Church's Thesis All Computers Are Created Equal

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Church's Thesis All Computers Are Created Equal

• By Patrick Goergen
• COT 4810
• Date 2/12/08

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Outline

• Snapshot of Time Period
• Introduction to Church's Thesis
• Lambda (?) Calculus Examples
• General Recursive Functions and Turing Machines
• 3 in 1 w/ Recursion Proof Example
• Halting Problem
• Turing Example from Text

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Snapshot of Time Period

• Major Names of the Era
• Herbrand and Gödel - Recursion
• Alonzo Church ? Calculus
• Alan Turing Turing Machine
• Stephen Cole Kleene Equivalence

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Church's Thesis Introduction

• What does it mean to 'compute'?
• any process or procedure carried out stepwise by
  well defined rules (Dewdney, 434)?
• Church's Answer ''effective calculability''
• Lambda (?) Calculus was his way of explaining
• Church's thought was
• ''Anything that might fairly be called
  effectively calculable could be embodied in ?
  calculus.''(Dewdney, 434)?

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? Calculus

• ? calculus is a procedure for defining
  functions in terms of ? expressions(Dewdney,
  435)
• The smallest universal programming lang. of the
  world.(Rojas, 1)

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Rules of ? Calculus

• Productions of ? calculus
• ltexpressiongt ltnamegt ltfunctiongt
  ltapplicationgt
• ltfunctiongt ? ltnamegt.ltexpressiongt
• ltapplicationgt ltexpressiongtltexpressiongt
  (Rojas, 1)
• Two types of variables/names.

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? Calculus Expression

• Example of ? Expression
• ?x.x -gt where x is a ltnamegt
• Importance?
• Applied Example
• (?x.x)y y/xx y
• ?s.s ?sz.s(z)

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? Calculus Expression

• Successor Function
• S ?wyx.y(wyx)
• Counting
• 1 ?sz.s(z)?
• 2 ?sz.s(s(z))?
• 3 ?sz.s(s(s(z))) (Rojas, 1)

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? Calculus Example

• Question
• Given S ?wyx.y(wyx) 1 ?sz.s(z)Solve
  for S1

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? Calculus Example of Counting

• S1 (?wyx.y(wyx))(?sz.s(z)) ?yx.y((?sz.s(z))y
  x) ?yx.y(y(x)) 2 (Rojas,
  1)?

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General Recursive Functions And Turing Machines

• Turing Machines
• Alan Turing
• Recursive Functions
• Herbrand Gödel

(Dewdney, 208)?
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3 in 1
(Dewdney, 435)?
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3 in 1 cont...
Lambda
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Church's Thoughts

• Church showed that his own ''? definable
  functions yielded the same functions as the
  recursive functions of Herbrand and
  Godel'' (Turner, 518-519)?
• This was almost immediately proven by Kleene.
• Generality of the expression.

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A Proof that ? Calculus Recursion

• Recursive Function defined in ? Calculus
• Y (?y.(?x.y(xx))(?x.y(xx)))?
• YR (?x.R(xx))(?x.R(xx))?
• YR R((?x.R(xx))(?x.R(xx))))?
• meaning that YR R(YR) (Rojas, 5)?

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Halting Problem

• The Halting Theorem tells us that unboundedness
  of the kind needed for computational completeness
  is effectively inseparable from the possibility
  of non-termination. (Turner, 520)?

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Example

• Since we know that Church's ? Calculus is
  equivalent to Turing's Turing Machine let us take
  a look at how ''All Computers are Created
  Equal.''
• Lets represent a RAM Machine with a Turing Machine

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Example cont...
(Dewdney, 437)?
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Program 1 of 12
(Dewdney, 440)?
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References

• Dewdney, A.K.. The New Turing Omnibus. W.H.
  Freeman and Compant,1993.
• Rojas, Ra l. A Tutorsial Introduction to the
  Lambda Calculus. FU Berlin. 1998.
• Turner, David. Church's Thesis and Functional
  Programming. Church's Thesis after 70 Years.
  Transaction Book. Piscataway, NJ. 2006.

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Homework

• 1) What two other concepts are equivalent to
  Church's ? Calculus?
• 2) Who actually proved that ? Calculus was
  equivalent to a Turing Machine?