[PPT] - Overview Motivation Real Algebraic Numbers Well-Definedness PowerPoint Presentation

SLIDE 1

Algebraic Numbers in Isabelle/HOL1

Ren´ e Thiemann and Akihisa Yamada

Institute of Computer Science University of Innsbruck

ITP 2016, August 23, 2016

1Supported by the Austrian Science Fund (FWF) project Y757

SLIDE 2

Overview

Motivation Real Algebraic Numbers Well-Definedness Calculating Real Roots of Rational Polynomial Factorizing Rational Polynomial Arithmetic on Real Algebraic Numbers Complex Algebraic Numbers

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 2/22

SLIDE 3

Motivation

Certify Complexity of Matrix Interpretations

given: automatically generated complexity proof for program

χA = (x − 1) · (−39 + 360x − 832x2 + 512x3)

criterions
polynomial complexity if norms of all complex roots of χA 1
degree d in O(nd): more calculations with complex roots of χA

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 3/22

SLIDE 4

Motivation

Certify Complexity of Matrix Interpretations

given: automatically generated complexity proof for program

χA = (x − 1) · (−39 + 360x − 832x2 + 512x3)

criterions
polynomial complexity if norms of all complex roots of χA 1
degree d in O(nd): more calculations with complex roots of χA
problem: certifier crashed as numbers got too complicated

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 3/22

SLIDE 5

Motivation

Closed Form for Cubic Polynomials

−39 + 360x − 832x2 + 512x3 = 0 iff
x =

1 24

13 +

34

3

√

91+9i √ 383 +

3

91 + 9i

√ 383

,
x = 13

24 − 17(1+i √ 3) 24

3

√

91+9i √ 383 − 1 48

1 − i

√ 3)

3
91 + 9i

√ 383, or

x = 13

24 − 17(1−i √ 3) 24

3

√

91+9i √ 383 − 1 48

1 + i

√ 3)

3
91 + 9i

√ 383

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 4/22

SLIDE 6

Motivation

Closed Form for Cubic Polynomials

−39 + 360x − 832x2 + 512x3 = 0 iff
x1 =

1 24

13 +

34

3

√

91+9i √ 383 +

3

91 + 9i

√ 383

,
x2 = 13

24 − 17(1+i √ 3) 24

3

√

91+9i √ 383 − 1 48

1 − i

√ 3)

3
91 + 9i

√ 383, or

x3 = 13

24 − 17(1−i √ 3) 24

3

√

91+9i √ 383 − 1 48

1 + i

√ 3)

3
91 + 9i

√ 383

problem: calculate and decide

norm(xj) =

Re(xj)2 + Im(xj)2 1

for all j ∈ {1, 2, 3}

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 4/22

SLIDE 7

Motivation

Closed Form for Cubic Polynomials

−39 + 360x − 832x2 + 512x3 = 0 iff
x1 =

1 24

13 +

34

3

√

91+9i √ 383 +

3

91 + 9i

√ 383

,
x2 = 13

24 − 17(1+i √ 3) 24

3

√

91+9i √ 383 − 1 48

1 − i

√ 3)

3
91 + 9i

√ 383, or

x3 = 13

24 − 17(1−i √ 3) 24

3

√

91+9i √ 383 − 1 48

1 + i

√ 3)

3
91 + 9i

√ 383

problem: calculate and decide

norm(xj) =

Re(xj)2 + Im(xj)2 1

for all j ∈ {1, 2, 3}

problem: no closed form for roots of polynomials of degree 5

and higher

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 4/22

SLIDE 8

Motivation

Algebraic Numbers

number x ∈ R ∪ C is algebraic iff it is root of non-zero

rational polynomial

x1 = “root #1 of −39 + 360x − 832x2 + 512x3”
x2 = “root #2 of −39 + 360x − 832x2 + 512x3”
x3 = “root #3 of −39 + 360x − 832x2 + 512x3”

