Introduction Developing TTICAD TTICAD in Practice Conclusions etc. Cylindrical Algebraic Decomposition: from Polynomials to Formulae James Davenport: The University of Bath Joint work with: Bath Russell Bradford, Matthew England and David Wilson CIGIT Changbo Chen Macquarie Scott McCallum Western Ontario Marc Moreno Maza Rikkyo University: 31 July 2014 James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD TTICAD in Practice Conclusions etc. Outline Introduction 1 Cylindrical Algebraic Decomposition CAD for Boolean Combinations Developing TTICAD 2 Motivation New Projection Operator Important Technicalities TTICAD in Practice 3 Implementation in Maple Experimental Results Beyond equational constraints Regular Chains CAD Conclusions etc. 4 Conclusions Bibliography James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Cylindrical algebraic decomposition A Cylindrical Algebraic Decomposition (CAD) is a partition of R n into cells arranged cylindrically (meaning their projections are either equal or disjoint) such that each cell is defined by a semi-algebraic set. Defined by Collins who gave an algorithm to produce a sign-invariant CAD for a set of polynomials, meaning each polynomial had constant sign on each cell. In some sense, makes the induced geometry of R n explicit Originally motivated for use in quantifier elimination. Have also been applied directly on problems as diverse as algebraic simplification and (at least theoretically) robot motion planning. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Projection and lifting Collins algorithm has two main phases: Projection A projection operator is applied repeatedly to the polynomials, each time producing a new set of polynomials in one less variable. Lifting • A CAD of R is produced using the roots of the univariate polynomials and intervals between. • Over each cell: the bivariate polynomials are evaluated at a sample point, a stack is built consisting of sections (the roots) and sectors (the intervals). Together these are a CAD of R 2 . . . . • • Repeated until a CAD of R n is constructed. The projection operator is defined so the CAD is sign-invariant. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Projection example The projection operator applied to the sphere identifies the circle. The projection operator applied to the circle identifies two points on the real line. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Projection and lifting Collins algorithm has two main phases: Projection A projection operator is applied repeatedly to the polynomials, each time producing a new set of polynomials in one less variable. Lifting A CAD of R is produced using the roots of the univariate polynomials and intervals between. Over each cell: the bivariate polynomials are evaluated at a sample point, a stack is built consisting of sections (the roots) and sectors (the intervals). Together these are a CAD of R 2 . . . . Repeated until a CAD of R n is constructed. The projection operator is defined so the CAD is sign-invariant. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Lifting example A CAD of R 2 which is sign-invariant with respect to the circle. Each black dot represents a cell. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Lifting example A CAD of R 2 which is sign-invariant with respect to the circle. Each black dot represents a cell. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Lifting example A CAD of R 2 which is sign-invariant with respect to the circle. Each black dot represents a cell. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Lifting example A CAD of R 2 which is sign-invariant with respect to the circle. Each black dot represents a cell. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Lifting example A CAD of R 2 which is sign-invariant with respect to the circle. Each black dot represents a cell. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Lifting example A CAD of R 2 which is sign-invariant with respect to the circle. Each black dot represents a cell. James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Improvements to CAD There have been many improvements and extensions to CAD theory including but not limited to: Improvements to the sub-algorithms used by Collins. New projection operators. Results on complexity of CAD. CAD tailored to specific problems (notably Virtual Term Substitution). Results and algorithms on the adjacency of CAD cells. CAD via triangular decomposition (see later). James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. So how do we project? (Lifting is in fact relatively straight-forward) Given polynomials P n = { p i } in x 1 , . . . , x n , what should P n − 1 be? Naïve (Doesn’t work!) Every disc x n ( p i ) , every res x n ( p i , p j ) i.e. where the polynomials fold, or cross: misses lots of “special cases” [Col75] First enlarge P n with all its reducta, then naïve plus the coefficients of P n (with respect to x n ) the principal subresultant coefficients from the disc x n and res x n calculations [Hon90] a tidied version of [Col75]. [McC88] Let B n be a squarefree basis for the primitive parts of P n . Then P n − 1 is the contents of P n , the coefficients of B n and every disc x n ( b i ) , res x n ( b i , b j ) from B n [Bro01] Naïve plus leading coefficients (not squarefree!) James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Are these projections correct? [Col75] Yes, and it’s relatively straightforward to prove that, over a cell in R n − 1 sign-invariant for P n − 1 , the polynomials of P n do not cross, and define cells sign-invariant for the polynomials of P n [McC88] 52 pages (based on [Zar75]) prove the equivalent statement, but for order-invariance, not sign-invariance, provided the polynomials are well-oriented, a test that has to be applied during lifting. But what if they’re not known to be well-oriented? [McC88] suggests adding all partial derivatives In practice hope for well-oriented, and if it fails use Hong’s projection. [Bro01] Needs well-orientedness and additional checks James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. What about the complexity? If the McCallum projection is well-oriented, the complexity is ( 2 d ) n 2 n + 7 m 2 n + 4 l 3 (1) versus the original ( 2 d ) 2 2 n + 8 m 2 n + 6 l 3 ( ?? ) and in practice the gains in running time can be factors of a thousand, or, more often, the difference between feasibility and infeasibility “Randomly”, well-orientedness ought to occur with probability 1, but we have a family of “real-world” examples (simplification/ branch cuts) where it often fails James Davenport CAD: Polynomials ⇒ Formulae
Introduction Developing TTICAD Cylindrical Algebraic Decomposition TTICAD in Practice CAD for Boolean Combinations Conclusions etc. Need it be this hard? The Heintz construction Φ k ( x k , y k ) := � � y k − 1 = y k ∧ x k − 1 = z k ∨ y k − 1 = z k ∧ x k − 1 = x k ∃ z k ∀ x k − 1 y k − 1 ⇒ Φ k − 1 ( x k − 1 , y k − 1 ) If Φ 1 ≡ y 1 = f ( x 1 ) , then Φ 2 ≡ y 2 = f ( f ( x 2 )) , Φ 3 ≡ y 3 = f ( f ( f ( f ( x 3 )))) � 2 2 ( n − 2 ) / 5 � (using y R + iy I = ( x R + ix I ) 4 ) [DH88] shows Ω � 2 2 ( n − 1 ) / 3 � [BD07] shows Ω (using a sawtooth) Hence doubly exponential is inevitable, but there’s a lot of room! In fact, there are theoretical algorithms which are singly-exponential in n , but doubly-exponential in the number of ∃∀ alternations James Davenport CAD: Polynomials ⇒ Formulae
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