A Preprocessor Based on Clause Normal Forms and Virtual Substitutions to Parallelize Cylindrical Algebraic Decomposition

0.51 MB·English

by  Hari Krishna Malladi

A Parallel Cylindrical Algebraic Decomposition Algorithm
for Quantifier Elimination on Real Closed Fields
Hari Krishna Malladi and Ambedkar Dukkipati
IndianInstituteofScience
Bangalore
India
{harikrishnamalladi,ambedkar}@csa.iisc.ernet.in
ABSTRACT CylindricalAlgebraicDecomposition(CAD)[4],whicheven-
2
tually became the standard algorithm for quantifier elimi-
1 In this paper we propose a parallel approach to quantifier
nation over real closed fields. The time complexity of CAD
0 elimination on real closed fields through a modification in
algorithm is doubly exponential in the number of variables
2 thecylindricalalgebraicdecomposition(CAD)algorithmto
(both quantified and quantifier-free).
accommodate parallelism. In this approach the speed-up
n It is a recursive algorithm, which has a sequence of pro-
due to parallelism obtained on an input formula is a prop-
a jections, followedby a sequence of constructions, leading to
erty of that formula itself. Hence the time complexity of
J
the solution. CAD has undergone an extensive array of de-
algorithm depends on not just the size of the input but on
7 velopments,suchasHong’sPartialCAD[5],Hong’sprojec-
the input self. We classify prenex formulae depending on
2 tion operator [9], Scott McCallum’s projection operator [1],
theamountofinput-basedparallelismthatcanbeobtained
from it based on which we obtain some insights into the application of Gr¨obner Bases to CAD [3] and a first step
] towards parallelism in CAD [11].
M complexity of the algorithm. We demonstrate performance
This makes quantifier elimination through CAD impossi-
of the proposed algorithm with experimental results.
D ble for a wide range of real-world applications. The goal of
the paper is to address this issue and expand the class of
. Keywords
s problems where CAD could be used to perform quantifier
c CylindricalAlgebraicDecomposition,parallelism,quantifier elimination using a reasonable amount of resources.
[
elimination The present work builds on [11] and extends the idea of

2 parallelism to the entire algorithm. We show that paral-
v 1. INTRODUCTION lelism can be introduced in various aspects of CAD. They
2 are, (i) input to CAD can be made into independent units,
Thestudyofrealalgebraicgeometrydealswiththestudy
5 (ii)projectionscanbemadeparallel,and(iii)constructions
of the real roots of an equation as, more often than not,
3 can be made parallel to the maximum possible extent.
the real roots are the most desired solutions to a system of
5 TheinputtoCADcannotalwaysbesplitintoindependent
equations. It is important to note that for some problems
. units,becauseoftheconstraintsimposedbyfirst-orderlogic
2 there are no counterparts in complex algebraic geometry.
1 Three such examples are: (i) Hilbert’s 16th problem, (ii) itself. This motivate us to introduce Skolemization to elim-
1 Hilbert’s17thproblem,and(iii)Tarski-Seidenbergprinciple. inatetheexistentialquantifiers. Anad-hocimprovementto
1 Givenafirst-orderformulawithbothquantifiedandquan- theseschemesistheintroductionofchainsofsubstitutions,
: tifier - free variables, the process of finding an equivalent which have the potential to reduce the number of variables
v insomepolynomials,keepingthefinaltruthvalueconstant.
i first-orderformulainthesequantifier-freevariablesiscalled
X quantifier elimination. When the first-order formula is a These schemes benefit fromthefactthatspeed-up happens
atadoublyexponentialrate. Reductionofjustonevariable
r boolean combination of polynomial equations and inequali-
a ties, we can consider quantifier elimination as a problem in reduces the time complexity greatly. These improvements
are illustrated in the experimental results.
real algebraic geometry.
The paper is organized as follows. The preliminary nota-
ThealgorithmspecifiedbyTarski[12]forquantifierelim-
tionsandtheoriginalCADalgorithmarebrieflyintroduced
ination is highly resource intensive. So newer and more ef-
inSection2. Theideaofseparabilityandtheequi-satisfiable
ficient algorithms have come up and replaced it. The fact
properties of substitutions are described in Section 3. The
thatasemialgebraicsetcanbedividedintoafinitenumber
parallelCADalgorithmitselfisdescribedinSection4. Em-
ofsemialgebraicsets(Tarski-Seidenbergprinciple)hasledto
pirical results for separability and substitutability are de-
scribed in Section 5. A few directions sought for future re-
search are mentioned in Section 6.
Permissiontomakedigitalorhardcopiesofallorpartofthisworkfor
personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare 2. PRELIMINARIES
notmadeordistributedforprofitorcommercialadvantageandthatcopies
bearthisnoticeandthefullcitationonthefirstpage.Tocopyotherwise,to
republish,topostonserversortoredistributetolists,requirespriorspecific 2.1 BasicNotation
permissionand/orafee. Let k[x ...x ] denote the set of all polynomials in vari-
ISSAC’12Grenoble,France 1 n
ables, x ...x and coefficients from the field k. The base
Copyright2012ACMX-XXXXX-XX-X/XX/XX...$10.00. 1 n
field,kisassumedtobethesetofrealnumbers,Rthrough- polynomials in n−1 variables is obtained as output,
outthepaper. Afield,kiscalledalgebraicallyclosed ifevery upon applying a projection operator. This process is
polynomial, p ∈ k[x ,x ,...x ] with coefficients from k has continued till a set of univariate polynomials is ob-
1 2 n
a root in k itself. The best known example is the set of tained.
complex numbers, C.
