Models%20of%20Computation

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Models of Computation

• Turing Machines
• Finite store Tape
• transition function
• If in state S1 and current cell is a 0 (1) then
  write 1 (0) and move head left (right) and
  transition to state S2
• Halting problem undecidable!!!

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Models of Computation

• Turing machine variants do not add computational
  power
• Multiple tapes, multiple heads, 2-directional
  infinite tapes etc.
• RAM machines same as Turing, except that a MOVE
  can bring the head to an arbitrary position on
  the tape gt computers Von Neuman architecture

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3
A simple Machine Language

• MEMI memory
• ACC
• STORE I
• LOAD I
• LOADC X
• ADD I
• IFZ Label

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Computability vs. Expressiveness

• Computabilty can we compute something?
• Expressiveness how easy is to implement
  something we want to compute?

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Uniform and Non-Uniform models

• A computation defined by a Turing Machine is
  uniform solve problems of any size
• CIRCUITS non-uniform models
• Example 16-bit adder, 32-bit adder will not
  add (without extra tricks) 2 64-bit numbers

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Finite Functions as building blocks

• Finite Functions (in particular finite sets)
• 32-bit, 64-bit words are finite functions to
  0,1
• Arrays are finite functions
• Strings are arrays therefore finite functions
• Structures and Fields (name-gtvalue) are FF
• Functions about Functions gt
• A Turing Equivalent Model Lambda Calculus

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Sparseness

• String on V finite function (from 0..n) to V
• V an alphabet, V all strings on V
• A set S included in V is sparse iff it has a
  small number of strings of any given length N
• where small means something like a polynomial
  number of or something else, depending on the
  relevant complexity class
• Sparseness has a profound effect on computation
  models and computer architecture.
• Reading/writing a memory word at a time is
  efficient because, in most problems, relatively
  few words need to be changed at any given time gt
  Von Neuman computer, CPU etc.

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Programming Languages deal with sparse sets

• Intuitively only a small set of possible
  combinations of syntactic elements are meaningful
  and end up being used in an actual programming
  language
• Various conjectures (mostly by Hartmanis) state
  that if there are (complexity-wise) hard sparse
  sets than something in the complexity hierarchy
  collapses (i.e. Mahaneys theorem)
• Efficient representation of sparse sets gt
• A solution hashing represent large structures
  with small ones (small integers) knowing that
  only a few of them will be used gt dictionaries
  gt symbol tables gt memory allocators gt object
  and code sharing mechanisms

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Kolmogorov-Chaitin algorithmic complexity

• Size of the smallest program that generates a set
  of strings
• Undecidable
• Does not depend on the programming language
• Related to compressibility random sequences are
  harder to compress than regular sequences