Truth table invariant cylindrical algebraic decomposition

http://curve.coventry.ac.uk/open

Truth table invariant cylindrical algebraic

decomposition

Bradford, R. , Davenport, J. H. , England, M. , McCallum, S. and

Wilson, D.

Published PDF deposited in

Curve

February 2016

Original citation:

Bradford, R. , Davenport, J. H. , England, M. , McCallum, S. and Wilson, D. (2016) Truth table

invariant cylindrical algebraic decomposition . Journal of Symbolic Computation, volume 76 :

1-35

URL:

http://dx.doi.org/10.1016/j.jsc.2015.11.002

DOI: 10.1016/j.jsc.2015.11.002

ISSN: 0747-7171

Copyright © and Moral Rights are retained by the author(s) and/ or other copyright

owners. A copy can be downloaded for personal non-commercial research or study,

without prior permission or charge. This item cannot be reproduced or quoted extensively

from without first obtaining permission in writing from the copyright holder(s). The

content must not be changed in any way or sold commercially in any format or medium

without the formal permission of the copyright holders.

Contents lists available atScienceDirect

Journal

of

Symbolic

Computation

www.elsevier.com/locate/jsc

Truth

table

invariant

cylindrical

algebraic

decomposition

✩

Russell Bradford

a

,

James

H. Davenport

a

,

Matthew England

b

,

Scott McCallum

c

,

David Wilson

a

a_{DepartmentofComputerScience,UniversityofBath,Bath,BA27AY,UK}

b_{SchoolofComputing,ElectronicsandMaths,FacultyofEngineering,EnvironmentandComputing,}

Coventry University,Coventry,CV15FB,UK

c_{DepartmentofComputing,MacquarieUniversity,NSW2109,Australia}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received21December2014 Accepted21October2015 Availableonline4November2015

MSC:

68W30 03C10

Keywords:

Cylindricalalgebraicdecomposition Equationalconstraint

Whenusingcylindricalalgebraicdecomposition(CAD) tosolve a problemwithrespecttoasetofpolynomials,it islikelynot the signsofthosepolynomialsthatareofparamountimportancebut ratherthetruthvaluesofcertainquantifierfreeformulaeinvolving them.This observation motivates ourarticle and definition ofa TruthTableInvariantCAD(TTICAD).

InISSAC2013thecurrentauthorspresentedanalgorithmthatcan efficiently and directlyconstructa TTICADfora listof formulae inwhicheachhasanequationalconstraint.Thiswasachievedby generalisingMcCallum’stheoryofreducedprojectionoperators.In thispaper wepresent an extended version ofour theorywhich canbeapplied toanarbitrarylistofformulae, achievingsavings ifatleastonehasanequationalconstraint.Wealsoexplainhow thetheory ofreduced projectionoperators canallow for further improvementstotheliftingphaseofCADalgorithms,eveninthe contextofasingleequationalconstraint.

Thealgorithmisimplementedfullyin Maple andwepresentboth promisingresultsfromexperimentationandacomplexityanalysis showingthebenefitsofourcontributions.

©2015TheAuthors.PublishedbyElsevierLtd.Thisisanopen accessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

✩ _{ThisworkwassupportedbyEPSRCgrantEP/J003247/1.}

E-mailaddresses:[email protected](R. Bradford),

[email protected]

(J.H. Davenport),

[email protected](M. England),

[email protected]

(S. McCallum),

[email protected]

(D. Wilson).

http://dx.doi.org/10.1016/j.jsc.2015.11.002

0747-7171/©2015TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

A

cylindrical algebraic decomposition (CAD)

isadecompositionof

R

n_{intocellsarrangedcylindrically}

(meaningtheprojectionsofanypairofcellsareeitherequalordisjoint)eachofwhichis (semi-)alge-braic(describableusingpolynomialrelations).CADisakeytoolinrealalgebraicgeometry,offeringa methodforquantifiereliminationinrealclosedfields.Applicationsincludethederivationofoptimal numericalschemes(ErascuandHong,2014),parametricoptimisation(Fotiouetal.,2005),robot mo-tionplanning(SchwartzandSharir,1983),epidemicmodelling(Brownetal.,2006),theoremproving (Paulson,2012) andprogrammingwithcomplexfunctions(Davenportetal.,2012).

TraditionallyCADsareproduced

sign-invariant to

agivensetofpolynomials(thesignsofthe poly-nomials do not vary within each cell). However, this gives farmore information than required for mostapplications.Usually amoreappropriate objectis a

truth-invariant CAD

(the truth ofalogical formuladoesnotvarywithincells).

In this paper we generalise to define truth table invariant CADs (the truth values of a list of quantifier-free formulaedo not varywithin cells) andgive an algorithm to compute thesedirectly. This canbe a toolto efficientlyproducea truth-invariantCADforaparent formula(builtfromthe input list),orindeedtherequiredobjectforsolving aprobleminvolvingtheinputlist.Examplesof both such usesare provided following the formal definition inSection 1.2. We continue the intro-duction with some background on CAD, before defining our object of studyandintroducing some examplestodemonstrateourideaswhichwewillreturntothroughoutthepaper.Wethenconclude theintroductionbyclarifyingthecontributionsandplanofthispaper.

1.1. Background on CAD

A

Tarski

formula F

(x

1

,

. . . ,

xn

)

isaBooleancombination(

∧,

∨,

¬,

→

)ofstatementsaboutthesigns,

(

=

0

,

>

0

,

<

0,buttherefore

=

0

,

≥

0

,

≤

0 aswell),ofcertain polynomials fi

(x

1

,

. . . ,

xn

)

withinteger

coefficients.Suchstatementsmayinvolvetheuniversalorexistentialquantifiers(

∀,

∃

).Wedenoteby QFFa

quantifier-free Tarski formula.

GivenaquantifiedTarskiformula

Qk+1xk+1

. . .

QnxnF

(

x1

, . . . ,

xn

)

(1)

(where Qi

∈ {∀,

∃}

and F is a QFF)the

quantifier elimination problem is

toproduce

ψ(x

1

,

. . . ,

xk

)

,an

equivalentQFFto(1).

Collins developedCADasatoolforquantifiereliminationover thereals.He proposedto decom-pose

R

n _{cylindricallysuch thateach cellwas sign-invariantforallpolynomials f}

i usedtodefine F .

Then

ψ

wouldbethedisjunctionofthedefining formulaeofthosecells ci in

R

k suchthat (1)was

trueoverthewholeof

c

i,whichduetosign-invarianceisthesameassaying that(1)istrueatany

one

sample point of c

i.

A complete description of Collins’ original algorithm is given by Arnon et al. (1984a). The first phase,

projection,

appliesaprojectionoperatorrepeatedlytoasetofpolynomials,eachtimeproducing another setinone fewervariables. Togetherthesesets containthe projection polynomials. Theseare used in the second phase, lifting, to build theCAD incrementally.First

R

is decomposedinto cells which are points and intervalscorresponding to the real rootsof theunivariate polynomials. Then

R

2 _{isdecomposedbyrepeatingtheprocess overeachcellin}

_R

_{usingthebivariatepolynomialsata}

samplepoint.Overeachcellthereare

sections (where

apolynomialvanishes)and

sectors (the

regions between)whichtogether forma

stack.

TakingtheunionofthesestacksgivestheCADof

R

2.Thisis repeateduntilaCADof

R

n isproduced.Ateachstagethecellsarerepresentedby(atleast)asample pointandanindex:alistofintegerscorrespondingtotheorderedrootsoftheprojectionpolynomials whichlocatesthecellintheCAD.

