Outline

• Turing Machines
• Quiz
• Lecture short due to time spent on Midterm and HW3

General Models of Computation

• So far we have looked at machines that either have bounded memory or access to memory limited to stack operations.
• We would like to consider models of computation which correspond to general purpose computers.
• In 1936 ("On Computable Numbers", London Math Society), Alan Turing presented such a general model of a computer, the Turing Machine.
• In this model, the machine has a finite control and an arbitrarily long tape of data consisting of squares able to hold one symbol. The machine also has a read head which can read one square at a time. Initially, this tape is blank except for the n first squares under and to the right of the tape head which have the input. The machine in one step is allowed to read what's under its tape head, write a new symbol, move left or right one square and change its state.
• The machine has special states which cause it to halt. The contents of the tape when these states are entered is the output of the machine.
• The first actual computer developed for code-breaking during World War II was partially based on his model.
• It turns out this model is actually equivalent to what can be done on modern computers.

A Real-Life Turing Machine

[YouTube video of above machine in action]

Example

• Let `B` be the language `{w\#w | w in {0,1}^star }`. This language is not context-free.
• A Turing Machine `M` that could accept this language might operate as follows:
  

   On input `w`:
  (1) Zig-zag across the tape to the corresponding positions on either 
  side of the # symbol to check whether these positions contain the 
  same symbol. If they do not, or if no # is found do into the 
  reject state. If they have the same symbol change the symbol 
  to a new symbol X.
  (2) When all the symbols on the left side of the # have been 
  X'd out, check if there are any more symbols to the right of the #. 
  If yes reject; if not accept.
  

• By accept or reject, I mean we view the halting states as further partitioned into those which are accepting and those which are rejecting and we enter the appropriate kind of halting state.

Quiz

Which of the following is true?

1. The SEQUITUR algorithm gives a method to construct a CFG from a string.
2. We gave an algorithm in class to determine whether a PDA accepts all strings.
3. The CFLs are closed under intersection.