Churchs Thesis All Computers Are Created Equal By
Church's Thesis All Computers Are Created Equal By: Patrick Goergen COT 4810 Date: 2/12/08
Outline Snapshot of Time Period Introduction to Church's Thesis Lambda (λ) Calculus & Examples General Recursive Functions and Turing Machines 3 in 1 w/ Recursion Proof & Example Halting Problem Turing Example from Text
Snapshot of Time Period Major Names of the Era Herbrand Gödel - Recursion Alonzo Church – λ Calculus Alan Turing – Turing Machine Stephen Cole Kleene – Equivalence
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 (λ) Calculus was his way of explaining Church's thought was: ''Anything that might fairly be called effectively calculable could be embodied in λ calculus. ''(Dewdney, 434)
λ Calculus ”λ calculus is a procedure for defining functions in terms of λ expressions” (Dewdney, 435) “The smallest universal programming lang. of the world. ” (Rojas, 1)
Rules of λ Calculus Productions of λ calculus: <expression> : = <name> | <function> | <application> <function> : = λ <name>. <expression> <application> : = <expression> Two types of variables/names. (Rojas, 1)
λ Calculus Expression Example of λ Expression λx. x -> where x is a <name> Importance? Applied Example (λx. x)y = [y/x]x = y λs. s = λsz. s(z)
λ Calculus Expression Successor Function S = λwyx. y(wyx) Counting 1 = λsz. s(z) 2 = λsz. s(s(z)) 3 = λsz. s(s(s(z))) (Rojas, 1)
λ Calculus Example Question: Given: Solve for: S = λwyx. y(wyx) & 1 = λsz. s(z) S 1
λ Calculus Example of Counting S 1 = (λwyx. y(wyx))(λsz. s(z)) = λyx. y((λsz. s(z))yx) = λyx. y(y(x)) =2 (Rojas, 1)
General Recursive Functions And Turing Machines Alan Turing Recursive Functions Herbrand & Gödel (Dewdney, 208)
3 in 1 (Dewdney, 435)
3 in 1 cont. . . Lambda
Church's Thoughts Church showed that his own ''λ definable functions yielded the same functions as the recursive functions of Herbrand Godel'' (Turner, 518 -519) This was almost immediately proven by Kleene. Generality of the expression.
A Proof that λ Calculus ≡ Recursion Recursive Function defined in λ Calculus: Y = (λy. (λx. y(xx))) YR = (λx. R(xx)) YR = R((λx. R(xx)))) meaning that YR = R(YR) (Rojas, 5)
Halting Problem “The Halting Theorem tells us that unboundedness of the kind needed for computational completeness is effectively inseparable from the possibility of nontermination. ” (Turner, 520)
Example Since we know that Church's λ 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
Example cont. . . (Dewdney, 437)
Program 1 of 12 (Dewdney, 440)
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.
Homework 1) What two other concepts are equivalent to Church's λ Calculus? 2) Who actually proved that λ Calculus was equivalent to a Turing Machine?
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