ToconcludethataCADproducedinthiswayissign-invariantweneeddelineability.Apolynomial is delineable in a cellif theportion ofits zero setin thecell consistsof disjointsections.A setof polynomialsare

delineable in

acellifeachisdelineableandthesectionsofdifferentpolynomialsin the cell are either identicalordisjoint. The projection operator usedmust be definedso that over

eachcellofasign-invariantCADforprojectionpolynomialsin

r variables

(theword

over meaning

we arenowtalkingaboutan

(r

+

1

)

-dimspace)thepolynomialsin

r

+

1 variablesaredelineable.

The output of thisand subsequent CADalgorithms (including the one presented in thispaper) dependsheavily onthevariableordering.Weusually workwithpolynomialsin

Z[

x

]

= Z[

x1

,

. . . ,

xn

]

withthevariables,x,inascendingorder(sowefirstprojectwithrespectto

x

n andcontinuetoreach

univariatepolynomialsin

x

1).The

main variable of

apolynomial(mvar)isthegreatestvariablepresent

withrespecttotheordering.

CADhasdoublyexponentialcomplexityinthenumberofvariables(BrownandDavenport,2007; DavenportandHeintz, 1988). Therenow existalgorithms withbettercomplexity forsomeCAD ap-plications(seeforexampleBasuetal.,1996)butCADimplementationsoftenremainthebestgeneral purposeapproach. There have beenmany developmentsto the theory since Collin’s treatment, in-cludingthefollowing:

•

Improvementstotheprojectionoperator(Hong,1990;McCallum,1988, 1998;Brown,2001; Han etal.,2014),reducingthenumberofprojectionpolynomialscomputed.

•

Algorithmstoidentifytheadjacencyofcells ina CAD(Arnonetal., 1984b, 1988)andfollowing fromthistheideaofclustering(Arnon,1988) tominimisethelifting.

•

PartialCAD,introduced byCollinsandHong (1991),wherethestructure ofF is usedtoliftless ofthedecompositionof

R

k _to

_R

n_{,ifitissufficienttodeduce}

_ψ

_.

•

The theory ofequational constraints (McCallum, 1999, 2001; Brownand McCallum, 2005) also aimingtodeduce

ψ

itself,thistimeusingmoreefficientprojections.

•

Theuseofcertifiednumerics intheliftingphasetominimisetheamountofsymbolic computa-tionrequired(Strzebo ´nski,2006; Iwaneetal.,2009).

•

New approaches which break with the normal projection and lifting model: local projection (Strzebo ´nski,2014),thebuildingofsingleCADcells(Brown,2013; Jovanovicandde Moura,2012) andCADviatriangular decomposition (Chen etal.,2009b). Thelatteris nowused fortheCAD commandbuilt into Maple,andworksby firstcreatinga cylindricaldecompositionof complex space.

1.2. TTICAD

Brown (1998)defineda

truth-invariant CAD as

oneforwhichaformulahadinvarianttruthvalueon each cell.Givena QFF,asign-invariant CADforthedefining polynomialsistrivially truth-invariant. Brown consideredthe refinement ofsign-invariant CADswhilst maintaining truth-invariance,while some ofthedevelopmentslisted above canbe viewedasmethods toproduce truth-invariantCADs directly.WedefineanewbutrelatedtypeofCAD,thetopicofthispaper.

Definition1.Let

{φ

i

}

ti=1 refer toa listofQFFs.Wesaya cylindricalalgebraic decomposition

D

isa

Truth Table Invariant CAD fortheQFFs(TTICAD)iftheBooleanvalueofeach

φ

iisconstant(eithertrue

orfalse)oneachcellof

D

.

Asign-invariantCADforallpolynomialsoccurringinalistofformulaewouldclearlybeaTTICAD forthelist.However,weaimtoproducesmallerTTICADsformanysuchlists.Wewillachievethisby utilisingthepresenceofequationalconstraints,atechniquefirstsuggestedbyCollins (1998)withkey theorydevelopedbyMcCallum (1999).

Definition2.Supposesomequantifiedformulaisgiven:

φ

∗

= (

Qk+1xk+1

)

· · · (

Qnxn

)φ (

x

)

wherethe Qi are quantifiersand

φ

isquantifierfree. Anequation f

=

0 is an equational constraint

(EC)of

φ

∗if f

=

0 islogicallyimpliedby

φ

(thequantifier-freepartof

φ

∗).Suchaconstraintmaybe eitherexplicit(anatomoftheformula)orotherwiseimplicit.

InSections3and4wewilldescribehowTTICADscanbeproducedefficientlywhenthereareECs presentinthelistofformulae.Therearetworeasonstousethistheory.

(1) As

a tool to build a truth-invariant CAD efficiently: If

aparentformula

φ

∗ isbuiltfromtheformulae

{φ

i

}

thenanyTTICADfor

{φ

i

}

isalsotruth-invariantfor

φ

∗.

WenotethatforsuchaformulaaTTICADmayneedtocontainmorecellsthanatruth-invariant CAD. Forexample,consideracell inatruth-invariantCADfor

φ

∗

= φ1

∨ φ2

within which

φ

1 is

always true.If

φ

2 changedtruth value insucha cellthenit wouldneedtobe splitinorderto

achieveaTTICAD,butthisisunnecessaryforatruth-invariantCADof

φ

∗.

Nevertheless, we find that our TTICAD theory is often able to produce smaller truth-invariant CADsthan anyother availableapproach.We demonstratethesavings offeredvia worked exam-plesintroducedinthenextsubsection.

(2) When

given a problem for which truth table invariance is required: That

is,aproblemforwhichthe listofformulaearenotderivedfromalargerparentformulaandthusatruth-invariantCADfor theirdisjunctionmaynotsuffice.

For example,decomposing complexspace accordingto aset ofbranch cutsforthe purposeof algebraicsimplification(BradfordandDavenport,2002; Phisanbutetal.,2010).Heretheideaisto representeachbranchcutasasemi-algebraicsettogiveinputadmissibletoCAD(recentprogress onthishasbeendescribedbyEnglandetal.,2013).ThenaTTICADforthelistofformulaethese setsdefineprovidesthenecessarydecomposition.Example 33isfromthisclass.

1.3. Worked examples

To demonstrate our ideas we will provide details for two worked examples. Assume we have the variable ordering x

≺

y (meaning 1-dimensionalCADs are with respectto x) andconsider the followingpolynomials,graphedinFig. 1.

f1

:=

x2

+

y2

−

1 g1

:=

xy

−

14

f2

:= (

x

−

4

)

2

+ (

y

−

1

)

2

−

1 g2

:= (

x

−

4

)(

y

−

1

)

−

1₄

Supposewewishtofindtheregionsof

R

2 wherethefollowingformulaistrue:



:= (

f1

=

0

∧

g1

<

0

)

∨ (

f2

=

0

∧

g2

<

0

) .

(2)

Both Qepcad (Brown,2003) and Maple 16(Chenetal.,2009b) produceasign-invariantCADforthe polynomialswith317cells.Thenbytestingthesamplepointfromeachregionwecansystematically identifywheretheformulaistrue.

AtfirstglanceitseemsthatthetheoryofECsisnotapplicableto



asneither f1

=

0 nor f2

=

0

is logically implied by



. However, while there is no explicit EC we can observe that f1f2

=

0 is

an

implicit constraint

of



.Using Qepcad withthisdeclared(animplementationofMcCallum,1999) givesaCADwith249cells.Later,inSection3.3wedemonstratehowaTTICADwith105cellscanbe produced.

