Computation, Computers, and Programs Recursive functions

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CS20a Models (Nov 5, 2002)

• Turing Machines
• Churchs thesis TM are a general model of
  computation
• But (almost) everything is undecidable
• Today alternative models
• Primitive recursive functions
• Total recursive functions
• Partial recursive functions

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Primitive recursion
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Primitive recursion (conventional form)
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Arithmetic
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Exponentiation
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Hyper-exponentation
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Ackermans function
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Subtraction
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Equality
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For-programs
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For-loop for y do p done
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Interpretation of for-programs

• Some variables variables are designated as input
• Some are designated as output
• The program defines a function Ninput ? Noutput

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For programs and primitive recursion
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Encoding p.r. functions as for-loops
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Encoding for-loops as p.r. functions
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Partial-recursive functions
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Properties of p.r. programs

• The programs always terminate
• However, does not include all recursive
  computations
• Can code extremely large functions, but some
  functions are not definable (Ackermans)
• Well argue later that all tractable computations
  are primitive recursive

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Partial recursive functions (Godel)

• To capture r.e. computations, we need more
• A partial recursive computation includes the
  prim-rec computations, plus unbounded minimization

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Unbounded minimization
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While-programs
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Enumerating the partial-recursive funs
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Indexing partial recursive funs
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Universal functions
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Godel numbering
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Composition
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Constant functions
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Recursion theorem
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Recursion theorem proof (cont.)
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Rices theorem