polynomial hierarchy, betti numbers and a real analogue of toda's

(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation

Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem

Polynomial Hierarchy, Betti Numbers and areal analogue of Toda’s Theorem

Saugata Basu

Purdue/Georgia Tech

Geometry Seminar, Courant Institute, Feb 24, 2009(joint work with Thierry Zell)

Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem

Outline

1 (Discrete) Polynomial Hierarchy

2 Blum-Shub-Smale Models of Computation

3 Algorithmic Semi-algebraic Geometry

4 Real Analogue of Toda’s Theorem

5 ProofOutlineDetails

Outline

A quick primer of basic definitions and notation

Initially let k = Z/2Z = {0̄, 1̄}.A language L is a set⋃

Ln, Ln ⊂ kn

(abusing notation a little we will identify L with thesequence (Ln)n>0).A language

L = (Ln)n>0 ∈ P

if there exists a Turing machine M that given x ∈ kn

decides whether x ∈ Ln or not in nO(1) time.

Ln, Ln ⊂ kn

L = (Ln)n>0 ∈ P

Ln, Ln ⊂ kn

L = (Ln)n>0 ∈ P

Primer (cont.)

A languageL = (Ln)n>0 ∈ NP

if there exists a language L′ = (L′n)n>0 ∈ P such that

x ∈ Ln ⇐⇒ (∃ y ∈ km(n)) (y, x) ∈ L′m+n

where m(n) = nO(1) (such a y is usually called a“certificate” or a “witness” for x).A language

L = (Ln)n>0 ∈ coNPif there exists a language L′ = (L′n)n>0 ∈ P such that

x ∈ Ln ⇐⇒(∀ y ∈ km(n)

)(y, x) ∈ L′m+n

where m(n) = nO(1).Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem

Primer (cont.)

A languageL = (Ln)n>0 ∈ NP

x ∈ Ln ⇐⇒ (∃ y ∈ km(n)) (y, x) ∈ L′m+n

where m(n) = nO(1) (such a y is usually called a“certificate” or a “witness” for x).A language

L = (Ln)n>0 ∈ coNPif there exists a language L′ = (L′n)n>0 ∈ P such that

x ∈ Ln ⇐⇒(∀ y ∈ km(n)

)(y, x) ∈ L′m+n

where m(n) = nO(1).Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem

Discrete Polynomial Time Hierarchy– A QuickReminder

A languageL = (Ln)n>0 ∈ Σω

x ∈ Ln

m(Q1y1 ∈ km1)(Q2y2 ∈ km2) . . . (Qωyω ∈ kmω)

(y1, . . . , yω, x) ∈ L′m+n

where m(n) = m1(n) + · · ·+ mω(n) = nO(1) and for 1 ≤ i ≤ ω,Qi ∈ {∃,∀}, and Qj 6= Qj+1, 1 ≤ j < ω, Q1 = ∃.

x ∈ Ln

(y1, . . . , yω, x) ∈ L′m+n

x ∈ Ln

(y1, . . . , yω, x) ∈ L′m+n

Reminder (cont.)

Similarly a language

L = (Ln)n>0 ∈ Πω

x ∈ Ln

m(Q1y1 ∈ km1)(Q2y2 ∈ km2) · · · (Qωyω ∈ kmω)

(y1, . . . , yω, x) ∈ L′m+n

where m(n) = m1(n) + · · ·+ mω(n) = nO(1) and for 1 ≤ i ≤ ω,Qi ∈ {∃,∀}, and Qj 6= Qj+1, 1 ≤ j < ω, Q1 = ∀. Notice that

P = Σ0 = Π0,

NP = Σ1, coNP = Π1.

Reminder (cont.)

L = (Ln)n>0 ∈ Πω

x ∈ Ln

(y1, . . . , yω, x) ∈ L′m+n

P = Σ0 = Π0,

Reminder (cont.)