Wealsoconsidertherelatedproblemofidentifyingwhere



:= (

f1

=

0

∧

g1

<

0

)

∨ (

f2

>

0

∧

g2

<

0

)

(3)

istrue.Asabove,wecoulduseasign-invariantCADwith317cells,butthistimethereisnoimplicit EC.InSection3.3weproduceaTTICADwith183cells.

1.4. Contributions and plan of the paper

We reviewtheprojection operatorsofMcCallum (1998,1999)inSection2.Theformerproduces sign-invariantCADs1_{andthelatterCADstruth-invariantforaformulawithanEC.Thereviewis} nec-essarysinceweusesomeofthistheorytoverifyournewalgorithm.Italsoallowsustocompareour

Fig. 1. ThepolynomialsfromtheworkedexamplesofSection1.3.Thesolidcurvesare f1andg1whilethedashedcurvesare

f2andg2.

newcontributiontotheseexistingapproaches.Forthispurposeweprovidenewcomplexityanalyses oftheseexistingtheoriesinSection2.3.

Sections3and4presentournewTTICADprojectionoperatorandverifiedalgorithm.Theyfollow Sections2and3ofourISSAC2013paper(Bradfordetal.,2013a),butinsteadofrequiringallQFFsto haveanECthetheoryhereisapplicabletoallQFFs(producingsavingssolongasonehasanEC).The strengtheningofthetheory meansthataTTICADcannowbeproducedfor



inSection1.3aswell as



.This extension isimportantsince it means TTICADtheory nowapplied to caseswhere there can beno overall implicitECfora parent formula. Inthesecases theexisting theory ofECs isnot applicableandsothecomparativebenefitsofferedbyTTICADareevenhigher.

InSection5we discusshowthetheory ofreducedprojection operatorsalsoallowsfor improve-mentsintheliftingphase.Thisistruefortheexistingtheoryalsobutthediscovery wasonlymade duringthedevelopmentofTTICAD.InSection6wepresentacomplexity analysisofournew contri-butions fromSections3–5,demonstratingtheir benefitoverthe existingtheory fromSection2.We haveimplementedthenewideasina Maple package,discussedinSection7.Inparticular,Section7.3 summarises (Bradfordet al., 2013b) on the choicesrequired when using TTICADand heuristicsto help. Experimental results forour implementation (extending those in our ISSAC 2013 paper) are giveninSection8,beforewefinishinSection9withconclusionsandfuturework.

Dataaccessstatement: Data directlysupporting this paper(code, Maple and Qepcad input)are openlyavailablefromhttp://dx.doi.org/10.15125/BATH-00076.

2. ExistingCADprojectionoperators

2.1. Review: sign-invariant CAD

Throughoutthe paperwe let cont, prim,disc,coeff and ldcf denote thecontent, primitivepart, discriminant,coefficientsandleadingcoefficientofpolynomialsrespectively(ineachcasetakenwith respecttoagivenmainvariable).Similarly,weletres denote theresultant ofa pairofpolynomials. When applied to a set of polynomials we interpret theseas producing sets of polynomials, so for example

res

(

A

)

=



res

(

fi

,

fj

)

|

fi

∈

A

,

fj

∈

A

,

fj

=

fi



.

ThefirstimprovementstoCollinsoriginalprojectionoperatorweregivenbyMcCallum (1988)and Hong (1990).TheywerebothsubsetsofCollinsoperator,meaningfewerprojectionpolynomials,fewer cells inthe CADsproduced andquickercomputation time. McCallum’sisactually a strict subset of Hong’s,however,itcannotbeguaranteedcorrect(incorrectnessisdetectedintheliftingprocess)for acertainclassof(statisticallyrare)inputpolynomials,whereHong’scan.

Additional improvements havebeen suggested by Brown (2001)and Lazard (1994).The former required changes to the lifting phase while the latterhad a flawedproof ofvalidity (withcurrent unpublishedworksuggestingitcanstillbesafelyusedinmanycases).Inthispaperwewillfocuson McCallum’soperators,notingthatthealternativescouldlikelybeextendedtoTTICADtheoriestooif desired.McCallum’stheoryisbasedaroundthefollowingcondition,whichimpliessign-invariance.

Definition3. A CADis order-invariant with respect to a set of polynomialsif each polynomial has constantorderofvanishingwithineachcell.

Recall that a set A

⊂ Z[

x

]

is an irreducible basis if the elements of A are of positive degree in themainvariable,irreducibleandpairwise relativelyprime.Let A be a setofpolynomialsandB an irreduciblebasisoftheprimitivepartof A. Then

P

(

A

)

:=

cont

(

A

)

∪

coeff

(

B

)

∪

disc

(

B

)

∪

res

(

B

)

(4)

defines the operator of McCallum (1988). We can assume some trivial simplifications such as the removalof constantsandexclusionof entriesidenticalto aprevious one (uptoconstant multiple). Themaintheoremunderlyingtheuseof P follows.

Theorem4.

(See

McCallum,1998.) Let A be an irreducible basis in

Z[

x

]

and let S be a connected submanifold of

R

n−1_{. Suppose each element of P}

₍

_{A)is order-invariant in S.}

Then each element of A either vanishes identically on S or is analytic delineable on S, (a slight variant on traditional delineability, see McCallum,1998). Further, the sections of A not identically vanishing are pairwise disjoint, and each element of A not identically vanishing is order-invariant in such sections.

Theorem 4 means that we can use P in place of Collins’ projection operator to produce sign-invariantCADssolongasnoneoftheprojectionpolynomialswithmainvariable

x

kvanishesonacell

oftheCADof

R

k−1_{;aconditionthatcanbecheckedwhenlifting.Inputwiththispropertyisknown}

as well-oriented. Notethat although McCallum’s operator produces order-invariant CADs,a stronger property thansign-invariance,itisactuallymoreefficientthatthepre-existingsign-invariant opera-tors.WeexaminethecomplexityofCADusingthisoperatorinSection2.3.

2.2. Review: CAD invariant with respect to an equational constraint

ThemainresultunderlyingCADsimplificationinthepresenceofanECfollows.

Theorem5.

(See

McCallum,1999.) Let f

(

x

),

g(x

)

be integral polynomials with positive degree in xn, let r(x1

,

. . . ,

xn−1

)

be their resultant, and suppose r

=

0.

Let S be a connected subset of

R

n−1such that f is

delineable on S and r is order-invariant in S.

Then g is sign-invariant in every section of f over S.

Fig. 2givesagraphicalrepresentationofthequestionansweredbyTheorem 5.Hereweconsider polynomials f

(x, y,

z) and g(x, y, z) of positive degree in z whose resultant r is non-zero, and a connected subset S

⊂ R

2 _inwhich

_{r is}

_{order-invariant.We furthersuppose that f is delineableon}

S (noting that Theorem 4with n

=

3 and A

= {

f

}

provides sufficient conditions forthis). We ask whether

g is

sign-invariantinthesectionsof f over S. Theorem 5answersthisquestionaffirmatively: therealvarietyof

g either

alignswithagivensectionof f exactly (asforthebottomsectionof f in Fig. 2),orhasnointersectionwithsuchasection(asforthetop).Thesituationatthemiddlesection of f cannot happen.

Theorem 5 thus suggestsa reduction oftheprojection operator P relative to an EC f

=

0: take only P

(

f

)

together withtheresultantsof f with thenon-ECs.Let A be asetofpolynomials, E

⊂

A containonlythepolynomialdefiningtheEC,

F be

asquarefreebasisof A, and

B be

thesubsetof F whichisasquare-freebasisfor

E.

