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Problem formulation for truth-table invariant cylindrical algebraic decomposition.pdf

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a r X i v : 1 4 0 4 . 6 3 7 1 v 1 [ c s . S C ] 2 5 A p r 2 0 1 4 Problem formulation for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition Matthew England 1 , Russell Bradford 1 , Changbo Chen 2 , James H. Davenport 1 , Marc Moreno Maza 3 , and David Wilson 1 1 University of Bath, Bath, BA2 7AY, U.K. 2 Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green a
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    a  r   X   i  v  :   1   4   0   4 .   6   3   7   1  v   1   [  c  s .   S   C   ]   2   5   A  p  r   2   0   1   4 Problem formulation for truth-table invariantcylindrical algebraic decomposition byincremental triangular decomposition Matthew England 1 , Russell Bradford 1 , Changbo Chen 2 , James H. Davenport 1 ,Marc Moreno Maza 3 , and David Wilson 1 1 University of Bath, Bath, BA2 7AY, U.K. 2 Chongqing Key Laboratory of Automated Reasoning and Cognition, ChongqingInstitute of Green and Intelligent Technology, CAS, Chongqing, 400714, China. 3 University of Western Ontario, London, N6A 5B7, Canada. { R.J.Bradford, J.H.Davenport, M.England, D.J.Wilson }@bath.ac.uk,changbo.chen@hotmail.com, moreno@csd.uwo.ca Abstract.  Cylindrical algebraic decompositions (CADs) are a key toolfor solving problems in real algebraic geometry and beyond. We recentlypresented a new CAD algorithm combining two advances: truth-table in-variance, making the CAD invariant with respect to the truth of logicalformulae rather than the signs of polynomials; and CAD construction byregular chains technology, where first a complex decomposition is con-structed by refining a tree incrementally by constraint. We here considerhow best to formulate problems for input to this algorithm. We focuson a choice (not relevant for other CAD algorithms) about the order inwhich constraints are presented. We develop new heuristics to help makethis choice and thus allow the best use of the algorithm in practice. Wealso consider other choices of problem formulation for CAD, as discussedin CICM 2013, revisiting these in the context of the new algorithm. Keywords:  cylindrical algebraic decomposition, truth table invariance,regular chains, triangular decomposition, problem formulation 1 Introduction A  cylindrical algebraic decomposition   (CAD) is: a  decomposition   of  R n , meaninga collection of cells which do not intersect and whose union is  R n ;  cylindrical  ,meaning the projections of any pair of cells with respect to a given variableordering are either equal or disjoint; and,  (semi)-algebraic  , meaning each cellcan be described using a finite sequence of polynomial relations.CAD was introduced by Collins in [11], such that a given set of polynomialshad constant sign on each cell. This meant that a single sample point for eachcell was sufficient to conclude behaviour on the whole cell and thus it offered aconstructible solution to the problem of quantifier elimination. Since then a rangeof other applications have been found for CAD including robot motion planning  [23], epidemic modelling [8], parametric optimisation [18], theorem proving [22] and reasoning with multi-valued functions and their branch cuts [14].In [3] the present authors presented a new CAD algorithm combining tworecent advances in CAD theory: construction by first building a cylindrical de-composition of complex space, incrementally refining a tree by constraint [9];and the idea of producing CADs such that given formulae has invariant truth oneach cell [4]. Experimental results in [3] showed this new algorithm to be superior to its individual components and competitive with the state of the art. We nowinvestigate the choices that need to be made when using the new algorithm.We conclude the introduction with the necessary background theory and thenin Section 2 we demonstrate how constraint ordering affects the behaviour of thealgorithm. No existing heuristics discriminate between these orderings and so wedevelop new ones, which we evaluate in Section 3. In Section 4 we consider other issues of problem formulation, revisiting [6] in the context of the new algorithm. 1.1 Background on CAD The first CAD algorithm, introduced by Collins [11] with a full description in[1], works in two phases. First in the  projection   phase a projection operator isrepeatedly applied to the set of polynomials (starting with those in the input),each time producing another set in one fewer variables. Then in the  lifting   phaseCADs are built incrementally by dimension. First R 1 is decomposed according tothe real roots of the univariate polynomials. Then R 2 is decomposed by repeatingthe process over each cell in  R 1 using the bivariate polynomials evaluated at asample point, and so on. Collins’ srcinal projection operator was chosen so thatthe CADs produced could be concluded  sign-invariant   with respect to the inputpolynomials, meaning the sign of each polynomial on each cell is constant.Such decompositions can contain far more information than required for mostapplications, which motivated CAD algorithms which consider not just polyno-mials but their srcin. For example, when using CAD for quantifier eliminationpartial CAD [13] will avoid lifting over a cell if the solution there is already ap-parent. Another key adaptation is to make use of an  equational constraint   (EC):an equation logically implied by an input formula. The algorithm in [21] ensuressign-invariance for the polynomial defining an EC, any any other polynomialsonly when that constraint is satisfied. A discussion of the first 20 years of CADresearch is given in [12]. Some of the subsequent developments are discussednext, with others including the use of certified numerics when lifting [19,24]. 