[PPT] - Open Non-uniform Cylindrical Algebraic Decomposition Christopher W. PowerPoint Presentation

SLIDE 1

Open Non-uniform Cylindrical Algebraic Decomposition

Christopher W. Brown

Department of Computer Science

U. S. Naval Academy

ISSAC 2015

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 1 / 16

SLIDE 2

Context

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 2 / 16

SLIDE 3

Context

Tarski formula ↔ semi-algebraic set x2

1 +x2 2 −1 < 0∧

  x2 + x1 > 0 ∨ x2 − x1 < 0   ↔

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 2 / 16

SLIDE 4

Context

Tarski formula ↔ semi-algebraic set x2

1 +x2 2 −1 < 0∧

  x2 + x1 > 0 ∨ x2 − x1 < 0   ↔ Tarski formulas provide an implicit representation of semi-alg. sets

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 2 / 16

SLIDE 5

Context

Tarski formula ↔ semi-algebraic set x2

1 +x2 2 −1 < 0∧

  x2 + x1 > 0 ∨ x2 − x1 < 0   ↔ Tarski formulas provide an implicit representation of semi-alg. sets 75x + 15y2 + 16z − 18 > 0 ∧ −39x − 78yx − 91z − 70 > 0 ∧ −86x − 44y2 + 14z − 15 > 0 ∧ −27xz + 22y − z − 74 > 0 ∧ 55x + 87yz + 45z + 6 > 0 ∧ 2x2 + 4y − 13z + 34 > 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 2 / 16

SLIDE 6

Context

Tarski formula ↔ semi-algebraic set x2

1 +x2 2 −1 < 0∧

  x2 + x1 > 0 ∨ x2 − x1 < 0   ↔ Tarski formulas provide an implicit representation of semi-alg. sets CAD provides an explicit representation of semi-algebraic sets

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 2 / 16

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This paper’s contributions

1

introduces Open Non-uniform CAD,

2

provides an algorithm for constructing Open NuCADs from Tarski formulas, and

3

reports results of experiments with an initial implementaton.

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 3 / 16

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Outline

1

Open CAD,

2

Open Non-uniform CAD, and

3

Experimental results.

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 4 / 16

SLIDE 9

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 10

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 11

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 12

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 13

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 14

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 15

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 16

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 17

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 18

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 19

Open CAD: open cylindrical algebraic cell

An open cylindrical algebraic cell generalizes a box aligned with axes Explicit representation: sample point + upper & lower bound functions

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 5 / 16

SLIDE 20

Open CAD: a definition

Definition: An Open CAD is a (weak) decomposition of Rn into open cylindrical algebraic cells whose arrangement is uniformly cylindrical, meaning that for any two cells c1, c2, the projections πj(c1), πj(c2) onto Rj are either identical or disjoint.

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 6 / 16

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Open CAD: projection

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 7 / 16

SLIDE 22

Open CAD: projection

Idea 1: A set A of polynomials in x1, . . . , xn (weakly) decomposes Rn in a natural way — into connected regions in which ∀p ∈ A, p = 0 A = {x2

1 + x2 2 − 1, x1 + x2}

− →

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 7 / 16

SLIDE 23

Open CAD: projection

Idea 1: A set A of polynomials in x1, . . . , xn (weakly) decomposes Rn in a natural way — into connected regions in which ∀p ∈ A, p = 0 A = {x2

1 + x2 2 − 1, x1 + x2}

− → Idea 2: With the right set P of “lower level” polynomials, the natural decomposition defined by P ∪ A is an Open CAD. {x1 + 1, x1 − 1, 2x2

1 − 1}

P

∪A − →

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 7 / 16

SLIDE 24

Open CAD: an algorithm

Input: F, a Tarski formula in variables x1 . . . , xn Output: D, an Open CAD of Rn representing the set defined by F Step 1: A ← polynomials in F Step 2: P ← projection of A Step 3: D ← a list of the open cylindrical cells data structures that explicitly define the CAD given by the natural decomposition of P ∪ A

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 8 / 16

SLIDE 25

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 26

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 27

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 28

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 29

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 30

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 31

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 32

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 33

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 34

Example — constructing an open CAD

y < 0∧y + 1

2 > 0∧y2−(x + 1 2)(x − 1 2)2 > 0∧ 1 6(x − 1 2)2+(y + 1 2)2− 1 4 < 0

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 9 / 16

SLIDE 35

Open NuCAD: a definition

Definition: An Open Non-uniform CAD is a (weak) decomposition of Rn into open cylindrical cells.

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 10 / 16

SLIDE 36

Open NuCAD: merge

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 11 / 16

SLIDE 37

Open NuCAD: merge

Jovanovic & de Moura 2012 given point α and formula F, construct cell containing α in which F has constant truth value.

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 11 / 16

SLIDE 38

Open NuCAD: merge

Jovanovic & de Moura 2012 given point α and formula F, construct cell containing α in which F has constant truth value. ISSAC 2013 merge: given point α, cell c containing α, and polynomial p, construct cell c′ ⊆ c containing α such that p has constant non-zero sign in c′.

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 11 / 16

SLIDE 39

Open NuCAD: merge

Jovanovic & de Moura 2012 given point α and formula F, construct cell containing α in which F has constant truth value. ISSAC 2013 merge: given point α, cell c containing α, and polynomial p, construct cell c′ ⊆ c containing α such that p has constant non-zero sign in c′. Example: merge x2 + y2 − 1

α

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 11 / 16

SLIDE 40

Open NuCAD: merge

Jovanovic & de Moura 2012 given point α and formula F, construct cell containing α in which F has constant truth value. ISSAC 2013 merge: given point α, cell c containing α, and polynomial p, construct cell c′ ⊆ c containing α such that p has constant non-zero sign in c′. Example: merge x2 + y2 − 1

α α

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 11 / 16

SLIDE 41

Open NuCAD: merge

Jovanovic & de Moura 2012 given point α and formula F, construct cell containing α in which F has constant truth value. ISSAC 2013 merge: given point α, cell c containing α, and polynomial p, construct cell c′ ⊆ c containing α such that p has constant non-zero sign in c′. Example: merge x2 + y2 − 1

α α α

C. W. Brown (USNA)

Open Non-uniform CAD ISSAC 2015 11 / 16

SLIDE 42