Theoperator

Fig. 2. Graphical representation ofTheorem 5.

waspresentedbyMcCallum (1999)alongwithanalgorithmtoproduceaCADtruth-invariantforthe ECandsign-invariantfortheotherpolynomialswhentheECwassatisfied.Itworkedbyapplyingfirst PE

(A)

andthenbuildingan order-invariantCADof

R

n−1 using P . Wecallsuch CADs

invariant with

respect to an equational constraint. Note that aswithMcCallum (1999) thealgorithm only worksfor inputsatisfyinga well-orientednesscondition.Full detailsoftheverificationaregivenby McCallum (1999)andacomplexityanalysisisgiveninthenextsubsection.

2.3. New complexity analyses

We provide complexity analyses of the algorithms fromMcCallum (1998, 1999) forcomparison withournewcontributionslater.Ananalysisforthelatterhasnotbeenpublishedbefore,whilethe analysisfor the former differs substantially from the one in McCallum (1985): instead offocusing on computation time, we examine the numberof cells in theCAD of

R

n produced: the cell count. Wecomparethedominanttermsinacellcountboundforeachalgorithmstudied.Thisfocusavoids calculationswithlessrelevantparameters, identicalforallthealgorithms. Wenotethat allCAD ex-perimentationshowsastrongcorrelationbetweenthenumberofcellsproducedandthecomputation time.

Ourkeyparametersarethenumberofvariables

n,

thenumberofpolynomials

m and

their maxi-mumdegree

d (in

anyonevariable).Notethattheseareallrestrictedtopositiveintegervalues.We makemuchuseofthefollowingconcepts.

Definition6. Consider a set of polynomials pj. The combineddegree ofthe set is the maximum

degree (taken with respect to each variable) of the product of all the polynomials in the set: maxi



deg_xi



_jpj



.

Soforexample,theset

{

x2

₊

₁

_,

_x2

₊

_y3

_}

_{hascombineddegree 4 (sincetheproduct hasdegree 4}

in

x and

degree3 in y).

Definition7.(SeeMcCallum, 1985.)Asetofpolynomialshasthe

(

m

,

d

)

-property ifit canbe parti-tionedinto

m sets,

suchthateachsethasmaximumcombineddegree

d.

Soforexample,thesetofpolynomials

{

xy3

₋

_{x, x}4

₋

_{xy, x}4

₋

_y4

₊

₁

_}

_{hascombineddegree9 and}

thusthe

(

1

,

9

)

-property.However,bypartitioningitintothreesetsofonepolynomialeach,italsohas the

(

3

,

4

)

-property.Partitioninginto2setswillshowittohavethe

(

2

,

5

)

,

(

2

,

7

)

and

(

2

,

8

)

-properties also.Thefollowingresultfollowsfromthedefinitions.

Thiscontrastswiththefactsthattakingasquare-freebasismaynotreducethecombineddegree, butmaycauseexponentialblow-upinthenumberofpolynomials.

Proposition9.

Suppose a set has the (m,

d)-property. Then, by taking the union of groups of sets from the partition, it also has the



m



, d

-property.

Notethatinthecase



=

2 wehave



m₂

=



m+1 2



.

Example10. Let S

= {

x2_y4

₋

_x3

_,

_x2_y4

₊

_x3

_}

_{be a set of polynomials. Then S has the}

₍

₂

_,

₄

₎

_and

(

1

,

8

)

-properties. A squarefree basis of S is given by S

= {

x2

_{, y}

4

₋

_{x, y}4

₊

_x

_}

_{which hasthe}

₍

₃

_,

₄

₎

and

(

1

,

8

)

-properties.

Proposition 9statesthat Smustalsohavethe

(

2

,

8

)

-property,whichcanbecheckedby partition-ing

S

sothat

x

2 _{isinasetofitsown.However,fromProposition 8wealsoknowthat S}_musthave

the

(

2

,

4

)

-property,whichisobtainedfromeitheroftheotherpartitionsintotwosets.

S demonstratesthestrengthofthe

(m,

d)-property.Thetrivialpartitionintosetsofone polyno-mial isequivalent tothesimpleapproachofjusttrackingthenumberofpolynomialsandmaximum degree.Inthisexamplesuchanapproachwouldleadusto3polynomialsofdegree4,contributinga possible12realroots.However,byusingmoresophisticatedpartitionswereplacethisby2sets,for eachofwhichtheproductofpolynomialentrieshasdegree4,andsoatmost8realrootscontributed. Thoughnotusedinthispaper,wenoteanadvantageofthe

(m,

d)-propertyoverthe

(

1

,

md) -prop-ertyisabetterboundonrootseparation:anytworootsrequire

O

(

2d

)

subdivisionstoisolate,rather thanthe O(md)impliedbyconsideringtheproductofallpolynomials.

Wealsorecallthefollowingclassicidentitiesforpolynomials f

,

g, h:

res

(

f g

,

h

)

=

res

(

f

,

h

)

res

(

g

,

h

)

;

(6)

disc

(

f g

)

=

disc

(

f

)

disc

(

g

)

res

(

f

,

g

)

2

;

(7)

disc

(

f

)

= (−

1

)

12d(d−1)1

adres

(

f

,

f



₎

₍₈₎

where

d is

thedegree of f , f itsderivative and

a

d itsleading coefficient(alltakenwithrespectto

thegivenmainvariable).

Lemma11.

Suppose A is a set of polynomials in n variables with the (m,

d)property. Then P

(A)

has the

(M,

2d2

)

property with M

=

(

m

+

1

)

2 2



.

(9)

Proof. Partition A as S1

∪ · · · ∪

Sm accordingtoits

(m,

d)-property. LetB be asquare-freebasis for

prim

(A)

, T1 thesetofelementsof B which dividesomeelementof

S

1,andTi bethoseelementsof B which dividesomeelementof

S

i butwhichhavenotalreadyoccurredinsome

T

j

:

j <i.

(1) Wefirstclaimthateachset

cont

(

Si

)

∪

ldcf

(

Ti

)

∪

disc

(

Ti

)

∪

res

(

Ti

)

(10)

for

i

=

1

,

. . . ,

m has the

(

1

,

2d2

)

property.Let

c be

theproductoftheelementsofcont

(S

i

)

,Ti

=

{

F1

,

. . . , F

t

}

forsome

t

and F

:=

c F1

,

. . .

Ft. Then F divides the product of the elements of Si

and so hasdegree atmost

d.

Thus res

(F, F



)

musthave degree at most2d2 because it isthe determinant ofa

(

2d

−

1

×

2d

−

1

)

matrix inwhich each element hasdegree atmost d. Then

by (8) and repeated applicationof (6) and (7) we see res

(F

, F



)

is a (non-trivial) power of c multipliedby



t

j=1ldcf

(

Fj

)



tj=1disc

(

Fj

)



tj<kres

(

Fj

,

Fk

)

2

.

Sincethisincludesalltheelementsof(10)theclaimisproved.

(2) Wearestillmissingfrom

P

(A)

theres

(

f

, g)

where f

∈

Ti

, g

∈

Tjand

i

=

j. Forfixed

i, j consider

res



_f∈Ti f

,



_g∈Tjg



,whichby(6)istheproductofthemissingresultants.Thisistheresultant oftwo polynomialsofdegreeatmost

d and

hencewillhavedegree atmost2d2.Thus forfixed i, j the setofmissingresultantshasthe

(

1

,

2d2

₎

_{-property,andsotheunionofallsuchsets the}



1

2m(m

−

1

),

2d

2



_-property.

(3) Weare now missingfrom P

(A)

onlythe non-leadingcoefficientsof B. The polynomialsin the set Ti havedegree atmostd when multipliedtogether,andso,separately or together,haveat

most

d non-leading coefficients,

eachofwhichhasdegreeatmost

d.

