Master's Thesis: A Formal Study of Moessner's Sieve

Master’s Thesis

A Formal Study of Moessner’s Sieve

Peter Urbak

Department of Computer ScienceAarhus University

Advisor: Olivier Danvy October 28, 2015

Master’s Thesis

Peter Urbak

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 12

I Drop every kth element of an initial sequence

I Partially sum the remaining elements into a new sequence

I Decrease k by 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 12

Choose a k ≥ 2 (here, we choose k = 4)

I Decrease k by 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 12

I Decrease k by 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 54

I Decrease k by 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 54

I Decrease k by 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 54

I Decrease k by 1

repeat if k > 1 or stop if k = 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 541 4 15 32 65 108

I Decrease k by 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 541 4 15 32 65 108

I Decrease k by 1

Moessner’s Sieve

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 541 4 15 32 65 1081 16 81

I Decrease k by 1

Master’s Thesis

Peter Urbak

Master’s Thesis

Peter Urbak

Formal Study

I Formalization using a proof assistant

I Programs with types and function

I Mathematical theorems and their proofs

Formal Study

Master’s Thesis

Peter Urbak

Master’s Thesis

Peter Urbak

My thesis

“Moessner’s theoremcan be proved using

the Coq Proof Assistantin an elementary manner.”

Peter Urbak’s MSc Thesis 1/20

My thesis

Moessner’s theorem (1952)

Given the initial sequence of positive natural numbers

1, 2, 3, . . .

and a natural number k ≥ 2,we can obtain the sequence of powers

1k , 2k , 3k , . . .

by applying Moessner’s sieve on the initial sequence.

Moessner’s theorem illustrated

Example: If k = 2, the sequence is 12, 22, 32, . . . :

1 2 3 4 5 6 7 8 9 10 11 121 4 9 16 25 36

1 2 3 4 5 6 7 8 9 10 11 121 3 7 12 19 27 37 481 8 27 64

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 541 4 15 32 65 1081 16 81

Note: This is also the sieve from the introductory slides.

1 2 3 4 5 6 7 8 9 10 11 121 3 6 11 17 24 33 43 541 4 15 32 65 1081 16 81

Note: This is also the sieve from the introductory slides.

Moessner’s Sieve, to sum up

I Powers of successive numbers are obtained

I Not a single multiplication has been done

I Cute mathematics

I Practical relevance:this multiplication-free sieve has been used in hardware

I Cute mathematics

My thesis

An elementary manner

I Gian-Carlo Rota’s two kinds of mathematicians:“problem solvers” and “theory builders”

I Mathematics (“extension”) vs. computing (“intension”)

Our approach

I A minimal toolbox for building theories:equational reasoning, induction and coinduction

I Dogme95 (keeping things deliberately simple)

Our approach

My thesis

The solution

“Introduce a dual to Moessner’s sieveand a characteristic function of Moessner’s sieve

to state and provean idealized version of Moessner’s theorem.”

The solution

Dual to Moessner’s sieve (1/5)

1 2 3 41 3 61 41

5 6 7 811 17 2415 3216

1 2 3 41 3 61 41

5 6 7 811 17 2415 3216

See the triangles?

1 2 3 41 3 61 41

5 6 7 811 17 2415 3216

1 2 3 41 3 61 41

5 6 7 811 17 2415 3216

Note: The initial sequence is the result of a previous iteration.

Example: If k = 4 + 1, the sequence is 14, 24, 34, . . . :

1 1 1 1 11 2 3 41 3 61 41

1 1 1 1 15 6 7 8

11 17 2415 3216

1 1 1 1 11 2 3 41 3 61 41

1 1 1 1 15 6 7 8

11 17 2415 3216

Note: The initial sequence is the result of a previous iteration.

1 0 0 0 0 01 1 1 1 11 2 3 41 3 61 41

0 0 0 0 0 01 1 1 1 15 6 7 8

11 17 2415 3216

1 0 0 0 0 01 1 1 1 11 2 3 41 3 61 41

0 0 0 0 0 01 1 1 1 15 6 7 8

11 17 2415 3216

Note: The initial sequence is not the result of a previous iteration.

Moessner’s theorem, generalized, in passing (2013)

Given the initial sequence of 1 followed by 0s

1, 0, 0, . . . ←− was 1, 2, 3, . . .

and a natural number k + 2, ←− was k ≥ 2we can obtain the sequence of powers

1k , 2k , 3k , . . .