Figure: −39 + 360x − 832x2 + 512x3

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 5/22

SLIDE 9

Motivation

Problems with Algebraic Numbers

well-definedness

is there a “root #3 of −23 + x − 5x2 + x3”

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 6/22

SLIDE 10

Motivation

Problems with Algebraic Numbers

well-definedness

is there a “root #3 of −23 + x − 5x2 + x3”

representation

is there a simpler representation of “root #3 of −108 − 72x + 108x2 + 84x3 − 27x4 − 32x5 − 2x6 + 4x7 + x8”

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 6/22

SLIDE 11

Motivation

Problems with Algebraic Numbers

well-definedness

is there a “root #3 of −23 + x − 5x2 + x3”

representation

is there a simpler representation of “root #3 of −108 − 72x + 108x2 + 84x3 − 27x4 − 32x5 − 2x6 + 4x7 + x8”

comparisons

is “root #3 of −39 + 360x − 832x2 + 512x3” smaller than 1

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 6/22

SLIDE 12

Motivation

Problems with Algebraic Numbers

well-definedness

is there a “root #3 of −23 + x − 5x2 + x3”

representation

is there a simpler representation of “root #3 of −108 − 72x + 108x2 + 84x3 − 27x4 − 32x5 − 2x6 + 4x7 + x8”

comparisons

is “root #3 of −39 + 360x − 832x2 + 512x3” smaller than 1

arithmetic

calculate a polynomial representing “root #2 of −2 + x2” + “root #1 of −3 + x2”

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 6/22

SLIDE 13

Motivation

Problems with Algebraic Numbers

well-definedness

Sturm’s method is there a “root #3 of −23 + x − 5x2 + x3”

representation

factorization of rational polynomials is there a simpler representation of “root #3 of −108 − 72x + 108x2 + 84x3 − 27x4 − 32x5 − 2x6 + 4x7 + x8”

comparisons

Sturm’s method is “root #3 of −39 + 360x − 832x2 + 512x3” smaller than 1

arithmetic

matrices, determinants, resultants, . . . calculate a polynomial representing “root #2 of −2 + x2” + “root #1 of −3 + x2”

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 6/22

SLIDE 14

Motivation

Main Result: Formalization of Algebraic Numbers

common properties on algebraic numbers (R and C)
executable real algebraic numbers
executable complex algebraic numbers indirectly
easy to use via data-refinement for R and C

⌊norm(

3

√ 2 + 3 + 2i) · 100⌋

evaluate

֒ → 216

applicable inside (eval) and outside Isabelle (export-code)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 7/22

SLIDE 15

Motivation

Related work

Cyril Cohen (ITP 2012)
Coq
similar, but partly based on different paper proofs
our work: more focus on efficient execution

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 8/22

SLIDE 16

Motivation

Related work

Cyril Cohen (ITP 2012)
Coq
similar, but partly based on different paper proofs
our work: more focus on efficient execution
Wenda Li and Larry Paulson (CPP 2016)
independant Isabelle/HOL formalization, different approach
oracle (MetiTarski) performs computations
certified code validates results

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 8/22

SLIDE 17

Motivation

Related work

Cyril Cohen (ITP 2012)
Coq
similar, but partly based on different paper proofs
our work: more focus on efficient execution
Wenda Li and Larry Paulson (CPP 2016)
independant Isabelle/HOL formalization, different approach
oracle (MetiTarski) performs computations
certified code validates results
experimental comparison (examples of Li and Paulson)

MetiTarski 1.83 seconds (@ 2.66 Ghz) + validation of Li and Paulson 4.16 seconds (@ 2.66 Ghz) Our generated Haskell code 0.03 seconds (@ 3.5 Ghz)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 8/22

SLIDE 18

Real Algebraic Numbers Well-Definedness

Well-Definedness

“root #3 of −23 + x − 5x2 + x3”
“root #3 of −25 + 155x − 304x2 + 192x3”

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 9/22

SLIDE 19

Real Algebraic Numbers Well-Definedness

Sturm’s Method

input
polynomial over R
interval ( [2, 5], (−π, 7], (−∞, 3), or (−∞, +∞) )
output: count-roots p itval
number of distinct real roots of p in interval itval