Hencethissetof

non-leading

coefficientshas the

(

1

,

d2

₎

_{property. This is thecase for i from 1 to m and thus together the}

non-leadingcoefficientsof

B have

the

(m,

d2

₎

_{-property.Wecan thenpairupthesesetstogeta}

partitionwiththe

(



m/2

,

2d2

)

-property(Proposition 9). Hence P

(A)

canbepartitionedinto

m

+

m

(

m

−

1

)

2

+



_m 2



=

m

(

m

+

1

)

2

+

m

+

1 2



=

(

m

+

1

)

2 2



sets (where the final equality follows fromm(m

+

1

)

always beingeven) each withcombined de-gree 2d2.

2

This concerns a single projection, and we must apply it recursively to consider the full set of projectionpolynomials.Weakeningtheboundasinthefollowingallowsforaclosedformsolution.

Corollary 12. If A is a set of polynomials with the

(m,

d) property where m >1,

then

P

(A)

has the

(m

2

,

2d2

)-property.

Remark13.

(1) Notethatif

A has

the

(

1

,

d)-propertythen

P

(A)

hasthe

(

2

,

2d2

)

propertyandhencetheneedfor m >1 to applyCorollary 12.Asourpapercontinueswe presentnewtheory thatappliesto the firstprojectiononly.HenceforafairandaccuratecomplexitycomparisonwewilluseLemma 11 forthefirst projection andthenCorollary 12 forsubsequentones, (applicablesince evenifwe startwith

m

=

1 polynomialforthefirstprojection,wecanassume

m

≥

2 thereafter).

(2) Theanalysis sofar resemblesSection 6.1of McCallum (1985).However, that thesisleads usto the

(m

2_{d, 2d}2

₎

_{-propertyinplaceofCorollary 12.Theextradependencyon}

_{d was}

_avoidedbyan

improvedanalysisintheproofofLemma 11part(3).

Weconsiderthegrowthinprojectionpolynomialsandtheirdegreewhenusingtheoperator P in Table 1.Herethecolumnheadingsrefernottothenumberofpolynomialsandtheirdegree,buttothe numberofsetsandtheircombineddegreewhenapplyingDefinition 7.Westartwith

m polynomials

ofdegree

d and

afteroneprojectionhaveasetwiththe

(M,

2d2

₎

_{property,using}

_{M from}

_{Lemma 11.}

WethenuseCorollary 12tomodelthegrowthinsubsequentprojections,andasimpleinductionto fillinthetable.

ThesizeoftheCADproduceddependsonthenumberofrealrootsoftheprojectionpolynomials. Wecanhenceboundthenumberofrealrootsinasetofpolynomialswiththe

(m,

d)-propertywith md (in practicemanyofthemwillbestrictlycomplex).Wecan thereforeboundthenumberofreal rootsoftheunivariateprojectionpolynomialsbytheproductofthetwoentriesintherowofTable 1

Table 1

ExpressiongrowthforCADprojectionwhere:afterthefirstprojectionwehavepolynomialswiththe(M,2d2_{)-propertyand}

thereafterwemeasuregrowthusing

Corollary 12

.ThevalueofM couldbe

(9)

,

(13)

,

(18)

,

(24)

or

(29)

dependingonwhich projectionschemeweareanalysing.

Variables Number Degree Product

n m d md n−1 M 2d2 _2Md2 n−2 M2 _8d4 ₂3_M2_d4 n−3 M4 _128d8 ₂7_M4_d8 . . . . . . . . . . . . n−r M2r−1 22r₋₁ d2r 22r₋₁ d2r M2r−1 . . . . . . . . . . . . 1 M2n−2 22n−1₋₁ d2n−1 22n−1₋₁ d2n−1 M2n−2 Product M2n−1₋₁ m 22n_−1−n d2n₋₁ 22n_−n−1 d2n₋₁ M2n−1₋₁ m

for1variable.The numberofcells intheCADof

R

1 _{isboundedbytwice thisplus1.Similarly,the}

totalnumberofcellsintheCADof

R

n isboundedbytheproductof2K

+

1 where

K varies

through theProductcolumnofTable 1,i.e.by

(

2Md

+

1

)

n

_

−1 r=1



2



22r−1d2rM2r−1



+

1



.

Omittingthe

+

1 willleaveuswiththedominanttermofthebound,whichcanbecalculated explic-itlyas

22n−1d2n−1M2n−1−1m (11)

≤

22n−1d2n−1



1₂

(

m

+

1

)

2



2n−1−1

m

=

22n−1d2n−1

(

m

+

1

)

2n−2m

,

(12)

where the inequality was introduced by omitting thefloor function in (9).This may be compared with the bound inTheorem 6.1.5 of McCallum (1985), withthe main differencesexplained by Re-mark 13(2).

WenowturnourfocustoCADinvariantwithrespecttoanEC.Recallthatweuseoperator PE

(A)

for the first projection only and P

(A)

thereafter. Hence we use Corollary 12 for the bulk of the analysis,andthenextlemmawhenconsideringthefirstprojection.

Lemma14.

Suppose A is a set of m polynomials in n variables each with maximum degree d, and that E

⊆

A contains a single polynomial. Then the reduced projection PE

(

A)has the (M, 2d2

)-property with

M

=



1 2

(

3m

+

1

)



.

(13)

Proof. Since E contains asinglepolynomialitssquarefreebasis

F has

the

(

1

,

d)-property.

(1) Thecontents,leadingcoefficientsanddiscriminantsfromF form aset R1 withcombineddegree

2d2 (seeproofofLemma 11step1)andtheothercoefficientsaset R2 withcombineddegree

d

2

(seeproofofLemma 11step3).

(2) The set of remaining contents R3

=

cont

(A)

\

cont

(E)

has the

(m

−

1

,

d)-property and thus

trivially, the

(m

−

1

,

d2

)

-property. Then R2

∪

R3 has the

(m,

d2

)

-property and thus also the





m

2

,

2d2

-property(Proposition 9).

(3) Itremainstoconsiderthefinalsetofresultantsin(5).Followingtheapproachfromtheproofof Lemma 11step2,we concludethatforeachof

m

−

1 polynomialsin A

\

E there contributesa setwiththe

(

1

,

2d2

₎

_{-property.Sotogethertheyformaset R}

Hence PE

(A)

iscontainedin

R

1

∪ (

R2

∪

R3

)

∪

R4 whichmaybepartitionedinto 1

+



m₂

+ (

m

−

1

)

=



1 2

(

m

+

1

)



+

m

=



1 2

(

3m

+

1

)



setsofcombineddegree2d2.

2

WecanuseTable 1tomodelthegrowthinprojectionpolynomialsforthealgorithminMcCallum (1999) as well, since the only difference will be the numberof polynomialsproduced by thefirst projection,andthusthevalue ofM. Hence thedominanttermintheboundonthetotalnumberof cellsisgivenagainby(11),whichinthiscasebecomes(uponomittingthefloor)

22n−1d2n−1

(

1₂

(

3m

+

1

))

2n−1−1m

=

22n−1d2n−1

(

3m

+

1

)

2n−1−1m

.

(14)

Since

P

E

(A)

isasubsetof

P

(

A)aCADinvariantwithrespecttoanECshouldcertainlybesimpler

thanasign-invariantCADforthepolynomialsinvolved.Indeed,comparingthedifferentvaluesof M weseethat

1 2

(

m

+

1

)

2

_>

1

2

(

3m

+

1

)

(

strictly so for m

>

1

).

Comparingthe dominanttermsinthecell countbounds,(14) and(12),weseethemaineffectisa decreaseinoneofthedoubleexponentsby1.

