Hello World! An Invitation to Computer Science

Lecture Slides

Models of Computation
Thursday May 1, 2008

Recapping the Big Picture

• phrasing problems in way that computers can solve
• fundamental elements of algorithmic problem solving
• key concepts:
  • algorithms
  • bits
  • Boolean expressions and circuits (hardware realization)
  • low-level software realization
  • high-level software realization
  • abstraction!
      algorithms & programming
      circuit construction
      operating systems & networks
• we conclude by considering limits of computers
    and consider whether those limits apply to human brain

Models in general

• models are simplifications
• easier to reason about
• safety & ethical considerations
• ease/cost of modification/rerunning
• hard part: deciding what factors are important

Computing agents

1. input and output
2. storage and retrieval from memory (state)
3. carry out instructions according to memory and current input

models of computation
  allows us to ignore limits of computer hardware
  such as memory, numeric precision

Some historical background

• computer science before modern computers
• mathematicians searching for completeness
• Gödel's Incompleteness Theorem
• Turing machines and halting

Turing machines

• simple abstract model
    could almost build one (except infinite tape!)
• deterministic
• each machine "hardwired" to solve single problem
• tape head: read, write, move left/right
• fixed finite number of internal states
    independent of input
• unbounded input/output tape consisting of discrete cells
• each cell contains one of a finite set of symbols
  ("blanks" and at a least one-blank symbol)

Turing machine as computing agent

• Tm's follow criteria for algorithms
• halting only guaranteed when input appropriate
• (in some sense ability to fail to halt is good!)
• meta machines: executing Tm is algorithmic

Simple example

  current   current        value      direction   new
  state     cell content   to write   to move     state
  =====================================================
  START     *              0          right       A
  A         0              *          no move     HALT
  A         1              *          no move     HALT
  A         *              *          no move     HALT

What does this Turing machine do?

Another example

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

What does this Turing machine do?