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Models of Computation

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!!!. Models of Computation.

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Models of Computation

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  1. 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!!!

  2. 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 => computers – Von Neuman architecture 2

  3. A simple Machine Language • MEM[I] – memory • ACC • STORE I • LOAD I • LOADC X • ADD I • IFZ Label

  4. Computability vs. Expressiveness • Computabilty: can we compute something? • Expressiveness: how easy is to implement something we want to compute?

  5. 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

  6. 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->value) are FF • Functions about Functions => • A Turing Equivalent Model: Lambda Calculus

  7. 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 => Von Neuman computer, ‘CPU’ etc.

  8. 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. Mahaney’s theorem) • Efficient representation of sparse sets => • A solution: hashing – represent large structures with small ones (small integers) knowing that only ‘a few’ of them will be used => dictionaries => symbol tables => memory allocators => object and code sharing mechanisms

  9. 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

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