[Indigo]<QuantifierElimination>Cads.mesa!2

Cads.mesa

Last Edited by: Arnon, October 8, 1986 11:38:33 am PDT

Request for any AlgebraicSurfaces display op starts up the tool if not already started. Or have it running all the time, and each display request creates a new surfaceDisplay viewer.

AlgebraClasses has proc to add an op to a structures property list. Or MakeSequenceStructure proc takes procs to add to prop list as args.

Cad ops

induced cad

Alternate display

showTriangulationAll (valid only for cad's of 1-, 2- and 3-space)

showTriangulationSections

showTriangulation0Cells

showTriangulation1Cells

showTriangulation2Cells

rayTraceInputPolynomials (valid only for cad's of 3-space)

show cells

show statistics

Cad's should have names, statistics

Store only basis signatures with cells. Both clustering during cad construction, and clustering for disambiguation of signatures, are done with basis signatures (if basis signatures with ordinary projections don't suffice to get unambiguous clusters, then we have to go back and refine cad with respect to augmented projections).

Cluster ops

select containing cad

rayTrace (valid only for 2-cluster in 3-space)

showTriangulation (valid only for clusters in cad's of 1-, 2- and 3-space)

Cell ops

select containing cad

rayTrace (valid only for 2-section in 3-space)

showTriangulation (valid only for cells in cad's of 1-, 2- and 3-space)

Polynomial sequence ops, for base coeffs that are ints or bigrats

rayTraceZeros (valid only 3-variate polys)

decomposeZeros (forks Vax computation; pop up menu for options (including covering set parameters); inserts cad in Active MessageSet)

showZerosTriangulation (valid only for 1-, 2- and 3-variate polys)

New Polynomial ops, for base coeffs that are ints or bigrats

rayTrace (valid only for 3-variate poly)

decomposeZeros (forks Vax computation; pop up menu for options (including covering set parameters); inserts cad in Active MessageSet)

showZeroTriangulation (valid only for 1-, 2- and 3-variate poly)

Sequences

Basic display is one line: "sequence of n shortPNames"

Alternate display: Slider bar onto popup menu

popup menu entries are shortPNames of sequence elements

Insert creates a blank entry in popup menu; then select item to be inserted as an operand, it is type checked, inserted.

Delete modifies pop up menu

Selection of Insert or Delete forces Alternate display

DIRECTORY

Rope,

IO,

Graphs,

GraphOps,

AlgebraClasses,

BigRats,

RatIntervals,

Variables,

Polynomials,

AlgebraicNumbers,

Points,

Sequences,

Formulas,

SamplePoints,

CoveringSets,

Cells;

Cads: CEDAR DEFINITIONS

~ BEGIN OPEN AC: AlgebraClasses, BR: BigRats, RI: RatIntervals, VARS: Variables, AN: AlgebraicNumbers, PTS: Points, SEQ: Sequences, POL: Polynomials, QFF: Formulas, SP: SamplePoints, CS: CoveringSets;

Cad Representation

Cad: TYPE = AC.Object;

CadData: TYPE = REF CadDataRec;

CadDataRec: TYPE = RECORD [

dimension: CARDINAL, -- Define r = dimension

localizationFormula: QFF.Formula ← NIL, -- r-variate rational polynomials in inputVariables

inputPolynomials: SEQ.Sequence, -- of r-variate rational polynomials in inputVariables

minPolyVariable: VARS.VariableSeq,

basis: SEQ.Sequence, -- irreducible r-variate rational polynomials

signatures: LIST OF SIGNATURES

primaryPolynomials: SEQ.Sequence ← NIL, -- irreducible (<= r)-variate rational polynomials, all of which occur in the bases of some (possibly 0th) induced cad (and hence are invariant on all cells of the r-space cad), whose signs on each cell constitute its primarySignature

In other words, the polynomials in primaryPolynomials may be in different numbers of vars. Poly eval routines are assumed to be able to cope with this, i.e. if asked to eval a poly in i variables at a point with i+j coordinates, will use first i coordinates.