2. Theseunivariatepolynomialscaneasilybesolved(us-
Definition 2.1. A formula of the form ing Sturm’s theorem iteratively or otherwise). The
rootsoftheseunivariatepolynomialsandtheintervals
(Q1x1)...(Qnxn)[ψ(y1...ym,x1...xn)] between them are taken as the regions in the CAD
of one dimensional real space. Designate a randomly
is called a prenex formula, where Q ∈ {∃,∀} and ψ is a
i
selected point in each region as a sample point.
quantifier-free formula in x ,...x ,y ,...y .
1 n 1 m
3. For the two dimensional CAD, substitute the sample
Definition 2.2. V ⊆R is said to be a semialgebraic set
pointfoundineachone-dimensionalregion,R ineach
if ∃f ...f ∈ R[x ...x ] and g ...g ∈ R[x ...x ] such that
1 s 1 n 1 t 1 n of the polynomials in the 2-dimensional projection to
V = {(a ...a ) ∈ R : f (a ...a ) = 0,i = 1...s,g (a ...a ) ≥
1 n i 1 n j 1 n again get univariate polynomials. Find the roots of
0,i=1...t}.
these polynomials and create two dimensional regions
A connected subset of Rn is called a region in Rn. Con- thatformthestackovertheregionR. Designatesam-
sider a region R and functions f : Rn −→ R satisfying, ple points for these regions as well. Suppose we are
i
given a CAD for k-dimensional real space, we use the
f < f < ... < f. This ordering ensures that these func-
1 2 l
projected set for k+1-dimensions and substitute the
tions do not intersect each other. A f -section is the graph
i
sample points of the k-dimensional CAD in them to
ofthefunctionf . Inotherwords,it’sthesetcontainingall
i
points of the form (a,b), where a ∈Rn−1 and b=f(a). An get univariate polynomials. These can now be solved
(f ,f )-sector isthesetofallpoints(a,b),wherea∈Rn−1 to get k+1-dimensional CAD.
i i+1
and f (a) <b< f (a). A cylinder over a region R is the
i i+1 Now,thisinformationcanbeusedbyanotherrecursiveal-
setofallpoints(a,b),wherea∈Randb∈R. Astack over
gorithm to perform quantifier elimination [6]. Assume that
aregionRisthecollectionofsectionsandsectorsthatoccur
the variables x ,...,x are quantifier-free and x ,...,x
inthecylinderoverR. ApartitionofRn intosemialgebraic 1 k k+1 n
have quantifiers, and let Q be the corresponding quantifier
partitions is called an algebraic decomposition of Rn. i
for i∈{k+1,...,n}. Now we list the QE algorithm.
Definition 2.3. A partition of Rn satisfying the follow-
QE((D ,...,D ),(F ,...,F))
ing two properties is called a cylindrical algebraic decompo- 1 n 1 l
sition: Input: TheCADsofR1 toRn (D ,...,D )andtheformulae
1 n
1. If n = 1, then the CAD is a set of points and open describing the regions of the CAD of Rk (F1,...,Fl).
intervals. Output: The quantifier-free formula equivalent to the input
prenex formula.
2. If n>1, every region of the CAD of Rn−1 has a stack
overit,whichisadisjointsubsetofsomeoftheregions 1. For k≤i<n, we have
of the CAD of Rn.
(a) If Q is ∃, then C ∈ D is valid if at least one
i+1 i
Asetofpolynomials,F issaidtobesigninvariant onthe region in the stack over C is valid.
region R iff no polynomial in F changes its sign anywhere (b) If Q is ∀, then C ∈D is valid if all regions in
i+1 i
on R. the stack over C are valid.
Definition 2.4. A CAD of Rn in which each region is 2. Aregion,C inDn isvalidifψ(tC)isTRUE,wheretC
sign-invariant with respect to an input set of polynomials is is the sample point of that region.
called a sign-invariant CAD.
3. Obtain the cells of D which are TRUE. The dis-
k
The CAD algorithm has two phases in its description, junction of the formulae of these cells is the required
namely projection and construction. Projection is specified quantifier-free formula and is returned.
by a projection operator.
The projection operator takes as input a set of polyno-
2.2 Cylindrical Algebraic Decomposition Al- mials in n variables and returns another set of polynomials
gorithm in n−1 variables. Assume PSRC (P,Q) denotes the jth
j
principal subresultant coefficient of P and Q.
This algorithm takes as input, a set of polynomials in n
Collins’ projection operator takes as input the polynomi-
variables and generates a cylindrical algebraic decomposi-
als A={A ,...,A }. Then it returns (i) The coefficient of
tion of the n-dimensional real space. This algorithm works 1 m
each and every term of each A , (ii) PSRC (U ,U(cid:48)) for ev-
recursivelytoproduceaseriesofprojectionsonlesserdimen- i i j j
ery i and j and, (iii) PSRC (U ,U ) for every i and j <k.
sionsandbuildstheCADofeachdimensionwhilereturning i j k
U represents every element in the reducta set of A .
the recurring functions. An outline of CAD algorithm [7] is i i
presented below.