L = (Ln)n>0 ∈ Πω

x ∈ Ln

(y1, . . . , yω, x) ∈ L′m+n

P = Σ0 = Π0,

Reminder (cont.)

L = (Ln)n>0 ∈ Πω

x ∈ Ln

(y1, . . . , yω, x) ∈ L′m+n

P = Σ0 = Π0,

The polynomial time hierarchy

Also, notice the inclusions

Σi ⊂ Πi+1,Σi ⊂ Σi+1

Πi ⊂ Σi+1,Πi ⊂ Πi+1

The polynomial time hierarchy is defined to be

PH def=

⋃ω≥0

(Σω ∪ Πω) =⋃ω≥0

Σω =⋃ω≥0

The polynomial time hierarchy

Also, notice the inclusions

Σi ⊂ Πi+1,Σi ⊂ Σi+1

Πi ⊂ Σi+1,Πi ⊂ Πi+1

The polynomial time hierarchy is defined to be

PH def=

⋃ω≥0

(Σω ∪ Πω) =⋃ω≥0

Σω =⋃ω≥0

The Class #P

A sequence of functions

(fn : kn → N)n>0

is said to be in the class #P if there exists L = (Ln)n>0 ∈ Psuch that for x ∈ kn

fn(x) = card(Lm+n,x), m = nO(1),

where Lm+n,x is the fibre π−1(x) ∩ Lm+n, andπ : km+n → kn the projection map on the last nco-ordinates.

The Class #P

(fn : kn → N)n>0

The Class #P

(fn : kn → N)n>0

The Class #P

(fn : kn → N)n>0

Toda’s Theorem

Toda’s theorem is a seminal result in discrete complexity theoryand gives the following inclusion.

Theorem (Toda (1989))

PH ⊂ P#P

“illustrates the power of counting”

Toda’s Theorem

PH ⊂ P#P

Toda’s Theorem

PH ⊂ P#P

Toda’s Theorem

PH ⊂ P#P

Blum-Shub-Smale model

Generalized TM where k is allowed to be any ring (werestrict ourseleves to the cases k = C or R).Setting k = Z/2Z (or any finite field) recovers the classicalcomplexity classes.Informally, such a TM should be thought of as a programthat accepts as input x ∈ kn, and at each step

1 either makes a ring computation zi ← zj ∗ z`;2 or branches according to a test zj{=, 6=}0 in case k = C, or

the test zj{>,<,=}0 in case k = R;3 or accepts/rejects.

A B-S-S TM accepts for every n a subset Sn ⊂ kn.1 In case k = C, each Sn is a constructible subset of Cn,2 in case k = R, each Sn is a semi-algebraic subset of Rn.

Complexity Classes

Complexity classes Pk , NPk , coNPk and more generallyPHk are defined as before (for k = C, R).B-S-S developed a theory of NP-completeness.In case, k = C the problem of determining if a system ofn + 1 polynomial equations in n variables has a commonzero in Cn is NPC-complete.In case, k = R the problem of determining if a quarticpolynomial in n variables has a common zero in Rn isNPR-complete.It is unknown if PC = NPC (respectively, PR = NPR) just asin the discrete case.

Complexity Classes

Semi-algebraic sets

From now we assume k = R, and restrict ourselves to realTM in the sense of B-S-S.Such a machine accepts a sequence (Sn ⊂ Rn)n>0 whereeach Sn is a semi-algebraic subset of Rn.A semi-algebraic set, S ⊂ Rn, is a subset of Rn defined bya Boolean formula whose atoms are polynomial equalitiesand inequalities.

Semi-algebraic sets

Two classes of problems

The most important algorithmic problems studied in this areafall into two broad sub-classes:

1 the problem of quantifier elimination, and its special casessuch as deciding a sentence in the first order theory ofreals, or deciding emptiness of semi-algebraic sets.

