polynomial hierarchy, betti numbers and a real analogue of toda's
TRANSCRIPT
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
A quick primer of basic definitions and notation
Initially let k = Z/2Z = {0̄, 1̄}.A language L is a set⋃
n>0
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
A quick primer of basic definitions and notation
Initially let k = Z/2Z = {0̄, 1̄}.A language L is a set⋃
n>0
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
A quick primer of basic definitions and notation
Initially let k = Z/2Z = {0̄, 1̄}.A language L is a set⋃
n>0
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Discrete Polynomial Time Hierarchy– A QuickReminder
A languageL = (Ln)n>0 ∈ Σω
if there exists a language L′ = (L′n)n>0 ∈ P such that
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 = ∃.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Discrete Polynomial Time Hierarchy– A QuickReminder
A languageL = (Ln)n>0 ∈ Σω
if there exists a language L′ = (L′n)n>0 ∈ P such that
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 = ∃.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Discrete Polynomial Time Hierarchy– A QuickReminder
A languageL = (Ln)n>0 ∈ Σω
if there exists a language L′ = (L′n)n>0 ∈ P such that
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 = ∃.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Reminder (cont.)
Similarly a language
L = (Ln)n>0 ∈ Πω
if there exists a language L′ = (L′n)n>0 ∈ P such that
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Reminder (cont.)
Similarly a language
L = (Ln)n>0 ∈ Πω
if there exists a language L′ = (L′n)n>0 ∈ P such that
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Reminder (cont.)
Similarly a language
L = (Ln)n>0 ∈ Πω
if there exists a language L′ = (L′n)n>0 ∈ P such that
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Reminder (cont.)
Similarly a language
L = (Ln)n>0 ∈ Πω
if there exists a language L′ = (L′n)n>0 ∈ P such that
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
Πω.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
Πω.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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”
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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”
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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”
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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”
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
The compact real polynomial hierarchy (cont.)
We define
PHcR
def=
⋃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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
The compact real polynomial hierarchy (cont.)
We define
PHcR
def=
⋃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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
Main theorem
Theorem (B-Zell,2008)
PHcR ⊂ P#P†
RR .
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
OutlineDetails
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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 .
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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 .
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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 .
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
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
πY : Sk × S` → Sk
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
πY : Sk × S` → Sk
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
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
πY : Sk × S` → Sk
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
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Proof
OutlineDetails
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 ))).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
Iterated joins
For p ≥ 0, the (p + 1)-fold join Jp(X ) of X is
Jp(X )def= X × · · · × X︸ ︷︷ ︸
(p+1) times
×∆p/ ∼,
where
(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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
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Proof
OutlineDetails
Iterated joins
For p ≥ 0, the (p + 1)-fold join Jp(X ) of X is
Jp(X )def= X × · · · × X︸ ︷︷ ︸
(p+1) times
×∆p/ ∼,
where
(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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
Algorithmic Semi-algebraic GeometryReal Analogue of Toda’s Theorem
Proof
OutlineDetails
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
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Proof
OutlineDetails
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/ ∼,
where
(x0, . . . , xp, t0, . . . , tp) ∼ (x ′0, . . . , x ′p, t0, . . . , tp)
if for each i with ti 6= 0, xi = x ′i .
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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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).
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem
(Discrete) Polynomial HierarchyBlum-Shub-Smale Models of Computation
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Key Lemma
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.
Saugata Basu Polynomial Hierarchy, Betti Numbers and a real analogue of Toda’s Theorem