Limits of Finite State Machines

1
Limits of Finite State Machines

• We saw that FSMs are limited there is no
  recogniser for the palindrome language
• A similar argument can be used to prove other
  languages cannot be recognised by FSMs
• For example, the language of strings containing
  exactly twice as many 0s as 1s

2
No FSM for twice 0s for 1s

• Suppose there is such a machine, M
• M has finitely many states, say n of them
• Suppose we feed n 1s into M, then M must revisit
  some state call it SR and suppose it is first
  visited after x 1s and again after y 1s (x lt y)
• Then, after x 1s M is in SR and if we then feed
  in 2x 0s and finish, M should accept the string
• However, if we feed in y 1s M will be in SR and
  so will accept a string with 2x further 0s
• This is a problem, since that string is not in
  the language and M cannot accept it!

3
What is missing?

• FSMs have repetition can revisit states via
  transitions
• FSMs have conditional behaviour transitions
  depend on inputs
• But FSMs only have finite memory capacity
• The proofs of limitation all depend on
  overflowing the FSM memory

4
Adding Memory to FSMs

• Turing Machines extend FSMs by adding a memory
  unit
• Memory is modelled as an infinite tape of cells,
  with a read-write head over one of the cells
• Inputs now come from the tape and outputs are
  written into the cell over the top of the input
• In addition, each transition of the machine is
  coupled with a movement of the head, left or
  right, along the tape

5
Turing Machines
6
Simple Example

• A Turing Machine to check for palindromes

0/0/R 1/1/R
Read 1
1/_/R
_/_/R
Start
1?
_/_/L
0/0/R 1/1/R
1/_/L
Read 0
_/Y/.
0/N/. _/N/.
0/_/R
_/_/L
Back to start
0?
0/0/L 1/1/L
Stop
0/_/L
1/N/. _/N/.
7
What can Turing Machines do?

• Everything a FSM can do!
• Arbitrary simple arithmetic
• Palindrome checking
• Twice 0s for 1s
• Anything a computer can do!

8
Are Turing Machines really computers?

• Each Turing Machine computes one function its
  control is hardwired
• Turing showed that there is a family of Turing
  Machines that can simulate the behaviour of a
  Turing Machine from a description given on the
  tape
• A Universal Turing Machine

9
Universal Turing Machines

• A Universal Turing Machine is essentially a
  programmable computer the program is a Turing
  Machine description on its tape

10
How complex are UTMs?

• There is a Universal Turing Machine using only 10
  states (for a language of 0,1,_)
• The number of states can be reduced by using more
  special symbols
• With a language of 18 symbols (including _) there
  is a machine with just 2 states!

11
Is there anything a Turing Machine cannot do?

• Are there problems a computer cannot solve?
• This raises important questions about the nature
  of human problem-solving and its limits