Non-computable? 1/45

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The busy beaver game

A simple yet non-computable function

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A Hierarchy of Program Types

Recursively Enumerable Function

Recursive

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A Hierarchy of Program Types

A function that can be

de-recursed

i.e. that is the function can be computed by a computer

program whose loops are all "for" loops

Recursively Enumerable Function

Recursive

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A Hierarchy of Program Types

Informally, functions that you just have to define them

re-cursively!

A function for which a ‘yes’ answer can be verified by a Turing machine in a finite amount of time

Recursively Enumerable

Recursive

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A Hierarchy of Program Types

A subset of recursively enumerable (RE).

The Turing Machine will always halt in this case.

Recursive

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A Hierarchy of Program Types

Recursive

Primitive Recursive

(Non-computable)

Undecidable!

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Non-computable

• An non-computable function is a function

which can’t be computed by any algorithm.

• Equivalently, not by any

Turing machine

!

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The Busy Beaver Game

• The concept was first introduced by Tibor Radó in his 1961 paper, "On Non-Computable Functions".

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The Busy Beaver Game

•What is the maximum number of 1s that

get printed on your tape, for your

given

Turing machine

, when it

halts

?

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The Busy Beaver Game

•We shall use

binary Turing machines

;

that is, Turing machines with the binary

alphabet 0, 1.

•We shall use the term

“card”

instead of

“state”.

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Radó’s “Binary Turing Machine”

Divide a long tape into

squares, each square is a memory location

Read/write head (scanner)

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Radó’s “Binary Turing Machine”

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··· 0 0 0 0 0 0 0 0 ···

Difference Between Rado’s Turing Machine and what we’ve learned in the lecture:

• Initially an all-zero tape

• Start from the middle of the tape • Unbounded to both left and right

thus no left endmark symbol

• No blank symbol ▷

• Can’t “remain stationary”

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“Card”

C

1

0 1 0 2

1 1 1 3

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“Card”

C

1

0 1 0 2

1 1 1 3

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“Card”

C

1

0 1 0 2

1 1 1 3

Contains the alphabet 0,1

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“Card”

C

1

0 1 0 2

1 1 1 3

The “shift” column:

0 : left shift 1 : right shift

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“Card”

C

1

0 1 0 2

1 1 1 3

The “shift” column: 0 : left shift 1 : right shift Overprint by The index of next card 0: halt

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A 1-card Turing machine Example

C1

0 1 0 1 1 1 0 0

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A 1-card Turing machine Example

C1

0 1 0 1 1 1 0 0

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A 1-card Turing machine Example

C1

0 1 0 1 1 1 0 0

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A 1-card Turing machine Example

C1

0 1 0 1 1 1 0 0

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A 1-card Turing machine Example

C1

0 1 0 1 1 1 0 0

1 1 1 1 1 0 0 0 0 0

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Rules

•The contestant selects a positive number

n; and makes up his own n-card, binary

Turing machine.

•He starts his machine on the all-zero

tape, and satisfies himself that his

machine

stops

after a certain number of

shifts.

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Rules

• The BB-n champion is the contestant who

achieved the highest score (i.e. the number of 1s on the tape) in the BB-n game.

•The highest score in the BB-n game is

called

the busy beaver function (also

called Rado’s sigma function) Σ (n)

.

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How to compute Σ (n)?

The answer is –

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How to compute Σ (n)?

Overprint Shift Next card Read 0/1 n different cards

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Strategy

To prove Σ(x) is non-computable, we would like to prove that for

some x>x0, Σ(x)≠f(x) for any

computable function x.

In this proof, we will try to prove

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Notation: “-”

Let f(x), g(x) be two functions. We shall

write f(x)>-g(x) to state that f(x)>g(x) for x greater than a certain x0.

Using this notation, we shall now prove the following theorem.

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Theorem

Σ(n)>-f(n) for every computable (that is, general

recursive) function f(n) for n. Hence Σ(n) is non-computable.

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Proof

C1 0 1 1 2 1 1 1 0 C2 0 1 1 3 1 -C3 0 1 0 1 1 -0 0 0 0 0 0 0 0 0 0

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Proof

C1 0 1 1 2 1 1 1 0 C2 0 1 1 3 1 -C3 0 1 0 1 1 -0 0 0 0 1 1 1 0 0 0

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Proof

But note that C is decided by F, so n must _{be sufficiently large.}

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How big is Σ(n)?

Even though Σ(n) is non-computable, it is

entirely possible that Σ(n) can be effectively determined for particular values of n.

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How big is Σ(n)?

How many TMs? Cards Max score

64 1 1

20736 2 4

16777216 3 6

25.6 trillion 4 13

•For 5-card busy beaver game, the

maximum score

so far

is 4098.

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Summary

• The busy beaver function a famous function in

computation theory.

• We can prove its non-computability by proving that Σ(n)>f(n) for any computable function n for some

sufficiently large n.

• But we do can find Σ(n) for some n!

• Scientist are now working on 5-card and 6-card

busy beaver game.