1. Projecteachpolynomialontolesserdimensions. Poly-
nomials in n variables are taken as input and a set of
3. SOMEOBSERVATIONSTHATLEADUP and
TOPARALLELISM
Y =(Q x )(Q x )...(Q x )[g(x ,x ,...,x )]
r r r+1 r+1 n n g1 g2 gl
3.1 Separability then,
The concept of separability is used when we need to con- d(x ,...x )=X C Y.
1 n
vert one instance of quantifier elimination into multiple in-
Proof. SupposeF ={x ,x ,...,x }andG={x ,x ,...,x }
stances of quantifier elimination. f1 f2 fk g1 g2 gl
and H =F ∩G and H ={x ,x ,...,x }. Then we have
h1 h2 hj
Definition 3.1. Forprenexformulae,f andg,ifwehave, two cases, based on C:
d(x ,...x )=(Q x )...(Q x )f(x ,...,x )Cg(x ,...,x ), 1. C is ∨: The definition of separability gives us that
1 n r r n n f1 fk g1 gl
g can be separated if every x ∈ H does not have the
where Q ∈ {∃,∀},∀i ∈ {r,...,n} and C ∈ {∨,∧}, we say
i universal quantifier. We know that d is satisfied if
thatgisseparablefromdif,supposingF ={x ,x ,...,x }
and G={x ,x ,...,x } and H =F ∩G anf1d f2 fk we choose values for G = {xg1,xg2,...,xgl} in such a
g1 g2 gl way that g is satisfied. Some variables, namely H =
H ={x ,x ,...,x },
h1 h2 hj {xh1,xh2,...,xhj}, are common between f and g, and
1. C is ∨, then every x ∈ H should not be the universal do not have the universal quantifier. If we are able
quantifier, ∀. to choose values for the variables in G or F to satisfy
d, we implicitly chose values for the variables in H to
2. C is ∧, then every x∈H should not be the existential satisfy d. This choice is feasible due to the absence of
quantifier, ∃. universal quantifiers for the variables in H.
3. g does not evaluate to TRUE or FALSE upon quanti- 2. C is ∧: This is the dual of the previous case.
fier elimination, and g has quantifier-free variables.
Essentially, we would be converting
3.2 Parallelism in Projections and Construc-
(Q x )...(Q x )[ψ(x ...x )], Q ∈{∃,∀}
k+1 k+1 n n 1 n i
tions
intotwo(ormore,ifrecursivelyapplied)quantifierelimina-
Projections and constructions form the recursive steps of
tion problems,
the CAD algorithm. So, it’s natural to ask whether these
X C Y, can be parallelized. A step in this direction has been taken
X =(Q x )...(Q x )[ψ(cid:48)(x ...x )] and by Saunders et al. [11]. This paper proposed a way to find
k+1 k+1 n n 1 n
Y =(Q x )...(Q x )[ψ(cid:48)(cid:48)(x ...x )] parallelism in the construction phase. We borrow this idea,
k+1 k+1 n n 1 n
Q ∈{∃,∀},C ∈{∧,∨} and suitably modified, and call it ‘Pipelined Constructions’. A
i
ψ(cid:48)(x ...x )Cψ(cid:48)(cid:48)(x ...x )=ψ(x ...x ). recurring feature in most of the projection operators is the
1 n 1 n 1 n
factthattheydealwithpolynomialspair-wise. Thiscanbe
Thetheoryofseparabilityseekstoaddresstheconditions
utilized to work in a parallel fashion.
on ψ(cid:48) and ψ(cid:48)(cid:48), to allow such a splitting.
The projection operators utilize polynomials pair-wise to
TheprojectionandconstructionphasesofCADalgorithm
compute resultants and discriminants. This can be done in
are inherently sequential. A projection (construction) of
a parallel fashion, using the divide and conquer paradigm.
a set of polynomials in k variables happens only after the
Assuming there are n polynomials in r variables, we could
corresponding set of polynomials in k+1 dimensions (k− assign n of them to one processor and n to another pro-
1 dimensions for constructions) has been projected (con- 2 2
cessor. This goes on recursively till we get each proces-
structed). Hence, none of the major phases of the original
sor working on just two polynomials. These now generate
CAD algorithm can be executed in parallel. We can sep-
the discriminants and resultants simultaneously. Assuming
arate the input, leading to multiple instances of the CAD
algorithm with lesser number of variables. we have found the projections of X = {a1,a2,...,an2} and
Y = {an+1,...,an} separately, we now combine them by
2
pairing up each polynomial from X with every polynomial
Note that the concept of separability is derived from the
fromY. Thisprocedureensuresthatprojectionsaredoneat
following two well known properties of quantifiers:
effectivelyatalogarithmicfunctionofthetimetakenbythe
1. (∀x)[f(x)∧g(x)]↔(∀x)f(x)∧(∀x)g(x). originalprojectionoperator. Thiswayofprojectingshallbe
referred to as Parallel Projections.
