however, produced issues with “busy beaver numbers” on which a universal Turing
machine would halt while trying to produce the number because it was impossible to do so. How
could this outcome be predicted without actually running it through the machine? Could a
machine be devised which could tell just by looking at the other machine and a valid input
whether the code would be accepted? No. The presentation demonstrated that for such a machine
to exist would introduce the potential for a logical contradiction using a proof similar to
Russell’s Paradox.
He also demonstrated an example of “the game of life”, a mathematical game devised by
John von Neumann in which square cells of an infinite grid are filled or emptied based upon the
number of adjacent occupied squares. Some initial configurations lead to stable configurations
over many moves, some lead to cyclical configurations that fluctuate back and forth, some lead
to “extinction”, in which none are left filled, and others lead to complete chaos; any stability in
the system comes so late in the cycle that we couldn’t possibly notice. This idea was later tied-in
with the Turing machine idea… could a computer compute whether a given pattern would be
stable or not without running through it?
Again, no. The Turing Machine cannot predict such an outcome. The inevitable problem
we find over and over again is that the question of computability itself is not computable.