The Turing Machine
What Is Turing Machine?
• It is a conceptual machine put forward by
Allan Turing in 1938.
• It is composed of
– A unlimitedly long tape with 0/1;
– A device with a read/write head which moves
one step at a time;
– A process box with pre-stored rules.
What A Turing Machine Can
Do?
• A Turing Machine can carry out a
particular task, such as “adding one to
a number”, “multiply two numbers”,
“find the highest common factor
between two numbers”, “pick the
larger from two numbers”.
Church-Turing’s Thesis
• Any problem that has an algorithmic
solution can be solved on a Turing
Machine.
Algorithm
• An algorithm is a step-by-step procedure,
by following which we can get a problem
solved.
• The results of an algorithm is independent
of executor and times of executions. That
is, no matter who is execute the algorithm,
and how many time it is executed, the
results will be, if input data are same, either
identical or undistinguishable.
Any Algorithmic Solution Can
Be Realized on a Computer
• If a problem can be solved by an
algorithm, then a computer program
can be generated to solve the problem.
Therefore …
• Any task that is accomplished on a
computer can be accomplished on a
Turing Machine.
Universal Turing Machine
• If the pre-stored rules in the process
box are put on the tape, then individual
Truing Machines with particular tasks
can be combined into one Turing
Machine – Universal Turing Machine.
• It has been proved that those pre-stored
rules in process box can be coded onto
the tape.
Therefore …
• The computer as we have now can be
conceptually reduced to an equivalent
universal Turing Machine.
Implications
• We can predict some features of
what computers can do in the future,
even we do not know what
specifically computers can do and
how complicated it is.
• For example, what the computer
does in the future must be copiable.
Turing Machine May Set a Limit
for How Far Computer Can Go
• Since we can foresee some features of
future computers through Turing Machine,
we may figure out something that can never
be done by a computer.
• A rigorous proof may be there right at the
corner to show computers are not
omnipotent, or at least cannot as intelligent /
conscious as humans.
Theorem of Incompleteness
• The enthusiasm and fanaticism of
developing a computing machine able to
develop all theorems in a math system came
to a sudden stop after Godel published his
Incompleteness Theorem in 1936
• Godel showed it’s not possible to exhaust
all theorems of a math system if staying in
the system.
Be Humble
• Godel’s Theorem of Incompleteness
served a sharp warning against the
blind arrogance of humans.
• Can we humans understand ourselves
so well that we can make machines
more intelligent than us?