2. (∃x)[f(x)∨g(x)]↔(∃x)f(x)∨(∃x)g(x). Construction phase parallelism depends on many issues.
Clustering prevents any parallelism from entering construc-
In the above two cases, we effectively separated f(x) ( or
tion phase as some, if not all the regions would be inter-
g(x)) from the formula on the left side. We now need to
dependent. Hence, we consider a case without clustering.
generalize this notion of separation to an n variable case.
Thisideahasbeenexploredin[11]. Supposethen-dimensional
Theorem 3.2. Assume that CAD has regions, R = {r1,r2,...,rk}. In the construc-
tion stage, the stack over each region is constructed iter-
d(x1...xn)=(Qrxr)...(Qnxn)f(xf1...xfk)Cg(xg1...xgl) atively. Suppose the stack over the region r1 has regions,
R(cid:48) ={r ,r ,...r }. The construction of the stack over r
where C ∈ {∨,∧} and Qi ∈ {∀,∃}, is a prenex formula. If is follow1e1d b1y2 the1clonstruction of the stack over the region1
f is separable from g and
r . The observation to be made is that the construction of
2
X =(Q x )(Q x )...(Q x )[f(x ,x ,...,x )] thestackoverr isindependentoftheconstructionofstacks
r r r+1 r+1 n n f1 f2 fk 2
over the regions of R(cid:48). These can be done in parallel. This Proof. For any given n-tuple, (x ,...,x ), g(x ,...,x )
1 n 1 n
is analogous to the stages in a CPU pipeline. Hence, this is either true or false. If it were true,
work shall refer to this process as Pipelined Constructions.
f(x ,...,x )⇐⇒f(cid:48)(x ,...,x ,x ,...,x )
1 n 1 r−1 r+1 n
3.3 Substitutability
. If it were false, g(x ,...,x )∧f(cid:48)(x ,...,x ,x ,...,x )
1 n 1 r−1 r+1 n
Mostreal-worldproblemshavemorethanonepolynomial and h(x ,...x ) would both be false. Hence, h(x ,...x )
1 n 1 n
intheirprenexformulae,connectedbybooleanconnectives. andf(cid:48)(x ,...,x ,x ,...,x )∧g(x ,...,x )havethesame
1 r−1 r+1 n 1 n
Theobservationtobemadehereisthat,inmostcases,none truth table, and hence are equi-satisfiable.
of these polynomials have all the variables that are used in
the whole formula. The time complexity of the CAD al- 4. PARALLELCYLINDRICALALGEBRAIC
gorithm has an upper bound of (mn)krdk, where m is the
DECOMPOSITION
number of polynomials occurring in the input formula, r is
the total number of variables, n is the maximum degree of In this section we present a parallel CAD algorithm that
any polynomial in any variable, k is some constant and d is motivated by the observation we made above.
is the length of any integer coefficient of any polynomial.
4.1 PreliminaryConsiderations
We observe here, that in general, if we could decrease r by
even one variable, but add more number of polynomials to SkolemizationisaprocessdiscoveredbyThoralfSkolem[8],
m,we’dgetafasterrunningalgorithm. Thereareinstances which replaces all existentially quantified variables by new
wheretheeliminationofonevariablemakesaproblemcom- functions, called Skolem functions, which depend on the
putable in reasonable amounts of space and in reasonable presently quantified variables. This leaves only the univer-
time. But, there’s no closed form for the roots of quin- salquantifiersintact. WhenSkolemizationisperformedona
tic and higher degree equations, as proved by Niels Henrik CNFformula,eachtermoftheCNFbecomesseparablefrom
Abel. Hence, we can use only the variables, which have a therestoftheformula. Thisallowseachtermoftheresult-
degree of at most 4 to perform substitutions. ing CNF to be a separate parallel process. But, the newly
In our definition of substitutability, we classify polyno- introduced Skolem functions should also be accounted for,
mials into two classes, namely (i) substitutor and (ii) sub- intheinput. Thismayaddtothenumberofvariables,thus
stituend. increasing the running time.
Substitutorsarefurthersub-dividedintotwotypes,namely If separability was done before substitutions, we would
(i) substitutor-type I and (ii) substitutor-type II. not be considering the terms that would’ve become separa-
bleifsomevariableshavebeensubstitutedforothers. Thus,
Definition 3.3. A substitutor-type I polynomial should in the algorithms that follow, substitution is done prior to
form an equation and has to contain atleast one variable separations. This also derives from the fact that separabil-
whose exponent is at-most 4. itydependsexclusivelyonthevariables,andsubstitutability
neveraddsnewvariablestoanexistingformula(itcanonly
Definition 3.4. A substitutor-type II polynomial forms removevariablesandreplacethemwithexpressionscontain-
anequationandhasavariableofexponentat-most4inone ing other variables that occur in that formula).
of its factors, and is not a substitutor-type I. The proposed algorithms depend on the following defini-
tions of Conjunctive and Disjunctive Normal Forms,
Definition 3.5. Asubstituendpolynomialdoesnotcome
under the substitutor-types I and II. Definition 4.1. For polynomial equations and inequali-
ties f for some i,j, a prenex formula of the type,
ij
Definition 3.6. AssumethatP isanypolynomialandT [(f ∧f ∧...f )∨(f ∧f ∧...f )∨...∨(f ∧f ∧...f )]
is a substitutor-type I. We say that T is substitutable in P 11 12 1k1 21 22 2k2 n1 n2 nkn
if, the process of replacing a variable in P by substituting it is called a disjunctive normal form, and each (fi1 ∧fi2 ∧
withothervariablesfromT doesnotresultinanincreasein ...fiki), for some i is called a ‘term’.
the number of variables in the polynomial P.
