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

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

Context C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 2 / 16

Context Tarski formula ↔ semi-algebraic set   x 2 + x 1 > 0 x 2 1 + x 2 2 − 1 < 0 ∧ ∨ ↔   x 2 − x 1 < 0 C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 2 / 16

Context Tarski formula ↔ semi-algebraic set   x 2 + x 1 > 0 x 2 1 + x 2 2 − 1 < 0 ∧ ∨ ↔   x 2 − x 1 < 0 Tarski formulas provide an implicit representation of semi-alg. sets C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 2 / 16

Context Tarski formula ↔ semi-algebraic set   x 2 + x 1 > 0 x 2 1 + x 2 2 − 1 < 0 ∧ ∨ ↔   x 2 − x 1 < 0 Tarski formulas provide an implicit representation of semi-alg. sets 75 x + 15 y 2 + 16 z − 18 > 0 ∧ − 39 x − 78 yx − 91 z − 70 > 0 ∧ − 86 x − 44 y 2 + 14 z − 15 > 0 ∧ − 27 xz + 22 y − z − 74 > 0 ∧ 2 x 2 + 4 y − 13 z + 34 > 0 55 x + 87 yz + 45 z + 6 > 0 ∧ C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 2 / 16

Context Tarski formula ↔ semi-algebraic set   x 2 + x 1 > 0 x 2 1 + x 2 2 − 1 < 0 ∧ ∨ ↔   x 2 − x 1 < 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

This paper’s contributions introduces Open Non-uniform CAD, 1 provides an algorithm for constructing Open NuCADs from Tarski 2 formulas, and reports results of experiments with an initial implementaton. 3 C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 3 / 16

Outline Open CAD, 1 Open Non-uniform CAD, and 2 Experimental results. 3 C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 4 / 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

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

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

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

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

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

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

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

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

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

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

Open CAD: a definition Definition: An Open CAD is a (weak) decomposition of R n into open cylindrical algebraic cells whose arrangement is uniformly cylindrical, meaning that for any two cells c 1 , c 2 , the projections π j ( c 1 ) , π j ( c 2 ) onto R j are either identical or disjoint. C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 6 / 16

Open CAD: projection C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 7 / 16

Open CAD: projection Idea 1: A set A of polynomials in x 1 , . . . , x n (weakly) decomposes R n in a natural way — into connected regions in which ∀ p ∈ A , p � = 0 A = { x 2 1 + x 2 2 − 1 , x 1 + x 2 } − → C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 7 / 16

Open CAD: projection Idea 1: A set A of polynomials in x 1 , . . . , x n (weakly) decomposes R n in a natural way — into connected regions in which ∀ p ∈ A , p � = 0 A = { x 2 1 + x 2 2 − 1 , x 1 + x 2 } − → Idea 2: With the right set P of “lower level” polynomials, the natural decomposition defined by P ∪ A is an Open CAD. { x 1 + 1 , x 1 − 1 , 2 x 2 1 − 1 } ∪ A − → � �� � P C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 7 / 16

Open CAD: an algorithm Input: F , a Tarski formula in variables x 1 . . . , x n Output: D , an Open CAD of R n 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Example — constructing an open CAD y < 0 ∧ y + 1 2 > 0 ∧ y 2 − ( 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

Open NuCAD: a definition Definition: An Open Non-uniform CAD is a (weak) decomposition of R n into open cylindrical cells. C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 10 / 16

Open NuCAD: merge C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 11 / 16

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

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

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 x 2 + y 2 − 1 α C. W. Brown (USNA) Open Non-uniform CAD ISSAC 2015 11 / 16