2 the problem of computing topological invariants ofsemi-algebraic sets, such as the number of connectedcomponents, Euler-Poincaré characteristic, and moregenerally all the Betti numbers of semi-algebraic sets.

Analogy with Toda’s Theorem

The classes PH and #P appearing in the two sides of theinclusion in Toda’s Theorem can be identified with the twobroad classes of problems in algorithmic semi-algebraicgeometry;the class PH with the problem of deciding sentences with afixed number of quantifier alternations;the class #P with the problem of computing topologicalinvariants of semi-algebraic sets, namely their Bettinumbers, which generalizes the notion of cardinality forfinite sets;it is thus quite natural to seek a real analogue of Toda’stheorem.

Real Analogue of #P

In order to define real analogues of counting complexityclasses of discrete complexity theory, it is necessary toidentify the proper notion of “counting” in the context ofsemi-algebraic geometry.Counting complexity classes over the reals have beendefined previously by Meer (2000) and studied extensivelyby other authors Burgisser, Cucker et al (2006). Theseauthors used a straightforward generalization tosemi-algebraic sets of counting in the case of finite sets;namely

f (S) = card(S) if card(S) <∞= ∞ otherwise.

Real Analogue of #P

In order to define real analogues of counting complexityclasses of discrete complexity theory, it is necessary toidentify the proper notion of “counting” in the context ofsemi-algebraic geometry.Counting complexity classes over the reals have beendefined previously by Meer (2000) and studied extensivelyby other authors Burgisser, Cucker et al (2006). Theseauthors used a straightforward generalization tosemi-algebraic sets of counting in the case of finite sets;namely

f (S) = card(S) if card(S) <∞= ∞ otherwise.

An alternative definition

In our view this is not fully satisfactory, since the countgives no information when the semi-algebraic set is infinite,and most interesting semi-algebraic sets are infinite.If one thinks of “counting” a semi-algebraic set S ⊂ Rk ascomputing certain discrete invariants, then a naturalmathematical candidate is its sequence of Betti numbers,b0(S), . . . , bk−1(S), or more succinctlythe Poincaré polynomial of S, namely

PS(T )def=

∑i≥0

bi(S) T i .

In case card(S) <∞, we have thatb0(S) = PS(0) = card(S).

PS(T )def=

∑i≥0

bi(S) T i .

PS(T )def=

∑i≥0

bi(S) T i .

PS(T )def=

∑i≥0

bi(S) T i .

Definition of #P†R

We call a sequence of functions

(fn : Rn → Z[T ])n>0

to be in class #P†R if there exists a polynomial time real Turingmachine M which tests membership in the semi-algebraic sets(Sn ⊂ Rn)n>0 such that

fn(x) = PSm+n,x , m = nO(1)

for each x ∈ Rn, where Sm+n,x = Sm+n ∩ π−1(x) andπ : Rm+n → Rn is the projection on the last n coordinates.

Definition of #P†R

We call a sequence of functions

(fn : Rn → Z[T ])n>0

to be in class #P†R if there exists a polynomial time real Turingmachine M which tests membership in the semi-algebraic sets(Sn ⊂ Rn)n>0 such that

fn(x) = PSm+n,x , m = nO(1)

for each x ∈ Rn, where Sm+n,x = Sm+n ∩ π−1(x) andπ : Rm+n → Rn is the projection on the last n coordinates.

Real analogue of Toda’s theorem

It is now natural to formulate the following conjecture.

Conjecture

PHR ⊂ P#P†R

For technical reasons we are unable to prove this without afurther compactness hypothesis on the left hand-side.

