Register Machines

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Register Machines

• An Alternative Analysis of Sequential Computation

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Example 5.1
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Some Pseudocode
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Example 5.1.2 (Successor Function)
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Example 5.1.3 (Binary Addition Function)
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Other Examples of Function Computers

• Example 5.1.4 (Binary Multiplication Function)
• Example 5.1.5 (Unary Factorial Function)

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Example 5.1.6 (Ackermanns Function)

• H(0, m) m 1
• H(n 1, 0) H(n, 1)
• H(n 1, m 1) H(n, H(n 1, m))

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M computes k-ary f

• Ms registers R1, R2, , Rk contain arguments of
  f
• Ultimately, register Rk 1 contains f(n1, n2, ,
  nk)
• if f(n1, n2, , nk) is undefined, then Ms
  computation never terminates

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Formal Definition

• ? R1, R2, , Rm, possibly infinite set of
  registers
• ? ??1, ?2, , ?t? with t?2 a finite sequence of
  instructions
• Start instruction
• Halt instruction

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Instruction Types

• Ri 0 or Ri 0
• Ri or Ri
• Ri Rj? goto L or Ri Rj? goto L
• Ri Rj? goto L or Ri Rj? goto L or just
  goto L

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Register-Machine-Computability

• f is said to be register-machine-computable if
  there exists register machine that computes f
• Ackermanns function etc. is register-machine-comp
  utable

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An Equivalence Result

• Let h be Turing-computable number-theoretic
  function. Then h is register-machine-computable.
• If f is register-machine-computable, then f is
  Turing-computable.

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Summary

• Turing-computable function
• Markov-computable function
• register-machine-computable function
• The above notions are mutually equivalent

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Language Acceptance

• Input tape and output tape
• 1s represent as and 2s represent bs
• Initially, input word w on tape followed by
  terminating 0
• If w ? L, then 1 written to output tape
• Same Number of as as bs
• Register-machine-acceptable language

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Language Recognition

• If w ? L, then 0 written to output tape
• Palindromes (Example 5.3.2)

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Time Analysis

• Uniform Cost Assumption Size of register
  contents make no difference
• timeM(n)
• Palindromes computes in O(n2) steps

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Space Analysis

• spaceM(n) perhaps maximum of the base-2 log of
  register contents over the course of computation

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Edmonds Thesis

• Problem is computationally feasible provided it
  has polynomial-time solution.
• Language-acceptance problem is computationally
  feasible provided language is accepted in
  polynomially bounded time.
• Example Is w a palindrome?
• Computationally feasible

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Problems with the Thesis

• Computable by what?
• Which model of computation is the favored one?
• Does it make a difference?

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Polynomial Relatedness

• Suppose that multitape MTM accepts language L in
  O(f(n)) steps. Then there exists MRM that
  accepts L in time O(f(n)2) under the uniform
  cost assumption.
• Suppose that MRM accepts language L in O(g(n))
  steps under the assumption of uniform cost. Then
  there exists multitape MTM that accepts L in time
  O(g(n)3).

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Results Regarding Polynomial Relatedness of Models
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Conclusion

• In invoking Edmonds Thesis, it seems to make no
  difference which model we take Thesis to be
  talking about.