Definition 4.2. For polynomial equations and inequali-
ties f for some i,j, a prenex formula of the type,
Usingtheabovedefinedtermsandoperation,wecannow ij
statethefollowinglemma,whichessentiallystatesthatsub-
[(f ∨f ∨...f )∧(f ∨f ∨...f )∧...∧(f ∨f ∨...f )]
stitutabilityproducesanequi-satisfiableformulaforagiven 11 12 1k1 21 22 2k2 n1 n2 nkn
prenexformula. Wedenoteequi-satisfiabilitybythesymbol is called a conjunctive normal form, and each (fi1 ∨fi2 ∨
‘⇐⇒’ in the following lemma. ...fiki), for some i is called a ‘term’.
4.2 ProposedAlgorithms
Lemma 3.7. Assume a prenex formula,
We now list the proposed algorithms.
h(x ,...x )=f(x ,...,x ,...,x )∧g(x ,...,x ),
1 n 1 r n 1 n
PARALLEL-CAD(A)
where f and g are both prenex formulae. Then,
Input: A set of polynomials, A
h(x ,...x )⇐⇒f(cid:48)(x ,...,x ,x ,...,x )∧g(x ,...,x ), Output: The CAD for R1 to Rn, with respect to A
1 n 1 r−1 r+1 n 1 n
where the polynomial f(cid:48) is obtained by substituting x in f, 1. PerformparallelprojectionstoAtoobtainA inn−1
r 1
using g. variables.
2. Project the polynomials in the set A to get the set 1. Number of Processors: The number of processors is
i
A . ThiscontinuestillweobtainthesetA ,which assumed to be constant, denoted by an upper bound
i+1 n−1
consists of univariate polynomials. ofK. Thisassumptionisveryrigidasitdoesnotallow
the increase in the number of processing cores. This
3. Performpipelinedconstructionsonthesetofprojected
way, we can study the effects of parallelism in a case
polynomials, {A ,...,A }. We obtain the decom-
1 n−1 where we obtain more number of partitions than the
positions, D ,...,D from these constructions, corre-
1 n number of processors. This leads to an optimization
sponding to the CAD for R1 to Rn, with respect to
problem.
A.
2. Parallel Time Complexity: Assuming that we have a
4. Return D ,...,D .
1 n
collection of independent parallel processes, the time
This algorithm is called simultaneously for each set of takenbythelongestprocessiscalledtheparalleltime
polynomials,derivedfromseparations,afterperformingsub- complexity of the collection. A formula is separated
stitutions. These are described in the following algorithm. only when none of the resulting partitions have all
the variables present in the original formula. This
PREPROCESSOR(f)
ensures that, when separability happens, the paral-
Input: A prenex formula containing a boolean combination
lel time complexity is exponentially lower than the
of polynomial equations and inequalities, f
original CAD algorithm. This reduction in time com-
Output: Sets of polynomials, S ,...,S , where each S is a
1 k i plexity happens even though the total work done by
set of polynomials.
PARALLEL-CAD is greater than the original algo-
1. Considertheprenexformula, f. Convertf toaCNF, rithm.
c, or to a DNF, d, whichever has the most number of
3. Total Work Done: Given a collection of parallel in-
terms. If it was converted to a DNF, complement it
dependent processes, the amount of work done by all
twicetogetacomplementofaCNF.DenotethisCNF
theindependentprocessescombinediscalledthetotal
by c.
work done by the collection. Equational constraints
2. Identify the substitutor-type-I polynomials in c. Use in a prenex formula eliminate the need for many con-
them to substitute in all other polynomials and also structions, thus speeding up CAD. But, if these con-
among themselves. Retain the substitutor polynomi- straintsweretobeseparated,thentheresultingprenex
als, which weren’t substituted in, as part of the CNF. formulawithouttheconstraintswouldperformunnec-
essaryconstructions,thusincreasingtheresourcecon-
3. Considerallsubstitutor-type-Ipolynomialsassubstitutor-
sumption. This would also occur with inequalities.
type-II polynomials. Factorize them and substitute
among the factors, along with other substitutor-type-
Separability is a property of the prenex formula. Separa-
IIpolynomials. Theresultingformulashallbedenoted
bility might result in the loss of structure in a given prenex
by r.
formula. Bythiswemeanthattheinterdependenciesamong
4. Perform Skolemization on r to get r . Perform sepa- the formula’s constituent polynomials are negated, result-
s
rations on r to get the prenex formulae, r ,...,r. If ing in wasteful and redundant computations, which would
1 l
every term of r has at most as many variables as the not have been required if the formula was not separated.
s
number of variables in max(r ,...,r), use r for fur- We need a formal, quantitative definition of ‘structure’ in a
1 l s
ther computations, and denote it by r. Otherwise use givenformulatostudyit’seffectsontherunningtimeofthe
r. proposed algorithm.