Conjecture

PHR ⊂ P#P†R

Conjecture

PHR ⊂ P#P†R

The compact real polynomial hierarchy

We say that a sequence of semi-algebraic sets

(Sn ⊂ Sn)n>0 ∈ ΣcR,ω

with each Sn compact if there exists another sequence(S′

n)n>0 ∈ PR such that

x ∈ Sn

if and only if(Q1y1 ∈ Sm1)(Q2y2 ∈ Sm2) . . . (Qωyω ∈ Smω)

(y1, . . . , yω, x) ∈ S′m+n

where m(n) = m1(n) + · · ·+ mω(n) = nO(1) and for 1 ≤ i ≤ ω,Qi ∈ {∃,∀}, and Qj 6= Qj+1, 1 ≤ j < ω, Q1 = ∃ . The compactclass Πc

R,ω is defined analogously.Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem

x ∈ Sn

(y1, . . . , yω, x) ∈ S′m+n

x ∈ Sn

(y1, . . . , yω, x) ∈ S′m+n

x ∈ Sn

(y1, . . . , yω, x) ∈ S′m+n

The compact real polynomial hierarchy (cont.)

We define

⋃ω≥0

(ΣcR,ω ∪ Πc

R,ω) =⋃ω≥0

ΣcR,ω =

⋃ω≥0

cR,ω.

Notice that the semi-algebraic sets belonging to any languagein PHc

R are all semi-algebraic compact (in fact closedsemi-algebraic subsets of spheres). Also, notice the inclusion

PHcR ⊂ PHR.

The compact real polynomial hierarchy (cont.)

We define

⋃ω≥0

(ΣcR,ω ∪ Πc

R,ω) =⋃ω≥0

ΣcR,ω =

⋃ω≥0

cR,ω.

Notice that the semi-algebraic sets belonging to any languagein PHc

R are all semi-algebraic compact (in fact closedsemi-algebraic subsets of spheres). Also, notice the inclusion

PHcR ⊂ PHR.

Main theorem

Theorem (B-Zell,2008)

PHcR ⊂ P#P†

Remark about the compactness assumption

Even though the restriction to compact semi-algebraic setsmight appear to be only a technicality at first glance, this isactually an important restriction.For instance, it is a long-standing open question in realcomplexity theory whether there exists an NPR-completeproblem which belongs to the class Σc

1 (the compactversion of the class NPR i.e. where the certificates areconstrained to come from a compact set).

Remark about the compactness assumption

Even though the restriction to compact semi-algebraic setsmight appear to be only a technicality at first glance, this isactually an important restriction.For instance, it is a long-standing open question in realcomplexity theory whether there exists an NPR-completeproblem which belongs to the class Σc

1 (the compactversion of the class NPR i.e. where the certificates areconstrained to come from a compact set).

OutlineDetails

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OutlineDetails

Summary of the Main Idea

Our main tool is a topological construction which given asemi-algebraic set S ⊂ Rm+n, p ≥ 0, and πY : Rm+n → Rn

denoting the projection along (say) the Y-co-ordinates,constructs efficiently a semi-algebraic set, Dp

Y(S), suchthat

bi(πY(S)) = bi(DpY(S)), 0 ≤ i < p.

Notice that even if there exists an efficient (i.e. polynomialtime) algorithm for checking membership in S, the sameneed not be true for the image πY(S).A second topological ingredient is Alexander-Lefshetzduality which relates the Betti numbers of a compactsubset K of the sphere Sn with those of Sn − K .

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Y(S), suchthat

bi(πY(S)) = bi(DpY(S)), 0 ≤ i < p.

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Y(S), suchthat

bi(πY(S)) = bi(DpY(S)), 0 ≤ i < p.

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The case ΣcR,1

Consider a closed semi-algebraic set S ⊂ Sk × S` bedefined by a quantifier free formula φ(Y , X ) and let

πY : Sk × S` → Sk

be the projection map along the Y coordinates.Then the formula Φ(X) = ∃ Y φ(X, Y) is satisfied by x ∈ Sk

if and only if b0(Sx) 6= 0, where Sx = S ∩ π−1Y (x). Thus, the

problem of deciding the truth of Φ(x) is reduced tocomputing a Betti number (the 0-th) of the fiber of S over x.