3. AprojectionoperatorforTTICAD

3.1. New projection operator

InMcCallum (1999)thecentralconceptisthereducedprojectionofasetofpolynomials

A relative

toasubset E (defining the EC).Thefull projectionoperatorisapplied to E and then supplemented bytheresultantsofpolynomialsin

E with

thosein E

\

A, sincethelattergrouponlyeffectthetruth oftheformulawhen theysharea rootwiththeformer.We extendthisideato define aprojection foralist ofsetsof polynomials(derivedfromalist offormulae),some ofwhich mayhavesubsets (derivedfromECs).

ForsimplicityinMcCallum (1999)theconceptisfirstdefinedforthecasewhen

A is

anirreducible basis.Weemulatethisapproach,generalisingforothercasesbyconsideringcontentsandirreducible factorsofpositive degreewhen verifyingthealgorithm inSection4. Solet

A

= {

Ai

}

ti=1 bea listof

irreduciblebases Ai andlet

E = {E

i

}

t_i=1bealistofsubsets

E

i

⊆

Ai.Put

A

=



ti=1Aiand

E

=



ti=1Ei.

Notethat weusetheconventionofuppercaseRoman lettersforsetsofpolynomialsandcalligraphic lettersforlistsofthese.

Definition15.Withthenotationabovethe

reduced projection of A

with respect to E is

P_E

(

A

)

:=



t_i=1PEi

(

Ai

)

∪

RES×

(

E

)

(15)

whereRES×

(E)

isthecrossresultantset

RES×

(

E

)

= {

resxn

(

f

, ˆ

f

)

| ∃

i

,

j such that f

∈

Ei

, ˆ

f

∈

Ej

,

i

<

j

,

f

= ˆ

f

}

(16)

and PE

(

A

)

=

P

(

E

)

∪



resxn

(

f

,

g

)

|

f

∈

E

,

g

∈

A

,

g

∈

/

E



,

P

(

A

)

= {

coeffs

(

f

),

disc

(

f

),

resxn

(

f

,

g

)

|

f

,

g

∈

A

,

f

=

g

}.

Theorem16.

Let S be a connected submanifold of

R

n−1_{. Suppose each element of P}

E

(A)

is order invariant

in S. Then each f

∈

E either vanishes identically on S or is analytically delineable on S; the sections over S of the f

∈

E which do not vanish identically are pairwise disjoint; and each element f

∈

E which does not vanish identically is order-invariant in such sections.

Moreover, for each i, in 1

≤

i

≤

t every g

∈

Ai

\

Eiis sign-invariant in each section over S of every f

∈

Ei which does not vanish identically.

Proof. Thecrucialobservationforthefirstpartisthat

P

(E)

⊆

P_E

(

A)

.Toseethis,recallequation(15) andnotethatwecanwrite

P

(

E

)

=



_iP

(

Ei

)

∪

RES×

(

E

).

We canthereforeapply Theorem 4 totheset E and obtain thefirstthree conclusionsimmediately, leavingonlythefinalconclusiontoprove.

Let i be in the range1

≤

i

≤

t, let g

∈

Ai

\

Ei and let f

∈

Ei. Suppose that f does not vanish

identicallyon S. Nowresxn

(

f

, g)

∈

PE

(A)

,andsoisorder-invariant in S by hypothesis. Further,we

alreadyconcludedthat f is delineable.ThereforebyTheorem 5, g is sign-invariantineachsectionof f over S.

2

Theorem 16isthekeytoolfortheverificationofourTTICADalgorithminSection4.Itallowsus to concludetheoutputiscorrectsolongasno f

∈

E vanishes identicallyonthelowerdimensional manifold, S. Apolynomial f in r variables thatvanishesidenticallyatapoint

α

∈ R

r−1_issaidtobe

nullified at α.

The theory ofthis subsectionappears identical to the work in Bradfordet al. (2013a). The dif-ference isintheapplicationofthetheory inSection4.We supposethatthe inputisa listofQFFs,

{φ

i

}

,witheach Ai definedfromthepolynomialsineach

φ

i.InBradfordetal. (2013a)therewas an

assumption(nolongermade)thateachoftheseformulaehadadesignatedEC fi

=

0 fromwhichthe

subsets Ei are defined. Instead,we define Ei to bea basis for

{

fi

}

ifthereis sucha designatedEC

anddefine Ei

=

Ai otherwise.Thatis,we needtotreatallthepolynomialsinQFFswithnoECwith

theimportanceusuallyreservedforECs.

3.2. Comparison with using a single implicit equational constraint

It isclearthat ingeneralthe reducedprojection P_E

(

A)

willlead tofewer projection polynomi-als than usingthe full projection P . However, a comparisonwiththe existing theory ofequational constraintsrequiresalittlemorecare.

First,wenotethattheTTICADtheoryisapplicabletoasequenceofformulaewhilethetheoryof McCallum (1999)isapplicableonlytoasingleformula.HenceifthetruthvalueofeachQFFisneeded thenTTICADistheonlyoption;atruth-invariantCADforaparentformulawillnotnecessarilysuffice. Second wenote thateven ifthesequencedoforma parentformulathen thismusthavean overall ECtouseMcCallum (1999)whiletheTTICADtheoryisapplicableevenifthisisnotthecase.

Letusconsider thesituationwhereboth theoriesare applicable, i.e.we havea sequenceof for-mulae (formingaparentformula)forwhicheach hasanECandthustheparentformulaanimplicit EC (theirproduct). Inthe contextof Section1.2thiscorresponds to using



_ifi astheEC.The

im-plicitECapproachwouldcorrespondtousingthereducedprojection PE

(

A)ofMcCallum (1999),with E

= ∪

iEiandA

= ∪

iAi.Wemakethesimplifyingassumptionthat

A is

anirreduciblebasis.Ingeneral P_E

(A)

willstillcontain fewerpolynomialsthan PE

(A)

since PE

(A)

contains allresultantsres

(

f

, g)

where f

∈

Ei

,

g

∈

Aj(and

g

∈

/

E), while

P

E

(A)

containsonlythosewith

i

=

j (and g

∈

/

Ei).Thuseven

insituationswheretheprevious theoryappliesthereisan advantageinusingthenewTTICAD the-ory.Thesesavingsarehighlightedbytheworkedexamplesinthenextsubsectionandthecomplexity analysislater.

3.3. Worked examples

InSection4wedefineanalgorithmforproducingTTICADs.Firstweillustratethesavingswithour workedexamplesfromSection1.3,whichsatisfythesimplifyingassumptionsfromSection3.1.

Westartbyconsidering



fromequation(2).Inthenotationabovewehave:

A1

:= {

f1

,

g1

},

E1

:= {

f1

};

A2

:= {

f2

,

g2

},

E2

:= {

f2

}.

Fig. 3. Thepolynomialsfrominequation

(2)

alongwiththerootsofPE(A)(solidlines),PE(A)(dashedlines)and P(A)

(dottedlines).

Fig. 4. Magnified region ofFig. 3.

PE1

(

A1

)

=



x2

−

1

,

x4

−

x2

+

₁₆1



,

PE2

(

A2

)

=



x2

−

8x

+

15

,

x4

−

16x3

+

95x2

−

248x

+

3841₁₆



,

andthecross-resultantset

Res×

(

E

)

= {

resy

(

f1

,

f2

)

} = {

68x2

−

272x

+

285

}.

P_E

(

A)

is then the union ofthese three sets. InFig. 3 we plot the polynomials(solid curves) and identifythe12realsolutionsof P_E

(

A)

(solidverticallines).Wecanseethesolutionsalignwiththe asymptotesofthe fi’sandtheimportantintersections(thoseof f1 withg1and f2 with

g

2).