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 10/22

SLIDE 20

Real Algebraic Numbers Well-Definedness

Sturm’s Method

input
polynomial over R
interval ( [2, 5], (−π, 7], (−∞, 3), or (−∞, +∞) )
output: count-roots p itval
number of distinct real roots of p in interval itval
formalized in Isabelle by Manuel Eberl
nearly used as black-box

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 10/22

SLIDE 21

Real Algebraic Numbers Well-Definedness

Sturm’s Method

input
polynomial over R
interval ( [2, 5], (−π, 7], (−∞, 3), or (−∞, +∞) )
output: count-roots p itval
number of distinct real roots of p in interval itval
formalized in Isabelle by Manuel Eberl
nearly used as black-box
adapted functions to work over Q

number of distinct real roots of polynomial over Q in interval over Q reason: apply Sturm’s method to implement R formalization: locale for homomorphisms

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 10/22

SLIDE 22

Real Algebraic Numbers Well-Definedness

Sturm’s Method

input
polynomial over R
interval ( [2, 5], (−π, 7], (−∞, 3), or (−∞, +∞) )
output: count-roots p itval
number of distinct real roots of p in interval itval
formalized in Isabelle by Manuel Eberl
nearly used as black-box
adapted functions to work over Q

number of distinct real roots of polynomial over Q in interval over Q reason: apply Sturm’s method to implement R formalization: locale for homomorphisms

precompute first phase of Sturm’s method

⇒ query many intervals for same polynomial more efficiently

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 10/22

SLIDE 23

Real Algebraic Numbers Well-Definedness

Applying Sturm’s Method for Well-Definedness

“root #3 of −25 + 155x − 304x2 + 192x3” count-roots (−25 + 155x − 304x2 + 192x3) (−∞, +∞) = 2

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 11/22

SLIDE 24

Real Algebraic Numbers Well-Definedness

Representation of Real Algebraic Numbers – Part 1

quadruple
polynomial over Q: p
left and right interval bound ∈ Q: l and r
precomputation of Sturm for p: root-info
(flag for factorization information on p)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 12/22

SLIDE 25

Real Algebraic Numbers Well-Definedness

Representation of Real Algebraic Numbers – Part 1

quadruple
polynomial over Q: p
left and right interval bound ∈ Q: l and r
precomputation of Sturm for p: root-info
(flag for factorization information on p)
representing real number

THE x ∈ [l, r]. rpoly p x = 0

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 12/22

SLIDE 26

Real Algebraic Numbers Well-Definedness

Representation of Real Algebraic Numbers – Part 1

quadruple
polynomial over Q: p
left and right interval bound ∈ Q: l and r
precomputation of Sturm for p: root-info
(flag for factorization information on p)
representing real number

THE x ∈ [l, r]. rpoly p x = 0

five invariants
p = 0
root-info = count-roots p
root-info [l, r] = 1
. . .

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 12/22

SLIDE 27

Real Algebraic Numbers Well-Definedness

Representation of Real Algebraic Numbers – Part 1

quadruple
polynomial over Q: p
left and right interval bound ∈ Q: l and r
precomputation of Sturm for p: root-info
(flag for factorization information on p)
representing real number

THE x ∈ [l, r]. rpoly p x = 0

five invariants
p = 0
root-info = count-roots p
root-info [l, r] = 1
. . .
invariants are ensured via a subtype (typedef)
new type of quadruples satisfying invariants
lift-definition and transfer for function definitions and proofs

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 12/22

SLIDE 28

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Real Roots of Rational Polynomial

input: non-zero polynomial over Q

utput: list of real algebraic numbers representing the roots

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 13/22

SLIDE 29

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Real Roots of Rational Polynomial

input: non-zero polynomial over Q

utput: list of real algebraic numbers representing the roots
1. factorize input polynomial over Q into pm1