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The case ΣcR,1

Consider a closed semi-algebraic set S ⊂ Sk × S` bedefined by a quantifier free formula φ(Y , X ) and let

be the projection map along the Y coordinates.Then the formula Φ(X) = ∃ Y φ(X, Y) is satisfied by x ∈ Sk

if and only if b0(Sx) 6= 0, where Sx = S ∩ π−1Y (x). Thus, the

problem of deciding the truth of Φ(x) is reduced tocomputing a Betti number (the 0-th) of the fiber of S over x.

OutlineDetails

The case ΠcR,1

Using the same notation as before we have that theformula Ψ(X) = ∀ Y φ(X, Y) is satisfied by x ∈ Sk if andonly if b0(S` \ Sx) = 0 which is equivalent to b`(Sx) = 1 (byAlexander duality).Notice, that as before the problem of deciding the truth ofΨ(x) is reduced to computing a Betti number (the `-th) ofthe fiber of S over x.

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The case ΠcR,1

Using the same notation as before we have that theformula Ψ(X) = ∀ Y φ(X, Y) is satisfied by x ∈ Sk if andonly if b0(S` \ Sx) = 0 which is equivalent to b`(Sx) = 1 (byAlexander duality).Notice, that as before the problem of deciding the truth ofΨ(x) is reduced to computing a Betti number (the `-th) ofthe fiber of S over x.

OutlineDetails

Slightly more non-trivial case: ΠcR,2

Let S ⊂ Sk × S` × Sm be a closed semi-algebraic setdefined by a quantifier-free formula φ(X, Y, Z) and let

πZ : Sk × S` × Sm → Sk × S`

be the projection map along the Z variables, and

be the projection map along the Y variables as before.Consider the formula Φ(X) = ∀ Y∃ Z φ(X, Y, Z).For x ∈ Sk , Φ(x) is true if and only if πZ(S)x = S`, which isequivalent to b`(D`+1

Z (S)x) = 1.

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πZ : Sk × S` × Sm → Sk × S`

Z (S)x) = 1.

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πZ : Sk × S` × Sm → Sk × S`

Z (S)x) = 1.

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The case : ΠcR,2 (cont.)

Thus for any x ∈ Sk , the truth or falsity of Φ(x) isdetermined by a certain Betti number of the fiber D`+1

Z (S)x

over x of a certain semi-algebraic set D`+1Z (S) which can

be constructed efficiently in terms of the set S.

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In general ...

The idea behind the proof of the main theorem is a recursiveapplication of the above argument in case when the number ofquantifier alternations is larger (but still bounded by someconstant) while keeping track of the growth in the sizes of theintermediate formulas and also the number of quantifiedvariables.

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Key Proposition

Suppose there exists a real Turing machine M, and a sequenceof formulas

Φn(X0, . . . , Xn, Y0, . . . , Ym−1) :=

(Q1Z1 ∈ Sk1) · · · (QωZω ∈ Skω)φn(X, Y, Z1, . . . , Zω),

having free variables (X, Y) = (X0, . . . , Xn, Y0, . . . , Ym−1), with

Q1, . . . , Qω ∈ {∃,∀}, Qi 6= Qi+1,

where φn a quantifier-free formula defining a closed(respectively open) semi-algebraic subset of Sn, and such thatM tests membership in the semi-algebraic sets defined by φn inpolynomial time.

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Key Proposition (cont.)

Then, there exists a polynomial time real Turing machine M ′

which recognizes the semi-algebraic sets defined by asequence of quantifier-free first order formulas(Θn(X, V0, . . . , VN))n>0 such that for each x ∈ Sn, whereΘn(x, V ) describes a closed (respectively open) semi-algebraicsubset Tn ⊂ SN , with N = nO(1), and polynomial-timecomputable maps

Fn : Z[T ]≤N → Z[T ]≤m

such thatPR(Φn(x,Y)) = Fn(PR(Θn(x,V ))).