5. Assume r gives rise to k prenex formulae when sepa- Definition 4.3. The‘sharingfactor’existingamongtwo
rated. Collect these polynomials in sets, S1,...,Sk. given prenex formulae, f and g, is the number of variables
that are shared between f and g. It’s denoted by T .
6. Return the sets S ,...,S . f,g
1 k
Thealgorithm,PARALLEL-CAD(A)isexecutedforeach The sharing factor essentially captures the distribution of
A∈{S ,...,S }. Thesek instancesofParallel-CADarerun variablesamongtheconstituentpolynomialsofaprenexfor-
1 k
simultaneously. Quantifier elimination can be done by con- mula. If formulae f and g have k variables in common, it
sidering the connectives and the equalities and inequalities impliesthatf imposesconditionsonginkdimensions. The
specified among the member polynomials of each set, S , sharing factor gives a measure of structure present in a for-
i
i ∈ {1,...,k}. The resulting k quantifier-free formulae are mula,asthesharingofvariablesbetweenpolynomialscauses
united using conjunctions. interdependencies among them. Having defined the sharing
factor, we now define the operation, separation, where the
4.3 AnalysisoftheAlgorithms
sharing of variables comes into consideration.
In the following sections and subsections, ‘structure’ in a
prenex formula refers to the dependence of the constituent Definition 4.4. Consideraprenexformula,f,whichhas
polynomialsoneachother,resultinginconditions/constraints n variables, out of which, there are k quantified variables
beingimposedontheformulabecauseofthesepolynomials. and n−k quantifier-free variables. We say that the sepa-
For the analysis of the proposed algorithms, we use the ration of f is the ordered pair (F,C), if we obtain a set of
classicalnotionofparallelcomplexity[10]. Theparallelcom- prenex formulae, F = {f ,...f} and a set of boolean oper-
1 l
plexity analysis of PARALLEL-CAD can be studied using ators, C = {c ,...,c }, such that the following conditions
1 l−1
three parameters, hold.
1. f ⇐⇒f c f ...c f Definition 4.9. Given two k-separable prenex formulae
1 1 2 l−1 l
f and g and their separations, (F,C ) and (G,C ), we say
2. Assume that f = q f(cid:48),j ∈ {1,...,l}, where q is the 1 2
j j j j that f is closer to the centre than g (denoted by f (cid:22)g) if
stringofquantifiersandquantifiedvariables,Q x ...Q x ,
1 1 jk jk
Qi ∈{∃,∀}, i∈{1,...,k}, and fj(cid:48) is the formula with- 1. maxi#(fi) < maxj#(gj), fi ∈ F and gj ∈ G, 1 ≤
out the string of quantifiers. The condition for (F,C) i,j ≤k.
to be a separation of f is,
(cid:80) (cid:80)
2. Ifmax #(f )=max #(g ),then T ≤ T
i i j j i,j fi,fj m,l gm,gl
, f ,f ∈F and g ,g ∈G, 1≤i,j,m,l≤k.
i j m l
f ⇐⇒Q x ...Q x [f(cid:48)c f(cid:48)c ...c f(cid:48)].
1 1 k k 1 1 2 2 l−1 l
In the above definition, g is said to be farther away from
Lossofstructureinaformulaleadstoredundantcompu-
the centre, compared to f.
tations.
ThetimecomplexityofCADalgorithmisboundedabove
Definition 4.5. Consider a prenex formula, f and it’s by(mn)krdk,wheremisthenumberofpolynomialsoccur-
separation, (F,C). While executing PARALLEL-CAD on ringintheinputformula,risthetotalnumberofvariables,
thesetF,acomputationissaidtoberedundantifitneednot nisthemaximumdegreeofanypolynomialinanyvariable,
beperformedwhileexecutingtheoriginalCADalgorithmon k is some constant and d is the length of any integer coef-
f. The redundancy ratio is given by redundant computations. ficient of any polynomial. This motivates the definition of
total computations
comparability.
Thedefinitionofseparationdefinesseparabilityasaprop-
ertyoftheformula. Intheremainderofthissection,weas- Definition 4.10. Consider two prenex formulae, f and
sume that all prenex formulae are in variables, x1,...xn. It g. The maximum degrees of any polynomial in any variable
is natural to classify prenex formulae into classes based on in f and g are denoted by n and n . The number of vari-
1 2
thenumberofpartitionsitcanbeseparatedinto. Thus,we ables in f and g are denoted by r and r . Without loss of
1 2
have the following definition. generality, assume that r > r . Consider a function, α :
1 2
Definition 4.6. A prenex formula is defined to be k - N×N→N, such that, α(n,r)=(mn)krdk. The formulae,
f and g are said to be comparable if α(n ,r )>α(n ,r ).
separable if and only if it can be separated into k prenex 1 1 2 2
formulae, which are connected by boolean connectives. The
WenowcharacterizetherelationshipbetweenPARALLEL-
set of all k-separable formulae shall be denoted by S .
k CAD and the separable class. If the first condition in the
According to this definition, we can observe that S ⊆ definition4.9issatisfied,thentheformulaclosertothecen-
k
S . Thesetof1-separableformulaeisthesetofallprenex tre has fewer variables than the farther one, which might it
k−1
formulae,whichshallbedenotedbyP. Weassumethatwe execute faster (if both the formulae are comparable). On
have only a constant number of processors, and we denote the other hand, if the second condition is satisfied, then a
thisbyK. Thisleadstoadefinitionoftheclassofseparable lesseramountofstructureislostonseparations,thusneces-
problems. sitating a lower number of redundant computations. This
motivates the following observation.