secondaryPolynomials: SEQ.Sequence ← NIL, -- same specs as primaryPolynomials, signs on each cell constitute its secondarySignature. Typically a superset of primaryPolynomials.

contents: SEQ.Sequence, -- (r-1)-variate rational polynomials; dimension = 1 => NIL

basisProjection: SEQ.Sequence, -- (r-1)-variate rational polynomials; dimension = 1 => NIL

projectionType: ProjectionType, -- dimension = 1 => projectionType = none

abstractGraph: Graphs.Graph ← nullGraph,

cells: SEQ.Sequence ← NIL, -- of cells, in lexicographic order (for binary searching)

basisClusters: SEQ.Sequence ← NIL, -- of SignedRegions

otherClusters: SEQ.Sequence ← NIL, -- of SignedRegions

statistics: CadStatisticsRec, -- to be added later

inducedCad: Cad ← NIL, -- dimension = 1 => NIL

parentCad: Cad ← NIL -- if any

];

ProjectionType: TYPE = { standard, augmented, mccallum, none };

Adjacency: TYPE = RECORD[

lowerDimCell, higherDimCell: Cells.Cell

];

SignedRegion: TYPE = REF;

Stack: TYPE = REF;

nullGraph: Graphs.Graph = [NIL, NIL];

Class for Cad Structures

cadStructureClass: AC.StructureClass;

Instance Data for Cad Structures

CadStructureData: TYPE = REF CadStructureDataRec;

CadStructureDataRec: TYPE = RECORD [

];

Operations Unique to Cad Structures

CadOps: TYPE = REF CadOpsRec; -- prop key is $CadStructure.

CadOpsRec: TYPE = RECORD [

localizationFormula: AC.UnaryOp,

inputPolynomials: AC.UnaryOp,

basis: AC.UnaryOp,

contents: AC.UnaryOp,

basisProjection: AC.UnaryOp,

cells: AC.UnaryOp,

cellsWithSignature: AC.BinaryOp, -- cellsWithSignature(cad, sequence of FormulaOperators) -> sequence of cells

basisClusters: AC.UnaryOp,

otherClusters: AC.UnaryOp,

makeBasisClusters: AC.UnaryOp,

makeOtherClusters: AC.BinaryOp, -- cluster(cad, sequence of FormulaOperators) -> clusters

inducedCad: AC.UnaryOp,

parentCad: AC.UnaryOp,

coveringSetsFromLinearRope: AC.BinaryInPlaceOp

];

Cad Structure Constructor

MakeCadStructure: PROC [inputPolynomialRing, minPolyRing: AC.Structure] RETURNS [cadStructure: AC.Structure];

MakeCadStructure: PROC [] RETURNS [cadStructure: AC.Structure];

Extract Cad Operations from Class Property Lists

IsCadStructure: PROC [structure: AC.Structure] RETURNS [BOOL];

LocalizationFormula: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

InputPolynomials: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

Basis: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

Contents: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

BasisProjection: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

CadCells: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp]; -- Cells is interface name

CellsWithSignature: PROC [structure: AC.Structure] RETURNS [AC.BinaryOp];

BasisClusters: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

OtherClusters: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

Cluster: PROC [structure: AC.Structure] RETURNS [AC.BinaryOp];

InducedCad: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

ParentCad: PROC [structure: AC.Structure] RETURNS [AC.UnaryOp];

CoveringSetsFromLinearRope: PROC [structure: AC.Structure] RETURNS [AC.BinaryInPlaceOp];

Conversion and IO

Read: AC.ReadOp;

FromRope: AC.FromRopeOp;

ToRope: AC.ToRopeOp;

Write: AC.WriteOp;

Operations

LocalizFormula: AC.UnaryOp;

InputPolys: AC.UnaryOp;

Base: AC.UnaryOp;

Conts: AC.UnaryOp;

BasisProj: AC.UnaryOp;