1

· . . . · pmn

n

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 13/22

SLIDE 30

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Real Roots of Rational Polynomial

input: non-zero polynomial over Q

utput: list of real algebraic numbers representing the roots
1. factorize input polynomial over Q into pm1

1

· . . . · pmn

n

2. apply the following steps on each pi

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 13/22

SLIDE 31

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Real Roots of Rational Polynomial

input: non-zero polynomial over Q

utput: list of real algebraic numbers representing the roots
1. factorize input polynomial over Q into pm1

1

· . . . · pmn

n

2. apply the following steps on each pi
3. determine root-infoi for pi and initial bounds li, ri such that

root-infoi (−∞, +∞) = root-infoi [li, ri]

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 13/22

SLIDE 32

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Real Roots of Rational Polynomial

input: non-zero polynomial over Q

utput: list of real algebraic numbers representing the roots
1. factorize input polynomial over Q into pm1

1

· . . . · pmn

n

2. apply the following steps on each pi
3. determine root-infoi for pi and initial bounds li, ri such that

root-infoi (−∞, +∞) = root-infoi [li, ri]

4. perform bisection on [li, ri] to obtain intervals which all

contain a single root of pi

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 13/22

SLIDE 33

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

input: χA
computation:
39 − 399x + 1192x2 − 1344x3 + 512x4
output:

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 34

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

factorization
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
output:

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 35

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

factorization
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 36

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

initial bounds: 6 = degree pi · max{⌈|ci|⌉. ci coefficient of pi}
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[−6, 6]}
output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 37

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[−6, 0], [0, 6]}
output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 38

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[0, 6]}
output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 39

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[0, 3], [3, 6]}
output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 40

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[0, 3

2], [ 3 2, 3], [3, 6]}

output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 41

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[0, 3

4], [ 3 4, 3 2], [ 3 2, 3], [3, 6]}

output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 42

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[0, 3

8], [ 3 8, 3 4], [ 3 4, 3 2], [ 3 2, 3], [3, 6]}

output: (x − 1, [1, 1])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 43

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[ 3

8, 3 4], [ 3 4, 3 2], [ 3 2, 3], [3, 6]}

output: (x − 1, [1, 1]) , (pi, [0, 3

8])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 44

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[ 3

4, 3 2], [ 3 2, 3], [3, 6]}

output: (x − 1, [1, 1]) , (pi, [0, 3

8]) , (pi, [ 3 8, 3 4])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 45

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[ 3

2, 3], [3, 6]}

output: (x − 1, [1, 1]) , (pi, [0, 3

8]) , (pi, [ 3 8, 3 4]) , (pi, [ 3 4, 3 2])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 46

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {[3, 6]}
output: (x − 1, [1, 1]) , (pi, [0, 3

8]) , (pi, [ 3 8, 3 4]) , (pi, [ 3 4, 3 2])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 47

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Example

bisection
computation:
(x − 1) · (−39 + 360x − 832x2 + 512x3)
todo = {}
output: (x − 1, [1, 1]) , (pi, [0, 3

8]) , (pi, [ 3 8, 3 4]) , (pi, [ 3 4, 3 2])

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 14/22

SLIDE 48

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Bisection Algorithm

bisection takes root-info as parameter for efficiency
does not terminate (e.g., pass λx.2)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 15/22

SLIDE 49

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Bisection Algorithm

bisection takes root-info as parameter for efficiency
does not terminate (e.g., pass λx.2)
definition as partial-function to obtain code-equations

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 15/22

SLIDE 50

Real Algebraic Numbers Calculating Real Roots of Rational Polynomial

Bisection Algorithm

bisection takes root-info as parameter for efficiency
does not terminate (e.g., pass λx.2)
definition as partial-function to obtain code-equations
soundness via well-founded induction;
rder based on minimal distance between roots of pi