1.2 TTICAD by regular chains In [3] we presented a new CAD algorithm, referred to from now on as  RC-TTICAD .It combined the following two recent advances. Truth-table invariant CAD:  A TTICAD is a CAD produced relative to alist of formulae such that each has constant truth value on every cell.The first TTICAD algorithm was given in [4], where a new projection operatorwas introduced which acted on a set of formulae, each with an EC.  TTICADs are useful for applications involving multiple formulae like branchcut analysis (see for example Section 4 of  [16]), but also for building truth-invariant CADs for a single formula if it can be broken into sub-formulae withECs. The algorithm was extended in [5] so that not all formulae needed ECs,with savings still achieved if at least one did. These algorithms were implementedin the freely available  Maple  package  ProjectionCAD  [17]. CAD by regular chains technology:  A CAD may be built by first forminga  complex cylindrical decomposition   (CCD) of  C n using triangular decompo-sition by regular chains, which is refined to a CAD of   R n .This idea to break from projection and lifting was first proposed in [10]. In [9] the approach was improved by building the CCD incrementally by constraint,allowing for competition with the best projection and lifting implementations.Both algorithms are implemented in the  Maple RegularChains  Library, withthe algorithm from [10] currently the default CAD distributed with  Maple . RC-TTICAD  combined these advances by adapting the regular chains compu-tational approach to produce truth-table invariant CCDs and hence CADs. Thisnew algorithm is specified in [3] where experimental results showed a  Maple  im-plementation in the  RegularChains  Library as superior to the two advancesindependently, and competitive with the state of the art. The CCD is built usinga tree structure which is incrementally refined by constraint. ECs are dealt withfirst, with branches refined for other constraints in a formula only is the ECsare satisfied. Further, when there are multiple ECs in a formula branches can beremoved when the constraints are not both satisfied. See [3,9] for full details. The incremental building of the CCD offers an important choice on prob-lem formulation: in what order to present the constraints? Throughout we use A  →  B  to mean that  A  is processed before  B , where  A  and  B  are polynomi-als or constraints defined by them. Existing CAD algorithms and heuristics donot discriminate between constraint orderings [6,15] and so a new heuristic is required to help make an intelligent choice. 2 Constraint ordering The theory behind  RC-TTICAD  allows for the constraints to be processed in anyorder. However, the algorithm as specified in [3] states that  equational con-straints should be processed first . This is logical as we need only considerthe behaviour of non-ECs when corresponding ECs are satisfied, allowing forsavings in computation.We also advise  processing all equational constraints from a formula inturn , i.e. not processing one, then moving to a different formula before returningto another in the first. Although not formally part of the algorithm specification,this should avoid unnecessary computation by identifying when ECs have amutual solution before more branches have been created.There remain two questions to answer with regards to constraint ordering: Q1)  In what order to process the formulae? Q2)  In what order to process the equational constraints within each formula?  Fig.1.  Visualisations of the four TTICADs which can be built using  RC-TTICAD  forExample 1. The figures on the top have  φ 1 → φ 2  and those on the bottom  φ 2 → φ 1 .The figures on the left have  f  1 → f  2  and those on the right  f  2 → f  1 . 2.1 Illustrative example The following example illustrates why these questions matter. Example 1.  We assume the ordering  x  ≺  y  and consider f  1  :=  x 2 + y 2 − 1 , f  2  := 2 y 2 − x, f  3  := ( x − 5) 2 + ( y − 1) 2 − 1 ,φ 1  :=  f  1  = 0 ∧ f  2  = 0 , φ 2  :=  f  3  = 0 . The polynomials are graphed within the plots of Figure 1 (the circle on the leftis  f  1 , the one on the right  f  3  and the parabola  f  2 ). If we want to study the truthof   φ 1  and  φ 2  (or a parent formula  φ 1 ∨ φ 2 ) we need a TTICAD to take advantageof the ECs. There are two possible answers to each of the questions above andso four possible inputs to  RC-TTICAD . The corresponding outputs are 1 : φ 1  →  φ 2  and  f  1  →  f  2 :  37 cells in 0.095 seconds. φ 1  →  φ 2  and  f  2  →  f  1 :  81 cells in 0.118 seconds. φ 2  →  φ 1  and  f  1  →  f  2 :  25 cells in 0.087 seconds. φ 2  →  φ 1  and  f  2  →  f  1 :  43 cells in 0.089 seconds.The plots in Figure 1 show the two-dimensional cells in each of these TTICADs.First compare the induced CADs of   R 1 (how the real line is dissected). Ob-serve the following similarities in all four images: –  The points  14 ( − 1 ∓√  17) (approximately -1.28 and 0.78) are always identified.The latter is at the intersection of   f  1  and  f  2  and so is essential for the output tobe correct as the truth of   φ 1  changes here. The former is the other root of theresultant of   f  1  and  f  2  and so marks an intersection with complex  y -value. –  The points 4 and 6 are always identified. These mark the endpoints of   f  3 ,required for cylindricity and obtained as roots of the discriminant of   f  3 . 1 All timings in this paper were obtained on a Linux desktop (3.1GHz Intel processor,8.0Gb total memory) using  Maple 18 .
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