Cels: AC.UnaryOp;

CellsWithSig: AC.BinaryOp;

BasClusters: AC.UnaryOp;

OthClusters: AC.UnaryOp;

Clust: AC.BinaryOp;

IndCad: AC.UnaryOp;

ParCad: AC.UnaryOp;

LookupCell: AC.BinaryOp;

ReadCSets: PROC [in: IO.STREAM, cad: Cads.Cad];

CSetsFromLinRope: AC.BinaryInPlaceOp;

Main Algorithm

BuildCad: PROC [inputPolynomials: SEQ.Sequence, cadStructure: AC.Structure] RETURNS [cad: Cad];

Projection Phase

CoarsestSqFreeBasis: PROC [inputPolynomials: SEQ.Sequence, localizationFormula: QFF.Formula ← NIL] RETURNS [contents, basis: SEQ.Sequence];

Projection: PROC [basis: SEQ.Sequence] RETURNS [basisProjection: SEQ.Sequence];

The requirements are:

(1) we have a certain set K of polynomials in E^r which we need to make delineable on each cell and each signature-cluster in E^(r-1).

(2) The collection of cells we end up with in E^r must have the boundary property: the boundary of any cell must equal the union of some collection of lower-dimensional cells.

Question: why does the boundary property really matter?

Possible answer: then it doesn't matter where you probe when slicing an adjacency. To be more precise, boundary property is required for the "decoding" phase of our (0,1) adjacency determination in 3-space to work - we have to know that the 0-cells we see in our 2-space projection are images of 0-cells in the 3-space decomposition.

Given these, our approach is:

(1) Compute contents and basis(pp) = K of inputs, pull off initial sequences of non-constant coeffs of all basis elts, compute basis for contents union initial sequences, do a signature-invariant decomp D of E^(r-1) to get regions on which each element of K has constant degree. Note that if any element of K is identically zero at some point of E^r-1, D will have maximal regions for this property.

(2) using the appropriate reducta of elements of K (ignore identically zero elements), compute projection sets for each reducta pattern that occurs on a positive-dimensional regions, compute basis. For each positive-dimensional region, use this basis to break it up to be signature-invariant with respect to its projection basis. We end up with a new decomposition D'.

(3) Assume now r <= 3; maybe works for general r.

(4) On each region in E^(r-1), evaluate the appropriate non-identically zero reducta of elements of K to get a set of algebraic polys; compute its basis. If any element of K is
identically zero on this region, determine the appropriate algebraic polynomials whose roots we need to take account of to insure boundary property, compute their basis, merge the two bases. Isolate roots of the final basis to determine the stack over this region.

Simpler version, to get a single basis set in each E^i:

(1) Compute P1 = contents and K = basis(pp) of inputs.

Compute P2 = initial sequences of non-constant coeffs of all basis elts,

Compute each Pijk, where suppose K has three elements, 0 <= i < length of initial sequences of non-constant coeffs of K1, 0 <= j < etc., Qijk is projection computed using ith, jth, and kth reducta of respective elements of K, and Pijk is the result of simplifying Qijk with respect to the side relations that the appropriate initial non-constant coeffs of the elements of K are equal to zero.

(Possible precomputation for each Pijk: do a heuristic test (e.g. Groebner basis to check emptiness of complex variety), or cad, to check that real varieties stay nonempty as you take progressively larger sequences of initial non-constant coeffs; if such a variety is empty, then you don't need to continue. It could happen that appropriate tails of initial sequences of elements of K could be thrown out of P2).

Set P = P1 un P2 un (all Pijk's).

Compute P-invariant decomposition of E^(r-1).

(2) Assume now r <= 3; maybe works for general r.

(3) Extension phase: On each region in E^(r-1), evaluate the appropriate non-identically zero reducta of elements of K to get a set of algebraic polys; compute its basis. If any element of K is identically zero on this region, determine the appropriate algebraic polynomials whose roots we need to take account of to insure boundary property, compute their basis, merge the two bases. Isolate roots of the final basis to determine the stack over this region. Record K-signature of each element of the stack.