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 15/22

SLIDE 51

Real Algebraic Numbers Factorizing Rational Polynomial

Factorization of Rational Polynomials

important for efficiency: keep polynomials small

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 16/22

SLIDE 52

Real Algebraic Numbers Factorizing Rational Polynomial

Factorization of Rational Polynomials

important for efficiency: keep polynomials small
time/degree of representing polynomials for n

i=1

√ i factorization n = 8 n = 9 n = 10 none 2m11s/256 22m19s/512 12h19m/1024 square-free 2m14s/256 15m31s/384 9h31m/768 full 0.35s/16 0.35s/16 0.59s/16

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 16/22

SLIDE 53

Real Algebraic Numbers Factorizing Rational Polynomial

Factorization of Rational Polynomials

important for efficiency: keep polynomials small
time/degree of representing polynomials for n

i=1

√ i factorization n = 8 n = 9 n = 10 none 2m11s/256 22m19s/512 12h19m/1024 square-free 2m14s/256 15m31s/384 9h31m/768 full 0.35s/16 0.35s/16 0.59s/16

algorithm

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 16/22

SLIDE 54

Real Algebraic Numbers Factorizing Rational Polynomial

Factorization of Rational Polynomials

important for efficiency: keep polynomials small
time/degree of representing polynomials for n

i=1

√ i factorization n = 8 n = 9 n = 10 none 2m11s/256 22m19s/512 12h19m/1024 square-free 2m14s/256 15m31s/384 9h31m/768 full 0.35s/16 0.35s/16 0.59s/16

algorithm
apply certified square-free factorization

− 108 − 72x + 108x2 + 84x3 − 27x4 − 32x5 − 2x6 + 4x7 + x8 = ( − 3 + x2)2 · ( − 12 − 8x + 4x2 + 4x3 + x4)1

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 16/22

SLIDE 55

Real Algebraic Numbers Factorizing Rational Polynomial

Factorization of Rational Polynomials

important for efficiency: keep polynomials small
time/degree of representing polynomials for n

i=1

√ i factorization n = 8 n = 9 n = 10 none 2m11s/256 22m19s/512 12h19m/1024 square-free 2m14s/256 15m31s/384 9h31m/768 full 0.35s/16 0.35s/16 0.59s/16

algorithm
apply certified square-free factorization

− 108 − 72x + 108x2 + 84x3 − 27x4 − 32x5 − 2x6 + 4x7 + x8 = ( − 3 + x2)2 · ( − 12 − 8x + 4x2 + 4x3 + x4)1

invoke oracle on each factor

−12 − 8x + 4x2 + 4x3 + x4 oracle = (−2 + x2) · (6 + 4x + x2)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 16/22

SLIDE 56

Real Algebraic Numbers Factorizing Rational Polynomial

Factorization of Rational Polynomials

important for efficiency: keep polynomials small
time/degree of representing polynomials for n

i=1

√ i factorization n = 8 n = 9 n = 10 none 2m11s/256 22m19s/512 12h19m/1024 square-free 2m14s/256 15m31s/384 9h31m/768 full 0.35s/16 0.35s/16 0.59s/16

algorithm
apply certified square-free factorization

− 108 − 72x + 108x2 + 84x3 − 27x4 − 32x5 − 2x6 + 4x7 + x8 = ( − 3 + x2)2 · ( − 12 − 8x + 4x2 + 4x3 + x4)1

invoke oracle on each factor

−12 − 8x + 4x2 + 4x3 + x4 oracle = (−2 + x2) · (6 + 4x + x2)

check equality at runtime, not irreducibility

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 16/22

SLIDE 57

Real Algebraic Numbers Factorizing Rational Polynomial

Simplification of Representation

Consider p(x) = −108−72x +108x2 +84x3 −27x4 −32x5 −2x6 +4x7 +x8 THE root of p in [1, 1.5] = THE root of (−3 + x2) · (6 + 4x + x2) · (−2 + x2) in [1, 1.5] = THE root of −2 + x2 in [1, 1.5] (= √ 2)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 17/22

SLIDE 58

Real Algebraic Numbers Factorizing Rational Polynomial

Comparison of Real Algebraic Numbers

1. decide whether (p, [l, r]) and (q, [l′, r′]) encode same number

THE root of p in [l, r] = THE root of q in [l′, r′] ⇐ ⇒ gcd p q has real root in [l, r] ∩ [l′, r′]

2. if not, tighten bounds of both numbers via bisection until

intervals are disjoint (bisection algorithm again defined by partial-function)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 18/22

SLIDE 59

Real Algebraic Numbers Arithmetic on Real Algebraic Numbers

Arithmetic on Real Algebraic Numbers

how to perform operations like +, −, ×, /,

n

√·, etc. on algebraic numbers (p, [l, r]) and (q, [l′, r′])?