Note: induction on the rank k , instead of k + 2.

Moessner’s theorem, generalized, in passing (2013)

Given the initial sequence of 1 followed by 0s

1, 0, 0, . . . ←− was 1, 2, 3, . . .

and a natural number k + 2, ←− was k ≥ 2we can obtain the sequence of powers

1k , 2k , 3k , . . .

Note: induction on the rank k , instead of k + 2.

Example: Letting k = 4 + 2 we get a sequence of powers:

1 0 0 0 0 01 1 1 1 11 2 3 41 3 61 41

0 0 0 0 0 01 1 1 1 15 6 7 8

11 17 2415 3216

1 0 0 0 0 01 1 1 1 11 2 3 41 3 61 41

0 0 0 0 0 01 1 1 1 15 6 7 8

11 17 2415 3216

Seed tuples:

I horizontal seed tuple

I vertical seed tuple

1 0 0 0 0 0

0 1 1 1 1 10 1 2 3 40 1 3 60 1 40 10

0 0 0 0 0 0

1 1 1 1 1 14 5 6 7 86 11 17 244 15 321 160

The solution

Characteristic function of Moessner’s sieve

1 0 0 0 0 0

0 1 1 1 1 10 1 2 3 40 1 3 60 1 40 10

0 0 0 0 0 0

1 1 1 1 1 14 5 6 7 86 11 17 244 15 321 160

Note: The first triangle is Pascal’s triangle

Characteristic function of Moessner’s sieve

0 0 0 0 0 0↙↗

1 1 1 1 1 10 1 2 3 40 1 3 60 1 40 10

0 0 0 0 0 0

1 1 1 1 1 14 5 6 7 86 11 17 244 15 321 160

Note: The first triangle is Pascal’s triangle, which is symmetric

The solution

Moessner’s idealized theorem (2014)

Given an initial configuration of two seed tuples of length k + 2,

(0, 0, 0, . . . , 0) and (1, 0, 0, . . . , 0),

we obtain the sequence of Moessner triangles of rank k .Their bottom-most elements enumerate

1k , 2k , 3k , . . .

Moessner’s idealized theorem generalized (1/3)

0 0 0 0 0 0

1 1 1 1 1 10 1 2 3 40 1 3 60 1 40 10

0 0 0 0 0 0

1 1 1 1 1 14 5 6 7 86 11 17 244 15 321 160

Note: we can now generalize 1 to any value c .

0 0 0 0 0 0

1 1 1 1 1 10 1 2 3 40 1 3 60 1 40 10

0 0 0 0 0 0

1 1 1 1 1 14 5 6 7 86 11 17 244 15 321 160

Note: we can now generalize 1 to any value c .

0 0 0 0 0 0

c c c c c c0 c 2c 3c 4c0 c 3c 6c0 c 4c0 c0

0 0 0 0 0 0

c c c c c c4c 5c 6c 7c 8c6c 11c 17c 24c4c 15c 32cc 16c0

Note: we can even add another value, d , to the seed tuple.

0 0 0 0 0 0

c c c c c c0 c 2c 3c 4c0 c 3c 6c0 c 4c0 c0

0 0 0 0 0 0

c c c c c c4c 5c 6c 7c 8c6c 11c 17c 24c4c 15c 32cc 16c0

Note: we can even add another value, d , to the seed tuple.

0 0 0 0 0 0

c c c c c cd c + d 2c + d 3c + d 4c+d0 c + d 3c + 2d 6c+3d0 c + d 4c+3d0 c+d0

The contributions

I A new formalization of Moessner’s sieve.

I A new characteristic function of Moessner’s sieve.

I A formal proof of Moessner’s idealized theorem.

I A new property of Moessner’s sieve that establishesa connection between Moessner triangles of different rank.

I A formal proof of Long’s idealized theorem.

I A new conjecture about Long’s idealized theoremthat establishes a connection to polynomial evaluation.

I A connection between Moessner trianglesand Horner’s method for polynomial division.

I Program-generated dependency graphs of all proofs in the thesis.

I An extensive glossary of all concepts mentioned in the thesis.

Thank you.

The contributions

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The contributions

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The contributions

Thank you.

The contributions

Thank you.

The contributions

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The contributions

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The contributions

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The contributions

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The contributions

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The contributions

Thank you.

Master's Thesis: A Formal Study of Moessner's Sieve

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