Base Phase

BasePhase: PROC [cad: Cad];

cad is of r>=1 space; follow chain of induced cad's down to 1-space, contruct cells there. Fill in basisSignature's = primarySignature's of cells. If 1-space cad.localizationFormula # NIL, then discard cells which don't satisfy it.

Adjacencies and Clusters

ExtractClusters: PROC [cad: Cad, numComponents: CARDINAL] RETURNS [clusters: SEQ.Sequence];

components assumed to be of cad, and not some induced cad thereof.

Components: Menus.MenuProc = {

User specifies polynomials with respect to which to cluster by (*,*,@,*,*,@) tuples (@ is new FormulaOperators.Op).

edgePredicate: GraphOps.EdgePredicate ~ {

cell1: QET.Cell ← NARROW[v.rep];

cell2: QET.Cell ← NARROW[w.rep];

RETURN[cell1.basisSignature.structure.class.equal[cell1.basisSignature, cell2.basisSignature] ];

};

inducedCad: QET.Cad ← QEIO.GetInducedCad[cad, dimension];

numComponents: CARDINAL ← GraphOps.DetermineComponents[cad.abstractGraph, Undirected, edgePredicate];

clusters: QET.SignedRegionSeq ← Cads.ExtractClusters[inducedCad, numComponents];

QEIO.WriteSignedRegionSeq[clusters, out];

};

MaximalSignedRegions: PROC [dimension: CARDINAL, cad: Cad, primary: BOOL];

Depth-First Search connected components computation in the dimension-space cad graph, filling in the (IF primary THEN primaryRegion ELSE secondaryRegion) fields of cells.

ExtractSignedRegions: PROC [dimension: CARDINAL, cad: Cad, primary: BOOL] RETURNS [SEQ.Sequence];

Extract from the dimension-space cad graph, the (IF primary THEN primaryRegion ELSE secondaryRegion) cell-field-defined regions.

AdjacenciesOverAdjacency: PROC [dimension: CARDINAL, adjacency: Adjacency, cad: Cad] RETURNS [newAdjacencies: LIST OF Adjacency];

For i = 2 or 3 only, returns all interstack adjacencies in i-space over an adjacency in dimension-1 space. cad is i-space cad. newAdjacencies = NIL if no adjacencies in dimension space can be built, either because needed auxiliary E^(i-1) cell-cell adjacencies, or non-NIL sample points, cannot be found.

Converts extended to primitive sample points as needed.

May build adjacencies over appropriate auxiliary E^(i-1) cell-cell adjacencies if necessary, and/or query i-space cad for existing adjacency information.

Extension Phase

Information to save to pass to adjacency algorithms:

E1 to E2: For each 0-cell in E1: isolating intervals (perhaps expanded to be strictly open) and corresponding algebraic basis elements over it; for each element F of integral bivariate basis, a sequence of length m of positive integers, such that F has m real roots over it, and ith element of sequence is section (or element) of overall stack which is ith real root of F.

Use this list to modify cell indices of section-section adjacencies produced by adjacency algorithms.

E2 to E3: For each 0-cell and 1-cell in E2: isolating intervals (perhaps expanded to be strictly open) and corresponding algebraic basis elements over it; for each element F of integral trivariate basis, a sequence of length m of positive integers, such that F has m real roots over it, and ith element of sequence is section of overall stack which is ith real root of F.

RegionExtendCadToCad: PROC [cad: Cad];

Extend cad of dimension-1 space to dimension space, using the largest signed regions (wrt primarySignatures) that can be constructed in the graph for dimension-1 space.

Let i = dimension.

Assume that in E^(i-1), with primaryPolynomials ← CompleteFactors(inputPolynomials), primarySignatures of cells are set.

1. MaximalSignedRegions[i-1, cad, TRUE]; regions ← ExtractSignedRegions[i-1, cad, TRUE]; sort regions in order of decreasing dimension (for i-1 = 2, put degenerate 0-dimensional regions after nondegenerate).