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 19/22

SLIDE 60

Real Algebraic Numbers Arithmetic on Real Algebraic Numbers

Arithmetic on Real Algebraic Numbers

how to perform operations like +, −, ×, /,

n

√·, etc. on algebraic numbers (p, [l, r]) and (q, [l′, r′])?

1. determine polynomial
negation: p(−x)
n

√·: p(xn)

addition: resultant(p(x − y), q(y))

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 19/22

SLIDE 61

Real Algebraic Numbers Arithmetic on Real Algebraic Numbers

Arithmetic on Real Algebraic Numbers

how to perform operations like +, −, ×, /,

n

√·, etc. on algebraic numbers (p, [l, r]) and (q, [l′, r′])?

1. determine polynomial
negation: p(−x)
n

√·: p(xn)

addition: resultant(p(x − y), q(y))
2. compute initial interval, e.g. [l + l′, r + r′] for addition

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 19/22

SLIDE 62

Real Algebraic Numbers Arithmetic on Real Algebraic Numbers

Arithmetic on Real Algebraic Numbers

how to perform operations like +, −, ×, /,

n

√·, etc. on algebraic numbers (p, [l, r]) and (q, [l′, r′])?

1. determine polynomial
negation: p(−x)
n

√·: p(xn)

addition: resultant(p(x − y), q(y))
2. compute initial interval, e.g. [l + l′, r + r′] for addition
3. tighten intervals [l, r] and [l′, r′] until resulting interval

contains unique root

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 19/22

SLIDE 63

Real Algebraic Numbers Arithmetic on Real Algebraic Numbers

Arithmetic on Real Algebraic Numbers

how to perform operations like +, −, ×, /,

n

√·, etc. on algebraic numbers (p, [l, r]) and (q, [l′, r′])?

1. determine polynomial
negation: p(−x)
n

√·: p(xn)

addition: resultant(p(x − y), q(y))
2. compute initial interval, e.g. [l + l′, r + r′] for addition
3. tighten intervals [l, r] and [l′, r′] until resulting interval

contains unique root

4. optimize representation

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 19/22

SLIDE 64

Real Algebraic Numbers Arithmetic on Real Algebraic Numbers

Increase Efficiency, Representation – Part 2

factorizations in between to simplify representations
efficient (not optimal) computation of resultants and GCDs
tuned algorithms on polynomials
special treatment for rational numbers

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 20/22

SLIDE 65

Real Algebraic Numbers Arithmetic on Real Algebraic Numbers

Increase Efficiency, Representation – Part 2

factorizations in between to simplify representations
efficient (not optimal) computation of resultants and GCDs
tuned algorithms on polynomials
special treatment for rational numbers

datatype real_alg_3 = Rational rat | Irrational "quadruple with invariants" typedef real_alg_4 = "real_alg_3 with invariant" definition real_of_4 :: real_alg_4 => real quotient_type real_alg = real_alg_4 / "% x y. real_of_4 x = real_of_4 y"

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 20/22

SLIDE 66

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 67

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 68

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))
only new difficulty: find complex roots, e.g., of p = 1 + x + x3