If we were to instead use a projection operator based on an implicitEC f1f2

=

0 then in the

notationabovewewouldconstructPE

(

A)fromA

= {

f1

, f

2

,

g1

, g

2}and

E

= {

f1

, f

2}.Thissetprovides

anextra4solutions(thedashedverticallines)whichalignwiththeintersectionsof f1 with g2 and

f2with

g

1.Finally,ifweweretoconsider

P

(A)

thenwegainafurther4solutions(thedottedvertical

lines) whichalign withthe intersectionsof g1 and g2 andtheasymptotes ofthe gi’s.In Fig. 4we

magnifyaregiontoshowexplicitlythatthepointofintersectionbetween f1 and

g

1 isidentifiedby

P_E

(A)

,whiletheintersectionsof g2 withboth f1and

g

1 areignored.

The 1-dimensionalCADproduced using P_E

(A)

has 25cells compared to 33when using PE

(A)

and41whenusing

P

(A)

.However,itisimportanttonotethatthisreductionisamplifiedafterlifting (usingTheorem 16andAlgorithm 1).The2-dimensionalTTICADhas105cells andthesign-invariant CADhas317.Using Qepcad tobuildaCADinvariantwithrespecttotheimplicitECgivesus249cells.

Fig. 5. The polynomials fromin equation(3)along with the roots of PE(A).

Fig. 6. Magnified region ofFig. 5.

Nextweconsiderdeterminingthetruthof



fromequation(3).Thistime

A1

:= {

f1

,

g1

},

E1

:= {

f1

},

A2

:= {

f2

,

g2

},

E2

:= {

f2

,

g2

},

andso

P

E1

(A

1

)

isasabovebut

P

E2

(A

2

)

containsanextrapolynomial

x

−

4 (thecoefficientofy in g2).

Thecross-resultantsetRES×

(E)

alsocontainsanextrapolynomial,

resy

(

f1

,

g2

)

=

x4

−

8x3

+

16x2

+

1₂x

−

31₁₆

.

These two extra polynomialsprovidethree extra realroots andhencethe 1-dimensionalCAD pro-ducedusing P_E

(

A)

thistimehas31cells.

In Fig. 5 we againgraphthe fourcurvesthistime withsolid vertical lineshighlighting the real solutions of P_E

(

A)

.BycomparingwithFig. 3weseethat morepointsintheCADof

R

1 havebeen identifiedfortheTTICADof



thantheTTICADof



(15insteadof12)butthatthereisstillasaving over the sign-invariant CAD(which had20,the five extrasolutions indicated by dotted lines). The lackofanECinthesecond clausehasmeant thattheasymptoteof g2 anditsintersectionswith f1

havebeenidentified.However, notethat theintersectionsof g1 with f2 and g2 andhavenot been.

Fig. 6magnifiesaregionofFig. 5.ComparewithFig. 4toseethedashedlinehasbecomesolid,while thedottedlineremainsunidentifiedbytheTTICAD.

Notethat weare unabletouseMcCallum (1999) tostudy



asthereisnopolynomial equation logicallyimplied(eitherexplicitlyorimplicitly)bythisformula.Hencetherearenodashedlinesand the choiceisbetweenthe sign-invariantCADwith317 cellsorthe TTICAD,whichforthisexample has183cells.

Algorithm 1: TTICADalgorithm.

Input : Alistofquantifier-freeformulae{φi}ti=1invariablesx1,. . . ,xn.EachφihasatmostonedesignatedEC fi=0. Output: Either• D:ACADofRn_{(describedbylistsI andS ofcellindicesandsamplepoints)whichistruthtable}

invariantforthelistofinputformulae;or •FAIL:IfAisnotwell-orientedwithrespecttoE(Definition 18).

1 for i=1. . .t do

2 IfthereisnodesignatedECthensetEi:=AiandotherwisesetEi:= {fi};

3 ComputethefinestsquarefreebasisFiforprim(Ei);

4 SetF← ∪t i=1Fi;

5 if n=1 then

6 IsolatetherealrootsofthepolynomialsinF andthusformcellindicesandsamplepointsforaCADofR;

7 return I andS forD;

8 else

9 for i=1. . .t do

10 ExtractthesetAiofpolynomialsinφi;

11 ComputethesetCiofcontentsoftheelementsofAi;

12 ComputethesetBi,thefinestsquarefreebasisforprim(Ai);

13 SetC:= ∪t

i=1Ci,B:= (Bi)it=1andF:= (Fi)ti=1;

14 ConstructtheprojectionsetP:=C∪PF(B);

15 Attempttoconstructalower-dimensionalCAD:w,I,S:= CADW(n−1,P);

16 if w=false then

17 return FAIL (sincePisnotwelloriented);

18 I← ∅;S← ∅;

19 for eachcellc∈ Ddo

20 Lc← {};

21 for i=1,. . .t do

22 if anyf∈Eiisnullifiedonc then

23 if dim(c)>0 then

24 return FAIL (since{φi}ti=1isnotwelloriented);

25 else

26 Lc←Lc∪Bi;

27 else

28 Lc←Lc∪Fi;

29 Generateastackoverc usingLc:constructcellindicesandsamplepointsforthestackoverc ofthe

polynomialsinLc,addingthemtoI andS ;

30 return I andS forD;

4. Algorithm

4.1. Description and proof

We describe carefullyAlgorithm 1. Thiswill createa TTICAD of

R

n fora listof QFFs

{φ

i

}

t_i=1 in

variables x

=

x1

≺

x2

≺ · · · ≺

xn, where each

φ

i has at mostone designated EC fi

=

0 of positive

degree(theremaybeothernon-designatedECs).

Itusesasubalgorithm

CADW

,whichwasvalidatedby McCallum (1998).Theinputof

CADW

is:

r,

apositiveintegerand A, asetof

r-variate

integralpolynomials.The outputisaboolean

w which

if trueisaccompanied byan order-invariant CADfor A (represented asa listofindices I and sample points S).

Let Ai bethesetofall polynomialsoccurringin

φ

i.If

φ

i hasadesignatedECthenput Ei

= {

fi

}

andifnotput

E

i

=

Ai.Let

A

and

E

bethelistsofthe

A

iand

E

irespectively.Ouralgorithmeffectively

definesthereducedprojectionof

A

withrespectto

E

intermsofthespecialcaseofthisdefinition fromtheprevioussection.Thedefinitionamountsto

Here

C is

thesetofcontentsofalltheelementsofall

A

i;

B

thelist

{

Bi

}

t_i=1suchthat

B

iisthefinest2

squarefreebasisforthesetprim

(A

i

)

ofprimitivepartsofelementsof

A

i whichhavepositivedegree;

and

F

is thelist

{

Fi

}

ti=1,such that Fi is thefinest squarefreebasis for prim

(E

i

)

.(Thereader may

notice thatthisnotation andthe definitionof P_E

(

A)

hereisanalogoustothework inSection5 of McCallum,1999.)

We shallprovethat,providedtheinputsatisfiesthecondition ofwell-orientedness givenin Def-inition 18, theoutput ofAlgorithm 1 is indeeda TTICADfor

{φ

i

}

. Wefirst recall themore general

notionofwell-orientednessfromMcCallum (1998).Thebooleanoutputof

CADW

isfalseiftheinput setwasnotwell-orientedinthissense.

Definition17.A set A of n-variate polynomialsissaid tobe

well oriented if

whenever n >1,every f

∈

prim

(A)

is nullified by at most a finite number of points in

R

n−1, and (recursively) P

(A)

is well-oriented.