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Outline

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Topological Join

The join J(X , Y ) of two topological spaces X and Y is definedby

J(X , Y )def= X × Y ×∆1/ ∼,

where(x , y , t0, t1) ∼ (x ′, y ′, t0, t1)

if t0 = 1, x = x ′ or t1 = 1, y = y ′.Intuitively, J(X , Y ) is obtained by joining each point of X witheach point of Y by a unit interval.Example:

J(Sm, Sn) ∼= Sm+n+1.

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Topological Join

J(X , Y )def= X × Y ×∆1/ ∼,

where(x , y , t0, t1) ∼ (x ′, y ′, t0, t1)

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Topological Join

J(X , Y )def= X × Y ×∆1/ ∼,

where(x , y , t0, t1) ∼ (x ′, y ′, t0, t1)

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Iterated joins

For p ≥ 0, the (p + 1)-fold join Jp(X ) of X is

Jp(X )def= X × · · · × X︸︷︷︸

(p+1) times

×∆p/ ∼,

(x0, . . . , xp, t0, . . . , tp) ∼ (x ′0, . . . , x ′p, t0, . . . , tp)

if for each i with ti 6= 0, xi = x ′i . It is easy to see that , Jp(S0), ofthe zero dimensional sphere is homeomorphic to Sp.

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Iterated joins

For p ≥ 0, the (p + 1)-fold join Jp(X ) of X is

Jp(X )def= X × · · · × X︸︷︷︸

(p+1) times

×∆p/ ∼,

(x0, . . . , xp, t0, . . . , tp) ∼ (x ′0, . . . , x ′p, t0, . . . , tp)

if for each i with ti 6= 0, xi = x ′i . It is easy to see that , Jp(S0), ofthe zero dimensional sphere is homeomorphic to Sp.

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p-equivalence

We call a map f : A→ B between two topological spaces to bea p-equivalence if the induced homomorphism

f∗ : Hi(A)→ Hi(B)

is an isomorphism for all 0 ≤ i < p, and an epimorphism fori = p. Observe that Jp(S0) ∼= Sp is p-equivalent to a point. Infact, this holds much more generally and we have that

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p-equivalence

We call a map f : A→ B between two topological spaces to bea p-equivalence if the induced homomorphism

f∗ : Hi(A)→ Hi(B)

is an isomorphism for all 0 ≤ i < p, and an epimorphism fori = p. Observe that Jp(S0) ∼= Sp is p-equivalent to a point. Infact, this holds much more generally and we have that

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Connectivity Property of Join Spaces

Theorem

Let X be a compact semi-algebraic set. Then, the (p + 1)-foldjoin Jp(X ) is p-equivalent to a point.

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Topological join over a map

Let f : A→ B be a map between topological spaces A and B.For p ≥ 0 the (p + 1)-fold join Jp

f (A) of A over f is

Jpf (A)

def= A×B · · · ×B A︸︷︷︸

(p+1) times

×∆p/ ∼,

(x0, . . . , xp, t0, . . . , tp) ∼ (x ′0, . . . , x ′p, t0, . . . , tp)

if for each i with ti 6= 0, xi = x ′i .

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Property of fibered join

Theorem

Let f : A→ B be a semi-algebraic map that is a semi-algebraiccompact covering (i.e. for every semi-algebraic compact subsetL ⊂ f (A) there exsists a semi-algebraic compact subset K ⊂ Awith f (K ) = L). Then for every p ≥ 0, the map f induces ap-equivalence

J(f ) : Jpf (A)→ f (A).

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Key Lemma

Let S ⊂ Sm × Sn be a compact semi-algebraic set and let πdenote the projection on the second sphere.Then there exists a semi-algebraic set DY(S) which ishomotopy equivalent to Jn+1

π (S) and such that membership inDY(S) can be checked in polynomial time if the same is true forS itself.

polynomial hierarchy, betti numbers and a real analogue of toda's

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