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 69

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))
only new difficulty: find complex roots, e.g., of p = 1 + x + x3
basic idea:
complex roots of rational polynomials come in complex

conjugate pairs ⇒ if x is root of p then ¯ x is root of p

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 70

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))
only new difficulty: find complex roots, e.g., of p = 1 + x + x3
basic idea:
complex roots of rational polynomials come in complex

conjugate pairs ⇒ if x is root of p then ¯ x is root of p ⇒ Re(x) = 1

2(x + ¯

x) is root of the polynomial for 1

2 ⊙ (p ⊕ p)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 71

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))
only new difficulty: find complex roots, e.g., of p = 1 + x + x3
basic idea:
complex roots of rational polynomials come in complex

conjugate pairs ⇒ if x is root of p then ¯ x is root of p ⇒ Re(x) = 1

2(x + ¯

x) is root of the polynomial for 1

2 ⊙ (p ⊕ p)

⇒ Im(x) = 1

2i (x − ¯

x) is root of the polynomial for

1 2i ⊙ (p ⊖ p)

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 72

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))
only new difficulty: find complex roots, e.g., of p = 1 + x + x3
basic idea:
complex roots of rational polynomials come in complex

conjugate pairs ⇒ if x is root of p then ¯ x is root of p ⇒ Re(x) = 1

2(x + ¯

x) is root of the polynomial for 1

2 ⊙ (p ⊕ p)

⇒ Im(x) = 1

2i (x − ¯

x) is root of the polynomial for

1 2i ⊙ (p ⊖ p)

⇒ compute all these roots

Re(x) =

SOME x. 8 − 24x − 88x3 + 96x4 + 288x5 + 384x6 + 768x7 + 512x9 = 0
Im(x) =
SOME x.

961 4096 x2 − 279 512 x4 + 453 256 x6 − 85 32 x8 + 27 8 x10 − 3x12 + x14 = 0

Thiemann and Yamada (Univ. Innsbruck)

Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 73

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))
only new difficulty: find complex roots, e.g., of p = 1 + x + x3
basic idea:
complex roots of rational polynomials come in complex

conjugate pairs ⇒ if x is root of p then ¯ x is root of p ⇒ Re(x) = 1

2(x + ¯

x) is root of the polynomial for 1

2 ⊙ (p ⊕ p)

⇒ Im(x) = 1

2i (x − ¯

x) is root of the polynomial for

1 2i ⊙ (p ⊖ p)

⇒ compute all these roots

Re(x) =

SOME x.
−1

8 + 1 4x + x3

·
1 + x + x3

= 0

Im(x) =
SOME x.
−31

64 + 9 16x2 − 3 2x4 + x6

· x = 0
Thiemann and Yamada (Univ. Innsbruck)

Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 74

Complex Algebraic Numbers

Complex Algebraic Numbers in Isabelle

missing: equivalent of Sturm’s method for C
take existing Cartesian representation: (Re(x), Im(x))
only new difficulty: find complex roots, e.g., of p = 1 + x + x3
basic idea:
complex roots of rational polynomials come in complex

conjugate pairs ⇒ if x is root of p then ¯ x is root of p ⇒ Re(x) = 1

2(x + ¯

x) is root of the polynomial for 1

2 ⊙ (p ⊕ p)

⇒ Im(x) = 1

2i (x − ¯

x) is root of the polynomial for

1 2i ⊙ (p ⊖ p)

⇒ compute all these roots

Re(x) =

SOME x.
−1

8 + 1 4x + x3

·
1 + x + x3

= 0

Im(x) =
SOME x.
−31

64 + 9 16x2 − 3 2x4 + x6

· x = 0
and filter: for all candidates z = Re(x) + Im(x)i, test p(z) = 0

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 21/22

SLIDE 75

Complex Algebraic Numbers

Summary

formalization of real and complex algebraic numbers,

implementing R and C +, −, ×, /,

n

√·, ⌊·⌋, =, <, i, Re, Im, show

factorization algorithms for rational polynomials over Q, R, C
heavily relying on Sturm’s method and matrix library
based on factorization oracle (S. Joosten @ Isabelle workshop)
∼ 20.000 loc, available in archive of formal proofs

Thiemann and Yamada (Univ. Innsbruck) Algebraic Numbers in Isabelle/HOL 22/22