Hello World! An Invitation to Computer Science

Lecture Slides

Turing Machines
Monday May 5, 2008

Key ideas from last time

• models are easier to reason about than real world
• simplifying assumptions, like unlimited memory
• Turing Machines:
    finite states, infinite tape
    help us reason about what is computable

What Turing machine program corresponds to this state diagram?

What state diagram corresponds to this Turing machine program?

  state     content      write        move    next state
  =====================================================
  START     0              0          right       B
  START     1              1          right       A
  A         0              0          right       C
  A         1              1          right       C
  B         0              0          right       D
  B         1              1          right       C
  C         *              1          no move     HALT
  C         0              1          no move     HALT
  C         1              1          no move     HALT
  D         *              0          no move     HALT
  D         0              0          no move     HALT
  D         1              0          no move     HALT

Building up our arsenal

• As with other models (pseudocode, circuits), once we build primitives
    we can build more complex machines/programs from those primitives
• we have seen how to represent simple arithmetic
• from book, we see how to build a NOT machine
• and now we can construct AND and OR

Making Turing machines more robust

• find first non-blank before starting
• reset head after main computation
• what to do if input is "poorly formatted"

Representing different kinds of data

• Counting in unary
    (+) : TMs are standard for reasoning about computability
    (-) : would not use either TMs or unary counting in practice
    (-) : unary requires exponentially more space than binary
  
• how could we represent lists with TMs?
• can we represent Turing machines themselves?