Definition 4.7. Consider a set S ⊆ P. S is called the
‘separable class’ if the following conditions hold. Observation 4.11. Given prenex formulae, f and g and
1. S ⊆S2 their separations, (F,C1) and (G,C2), if f (cid:22)g, then PAR-
ALLEL - CAD takes at least as much time and has at least
2. If f is a prenex formula and (F,C) is it’s separation,
as much redundancy ratio to execute g as it would take to
∀f ∈F, ∃x ∈{x ,...,x } such that f is independent
i 1 n execute f, provided that f and g remain comparable.
of x .
i
The speed-up obtained bya k-separable formuladepends 5. SIMULATIONRESULTS
on the distribution of variables among the k separated for- The simulation results have been divided into two cate-
mulaeandtheirinterdependencies. Thus,weneedtodefine gorieswhicharemutuallyindependent,namely(i)Thecases
asubsetofSk,whichhasthemostidealdistributionofvari- where only separations are considered and (ii) The cases
ables. where only substitutions are considered.
Definition 4.8. The centre of S is the set of all prenex TheseexperimentswererunonanInteli7quad-coreCPU
k
using 4GB of DDRAM. QEPCAD Version B 1.65, 10 May
formulae in n variables, which can be separated into k for-
mulae, f ,...f , each with n variables and every variable is 2011[2]isusedtoperformthecomputations. Inthefollow-
1 k k ing plots, time is in milliseconds and space is in cells used
present in exactly one formula in f ,...,f . The centre of
1 k
by QEPCAD.
S is denoted by C .
k k
4.3.1 PARALLEL-CADandSeparableClass 5.1 SimulationResultsforSeparability
Theabovedefinedtermshelpusinanalysingtheextentof Thesimulationhasbeenperformedfortwocases,depend-
improvementinrunningtimeandspaceaproblemcanpos- ing on the distribution of variables among the constituent
sibly achieve when PARALLEL-CAD is used. The notion polynomials of the prenex formula. This is based on the
ofthecentreofseparableclassindicatestheareaswherethe fact that the running time and the space required for CAD
maximum amount of speed-up is possible. We now define dependalmostentirelyonthenumberofvariables,thetime
the conditions which determine whether a given problem is being doubly exponential in the number of variables.
closer to the centre than another problem or not. Denote
the number of variables in a polynomial, p, by #(p).
Case Study 1 - Homogeneous Distribution of Variables
This case study involves four prenex formulae, each with
twopolynomialequationsandwithvaryingnumberofquan-
tifier - free variables being common between them. The
space and time analyses of this situation yields quite in-
teresting consequences, when separability is applied. The
polynomials considered were:
1. (∃x)[ax2+bx+c=0∨dx2+ex+f =0]
2. (∃x)[ax2+bx+c=0∨dax2+ex+f =0]
3. (∃x)[ax2+bx+c=0∨dax2+ebx+f =0]
4. (∃x)[ax2+bx+c=0∨dax2+ebx+fc=0]
Here, due to separations, the structure imposed by the
secondequationonthefirstoneislost,thusleadingtopos-
siblymorenumberofconstructions. But,asthedependency
reduces, the ‘amount of structure’ lost is also reduced, and
hence we observe that the separated case starts to perform
better.
Case Study 3 - Validity of Observation 4.11 Two case
studies have been performed demonstrating the validity of
observation 4.11. The inputs for the two case studies were
1. • ∃x[ax5+abx3+bc=0]∨[dx+ef =0]∨[gfx10+
hgx6+ix4+j =0]
• ∃x[ax5+abx3+bc=0]∨[dx+e=0]∨[gfx6+
hix+j =0].
2. • ∃x[ax3+bx+c=0]∨[cdx2+de=0]
The time complexity reduced nearly five-fold in all the • ∃x[ax3+bx+c=0]∨[dx2+de=0].
four cases. The space complexity remained same across all
Thetimetakenforexecutionandthespacerequiredwere:
the four cases if CAD was applied without separations, as
there would still be the same number of projection opera-
First Formula Second Formula
tions. But, when separations were applied, the space re-
Case 1 Time 1584ms 2128ms
quiredincreasedexponentiallyastheoverlapinthevariables
Space 15126564 cells 23304694 cells
of the two polynomials increased. This was expected, con-
Case 2 Time 40ms 36ms
sidering the doubly exponential nature of CAD algorithm.
Space 73998 cells 66287 cells
Case Study 2 - Heterogeneous Distribution of Variables
This case study considers a similar system as above, ex- In the first case, the time taken by the first formula was
ceptforthefactthatthevariablesareunevenlydistributed considerablylesser,eventhoughithadalargerexponentin
among the two polynomials and that there is a disparity it’sconstituentpolynomials. Thisisbecauseofamoreeven
among the degrees of the two polynomials. The formulae distributionofvariables. Accordingtoobservation4.11,the
considered are: first formula is closer to the centre of S than the second
3
formula.