2. cad.cells ← NIL; Go down sorted list of regions, for each:

2a. currentCells ← ExtendRegionToCells[i, currentRegion, cad]; newAdjacencies ← NIL;

If currentRegion is a 0-cell, then as each element of (trivariate) basis is evaluated at sample point, check for identically zero; if so, then add that basis element to identicallyZeroBasisElements field of the 0-cell. If identicallyZeroBasisElements#NIL when all basis elements have been evaluated, i.e. if this is a degenerate 0-cell, then do the extra work required to determine the elements of the stack over this cell (RES sets, etc.), otherwise just isolate the roots of the algebraic basis of evaluated (trivariate integral) basis.

2b. Do this step only if i = 2 or 3. If dimension of currentRegion < i-1, then determine adjacent regions of larger dimension. Sort these into order of increasing dimension. Go down this list, and for each region on it, find all cell-cell adjacencies between currentRegion and this region. Sort these adjacencies into decreasing sum of cell dimensions. For the next such adjacency, newAdjacencies ← AdjacenciesOverAdjacency[i, adjacency, cad]. If newAdjacencies = NIL, i.e. if needed auxiliary E^(i-1) cell-cell adjacencies, or non-NIL sample points, cannot be found, then go to the next cell-cell adjacency between current region and this region. If needed auxiliary adjacencies and/or non-NIL sample points cannot be found for any cell-cell adjacency between current region and this region, give up on building adjacencies in E^i between stacks over current region and this region.

Note that there is no need to worry about whether regions have the boundary property; Note the specification that we try to find an adjacency (that we can build adjacencies over) between each pair of adjacent regions; the lower dimensional cell in such an adjacency necessarily belongs to the portion of the lower dimensional region contained in the boundary of the larger dimensional region.

2c. Insert cells of currentCells into cad.cells in lexicographical order. IF newAdjacencies # NIL, then enter its adjacencies into the appropriate cells of cad.cells.

3. If i-space cad.localizationFormula # NIL, then delete all cells of cad.cells which don't satisfy it, and all adjacencies involving those cells .

4. For i = 1, 2 or 3, if no localization, we now have a graph for the cad of E^i in which a maximal signed regions computation will yield (signed) components (in 1 and 2-space, we can have all adjacencies, in 3-space, probably only a proper subset of all adjacencies.)

ExtendRegionToCells: PROC [dimension: CARDINAL, region: SignedRegion, cad: Cad] RETURNS [newCells: SEQ.Sequence];

Extend region in dimension-1 space to cells in dimension-space. cad is dimension-space cad.

Fill in basisSignatures, and, if desired, (with primaryPolynomials ← CompleteFactors(inputPolynomials)), primarySignatures, of newCells. Also fill in induced adjacencies, and intrastack adjacencies, among newCells.

(Let i = dimension). Method: Evaluate polys at base sample point, irreducible basis of algebraic polys, isolate roots, construct extended sample points, eval gsfd of i-variate polynomial at endpoints of isolating interval, or i-variate polynomial itself at easy primitive sector sample point, when we want its sign.

ExtendCellToStacks: PROC [i: CARDINAL, cell: Cells.Cell, cad: Cad] RETURNS [StacksInEi: SEQ.Sequence];

Return a StackSeq of length one consisting of the stack over cell in E^(i-1).

ExtendStacksToStacks: PROC [i: CARDINAL, StacksInEiM1: SEQ.Sequence, cad: Cad] RETURNS [StacksInEi: SEQ.Sequence];

StacksInEi returned in lexicographical order by base

RatApproxSamplePoint: PROC [cell: Cells.Cell, ratApproxBound: BR.BigRat] RETURNS [PTS.Point];

Return a rational approximation to cell sample point, each coordinate accurate to within ratApproxBound. Should have the property that always produces the same RatPoint for given RealFieldPoint and ratApproxBound inputs.

END.