Turing Machines

Abstract:

The finite-state machine is too limited to even be able to recognize simple sets like , where stands for n copies of a. In 1936 Alan Turing proposed a machine (now called a ``Turing Machine'') which is general enough to implement any algorithm to date. It is a simple modification of the finite-state machine, as we see below.

The Turing Machine

Definition: Let S be a finite set of states and I a finite set of tape symbols, including a special symbol b (for ``blank''). A Turing Machine is a set of quintuples of the form (s,i,i',s',d) where

• 
• 
• (R for Right, L for Left),

and no two quintuples begin with the same s and i symbols (ensuring that the machine is deterministic - that is, that it gives the same results every time, given the same initial state and input.

The machine acts on an ``infinite tape'' (so it's hard to implement...), with at most a finite number of non-blank entries (so it has unlimited memory, only a finite amount of which is actually non-blank).

The Turing machine processes the tape until it arrives at a state and input which is not found in the set SxI, at which point it terminates.

This is very similar to the finite state machine in some ways: we think of

• s as the present state;
• i as the current input;
• i' as the output, given the initial state s and present input i;
• s' as the next state (there is a next state function , since there are no two quintuples which have the same initial state s and input i);
• d as the direction in which the read-write head moves.

Only the last element d seems particularly different than before; but the output is also considerably different: it overwrites the input on the tape. Hence the machine can ``reread its input and ... erase and write over its input.''

``...[A] finite-state machine is a very special case of a Turing machine, one that always prints the old symbol on the cell read, always moves to the right, and always halts on the symbol b.'' p. 591.

Turing machines can be used as

• set recognizers,
• function computers,
• and, as ``promised'' by the Church-Turing Thesis, can be used to implement any algorithm.

Since any algorithm can be implemented in a Turing machine, the next questions to ask might be:

• Does an algorithm exist to decide whether individual statements from a large class of statements are true? (The Decision Problem)
• Under what circumstances will a Turing machine terminate, given fixed input? (The Halting problem)

You've got your work cut out for you: to tackle these and other difficult problems. Go to it!