Thisconditionisrequiredfor

CADW

sincethevalidityofthisalgorithmreliesonTheorem 4which holdsonlywhenpolynomialsdonotvanishidentically.Theconditionsallowsfora finitenumberof thesenullificationssincethisindicates aproblemonazerocell,thatisasinglepoint.Insuchcases it is possible to replacethe nullified polynomial by a so called delineating polynomial which is not nullifiedandcanbeusedinplacetoensurethedelineabilityoftheother.Theuseoftheseispartof theverifiedalgorithm

CADW

(McCallum,1998) andtheyarestudiedindetailbyBrown (2005).

Wenowdefineournewnotionofwell-orientednessforthelistsofsets

A

and

E

.

Definition18.Wesay that

A

is

well oriented with respect to

E

if,whenever

n >

1,every polynomial f

∈

E is nullifiedby atmosta finitenumberofpoints in

R

n−1,and P_F

(

B)

is well-orientedinthe senseofDefinition 17.

ItisclearthanAlgorithm 1terminates.Wenowprovethatitiscorrectusingthetheorydeveloped inSection3.

Theorem19.

The output of

Algorithm 1is as specified.

Proof. We must show that when the input is well-oriented the output is a TTICAD, (each

φ

i has

constanttruthvalueineachcellof

D

),and FAIL otherwise.

Iftheinputwasunivariatethenitistriviallywell-oriented.ThealgorithmwillconstructaCAD

D

of

R

1 _{usingtherootsoftheirreduciblefactorsofthepolynomialsin E (steps 6to7).Ateach0-cell}

all the polynomialsin each

φ

i trivially haveconstant signs,and henceevery

φ

i hasconstant truth

value. Ineach 1-cellno EC can change signand so every

φ

i hasconstant truth value false, unless

therearenoECsinanyclause.InthiscasethealgorithmwouldhaveconstructedaCADusingallthe polynomials andhence oneach 1-cellno polynomialchanges signandsoeach clause hasconstant truthvalue.

From nowonsuppose

n >

1.If

P

=

C

∪

P_F

(B)

isnotwell-orientedinthesense ofDefinition 17 then

CADW

returns

w

asfalse.InthiscasetheinputisnotwellorientedinthesenseofDefinition 18 and Algorithm 1 correctly returns FAIL in step 17. Otherwise, we have w

=

true with I and S specifying a CAD,

D

, which is order-invariant withrespect to

P

(by the correctness of

CADW

, as proved in McCallum, 1998). Letc, a submanifoldof

R

n−1_{,be a cell of}

_D

 _andlet

_α

_{be its sample}

point.

We supposefirstthatthedimensionof

c is

positive.Ifanypolynomial f

∈

E vanishes identically on c then the input is not well orientedin thesense ofDefinition 18 andthe algorithm correctly returns FAIL atstep24.Otherwise,weknowthattheinputlistwascertainlywell-oriented.Sinceno polynomial f

∈

E vanishes thennoelementofthebasis F vanishes identicallyon

c either.

Hence,by

Theorem 16,appliedwith

A

=

B

and

E = F

,eachelementof F is delineableon

c,

andthesections over

c of

theelementsofF are pairwisedisjoint.Thusthesectionsandsectorsover

c of

theelements of F comprise astack



over

c.

Furthermore,thelast conclusionofTheorem 16assures usthat,for each i, every element of Bi

\

Fi is sign-invariantineach section over c of every element of Fi.Let

1

≤

i

≤

t. Weshallshowthateach

φ

i hasconstanttruthvalueinboththesectionsandsectorsof



.

If

φ

ihasadesignatedECthenlet fi denotetheconstraintpolynomial;otherwiselet fi denotean

arbitraryelementofAi.

Consider firsta section

σ

of



. Now fi is aproduct ofits content cont

(

fi

)

andsome elements

ofthebasis Fi.Butcont

(

fi

)

,anelementof

P

,issign-invariant(indeedorder-invariant)inthewhole

cylinder

c

× R

andhence,inparticular, in

σ

.Moreover all oftheelements of Fi are sign-invariant

in

σ

,as was noted previously. Therefore fi issign-invariant in

σ

. If

φ

i has no constraint (andso fi denotesan arbitraryelement of Ai) then thisimpliesthat

φ

i hasconstant truth value in

σ

. So

considerfromnowonthecaseinwhich fi

=

0 isthedesignatedconstraintpolynomialof

φ

i.

If fiispositiveornegativein

σ

then

φ

ihasconstanttruthvalue

false in σ

.Sosupposethat fi

=

0

throughout

σ

.Itfollowsthat

σ

mustbeasectionofsomeelementofthebasis Fi.Let

g

∈

Ai

\

Eibe

anon-constraintpolynomialinAi.Now,bythedefinitionof

B

i,wesee g can bewrittenas

g

=

cont

(

g

)

hp1

1

· · ·

h

pk

k

where

h

j

∈

Bi

, p

j

∈ N

.Butcont

(g)

,in

P

,issign-invariant(indeedorder-invariant)inthewhole

cylin-der c

× R

, and hence in particular in

σ

. Moreover each hj is sign-invariant in

σ

, as was noted

previously.Hence g is sign-invariantin

σ

.(Notethat inthecasewhere g does nothavemain vari-able

x

n then

g

=

cont

(g)

andtheconclusionstillholds.) Since g was anarbitraryelementof Ai

\

Ei,

itfollowsthat allpolynomialsin Ai are sign-invariantin

σ

,hencethat

φ

i hasconstant truthvalue

in

σ

.

Nextconsider asector

σ

ofthestack



,andnotice that atleastone such sector exists.As ob-servedabove,cont

(

fi

)

issign-invariantin

c,

and fidoesnotvanishidenticallyon

c.

Hencecont

(

fi

)

is

non-zerothroughout

c.

Moreovereachelementofthebasis Fiisdelineableon

c.

Hence fiisnullified

bynopointof

c.

Itfollowsfromthisthatthealgorithmdoesnotreturn FAIL duringtheliftingphase. Itfollowsalsothat fi

=

0 throughout

σ

.Hence

φ

i hasconstanttruthvalue

false in

σ

.

It remains to consider the casein which the dimension ofc is 0. In thiscasethe roots of the polynomialsinthe liftingset Lc constructed by thealgorithm determinea stack



over

c.

Each

φ

i

triviallyhasconstanttruthvalueineachsection(0-cell)ofthisstack,andthesamecanroutinelybe shownforeachsector(1-cell)ofthisstack.

2

4.2. TTICAD via the ResCAD set

When no f

∈

E is nullified there is an alternative implementation of TTICAD which would be simpletointroduceintoexistingCADimplementations.Define

R

(

{φ

i

}) =

E

∪



ti=1



resxn

(

f

,

g

)

|

f

∈

Ei

,

g

∈

Ai

,

g

∈

/

Ei



tobethe

ResCAD set of

{φ

i

}

.

Theorem20.

Let A

= (

Ai

)

t_i=1be a list of irreducible bases Aiand let E = (Ei

)

t_i=1be a list of non-empty subsets Ei

⊆

Ai. Then we have

P

(

R

(

{φ

i

})) =

PE

(

A

).

Theproofisstraightforwardandsoomittedhere.

Corollary21.

If no

f

∈

E is nullified by a point in

R

n−1_{then inputting R({φ}

i

})

into any algorithm which pro-duces a sign-invariant CAD using McCallum’s projection operator P will result in the TTICAD for

{φ

i

}

produced by Algorithm 1.

Corollary 21givesasimple waytocompute TTICADsusingexisting CADimplementationsbased onMcCallum’sapproach,suchas Qepcad.