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 and 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><expression> (Rojas, 1)

Two types of variables/names.

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: S = λwyx.y(wyx) & 1 =

λsz.s(z)Solve for: S1

λ Calculus Example of Counting

S1 = (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

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 and 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))(λx.y(xx)))

YR = (λx.R(xx))(λx.R(xx))

YR = R((λx.R(xx))(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 non-termination.”(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 Ql. 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?