In the second case, the second condition of closeness to
1. (∃x)[dafx5+ebx3+ec=0∨fc+x=0]
the centre is considered. Thus, the time consumed by both
the formulae are almost identical. Due to sharing of vari-
2. (∃x)[dafx5+ebx3+ec=0∨f +x=0] ables in the first case, a greater amount of wasteful total
work has been done, than in the second case. According to
observation 4.11, the second formula is closer to the centre
3. (∃x)[dafx5+ebx3+ec=0∨x=0] of S than the first one.
2
5.2 SimulationResultsforSubstitutability for, and define a new projection operator, which embodies
The experiments have been done for 4 test cases. They thestructureimposedbyparallelism. Thisprojectionoper-
donotdynamicallyschedulenewinstancestodifferentcores. ator, when found, may speed-up CAD further.
The 4 cases are,
7. REFERENCES
1. (∀x)(∀y)[[b2x2+a2y2 =a2b2]=⇒[x2+y2 <1]]
[1] Caviness B.F. and J.R. Johnson. Quantifier
2. (∃y)(∃z)[[x3y −29x2 = 0]∧[x4z2 −101x3 −19x = Elimination and Cylindrical Algebraic Decomposition.
0]∧[x2+y2+z2 >0]∧[x+y>0]∧[y+z<0]] Springer Verlag, 1998.
[2] C.W. Brown. QEPCAD B: A Program for Computing
3. (∀v)[[x3y−29w2 = 0]∧[x4z2 −101w2 −19v = 0]∧ with Semi-Algebraic Sets using CADs. ACM SIGSAM
[x2+y2+z2 >0]∧[x+y>0]∧[w+v+x+z>0]] Bulletin, 37(4):97–108, 2003.
[3] B. Buchberger and H. Hong. Speeding-up Quantifier
4. (∃v)[[x5+4y2−3z=0]∧[x6+2w2−v=0]∧[x+y+
Elimination by Gr¨obner Bases. Johannes Kepler
w+z+v>0]]
University, Linz, 1991.
[4] G. Collins. Quantifier Elimination for Real Closed
This simulation is intended to demonstrate the extent
Fields by Cylindrical Algebraic Decompostion. In
till which substitutions can improve the time and space re-
Automata Theory and Formal Languages 2nd GI
quired.
Conference Kaiserslautern, May 20–23, 1975, pages
134–183. Springer, 1975.
[5] G.E. Collins and H. Hong. Partial Cylindrical
Algebraic Decomposition for Quantifier Elimination.
Journal of Symbolic Computation, 12(3):299–328,
1991.
[6] Andreas Dolzmann, Thomas Sturm, and Volker
Weispfenning. Real Quantifier Elimination in Practice.
In Algorithmic Algebra and Number Theory, pages
221–247. Springer, 1998.
[7] Collins G.E. Quantifier Elimination by Cylindrical
Algebraic Decomposition–Twenty Years of Progress.
Quantifier Elimination and Cylindrical Algebraic
Decomposition, pages 8–23, 1998.
[8] J Hintikka. Truth definitions, Skolem Functions and
Axiomatic Set Theory. 1998.
[9] H. Hong. An Improvement of the Projection Operator
inCylindricalAlgebraicDecomposition.InProceedings
of the International Symposium on Symbolic and
Algebraic Computation, pages 261–264. ACM, 1990.
[10] Parberry, I. Parallel Complexity Theory. 1987.
[11] B.D. Saunders, H.R. Lee, and S.K. Abdali. A Parallel
Implementation of the Cylindrical Algebraic
Decomposition Algorithm. In Proceedings of the
Intheaboveplots,weobservethatthespacerequiredfor
ACM-SIGSAM 1989 International Symposium on
computationsdecreasedtoagreaterextent,ascomparedto
Symbolic and Algebraic Computation, pages 298–307.
thetime. Thisisbecauseofthefactthatsubstitutionspre-
ACM, 1989.
serve the inter-dependency among the constituent polyno-
[12] A. Tarski. A Decision Method for Elementary Algebra
mialsinaprenexformula,unlikeseparations. Substitutions
and Geometry: Prepared for Publication with the
onlychangethewaythesepolynomialsarerepresented. This
Assistance of JCC McKinsey. 1951.
explains the difference between the space and time plots of
the separations case and the substitutions case. The plots
indicate that the space consumed and the time elapsed are
disproportionate,suggestingthatalargeamountofmemory
has been used on projecting and constructing operations,
which required small amounts of processing time.
6. CONCLUDINGREMARKS
InthispaperweproposedaparallelCylindricalAlgebraic
Decomposition algorithm. The ongoing work is to imple-
ment this algorithm for execution on a GPU. Based on the
conceptsofseparableclassandit’scentre,weproposetogive
the average case complexity analysis of PARALLEL-CAD.
Thisalgorithmstillusestheprojectionoperatorsdefinedfor
thesequentialCADalgorithm. Ourfutureworkistoexplore