On the PPA -completeness of the Combinatorial Nullstellensatz and - - PowerPoint PPT Presentation

On the PPA-completeness of the Combinatorial Nullstellensatz and the Chevalley-Warning Theorem

Miklos Santha

CNRS, Universit´ e Paris Diderot, France and Centre for Quantum Technologies, NUS, Singapore joint work with Alexander Belov G´ abor Ivanyos Youming Qiao Siyi Yang

• U. of Latvia, Riga

SZTAKI, Budapest

• U. Tech., Sydney

CQT, Singapore

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Overview of the talk

1 The class PPA 2 CNSS and Chevalley-Warning Theorem 3 Arithmetic circuits and parse subcircuits 4 The problems PPA-Circuit Chevalley and

PPA-Circuit CNSS

5 PPA-hardness and PPA-easiness

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The class PPA

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Functional NP (FNP)

NP-search problems are defined by binary relations R ⊆ {0, 1}∗ × {0, 1}∗ such that – R ∈ P, – for some polynomial p(n), R(x, y) = ⇒ |y| ≤ p(|x|). Search problem ΠR Input: x Output: A solution y such that R(x, y) if there is any, or “failure” ΠR is reducible to ΠS if there exist polynomial time computable functions f and g such that, for every positive x, S(f (x), y) = ⇒ R(x, g(x, y)).

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Total Functional NP (TFNP) [MP’91]

An NP-search problem is total if for all x there exists a solution y. Facts:

• If FNP ⊆ TFNP then NP = coNP.
• If TFNP ⊆ P then P = NP ∩ coNP.

TFNP is a semantic complexity class Syntactical subclasses of TFNP:

• Polynomial Local Search PLS

Examples: Local optima, pure equilibrium in potential games

• Polynomial Pigeonhole Principle PPP

Examples: Pigeonhole SubsetSum, Discrete Logarithm

• Polynomial Parity Argument classes PPA, PPAD.

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Polynomial Parity Argument [P’94]

Parity Principle: In a graph the number of odd vertices is even. Definition: PPA is the set of total problems reducible to Leaf Leaf Input: (z, M, ω), where

• z is a binary string
• M is a polynomial TM that defines a graph Gz = (Vz, Ez)
• Vz = {0, 1}p(|z|) for some polynomial p
• for v ∈ Vz, M(z, v) is a set of at most two vertices
• {v, v′} ∈ Ez if v′ ∈ M(z, v) and v ∈ M(z, v′)
• ω ∈ Vz is a degree one vertex, the standard leaf

Output: A leaf different from ω.

ω

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PPA with edge recognition and pairing

Graphs Gz = (Vz, Ez) of unbounded degree can be defined by two polynomial time algorithms ǫ and φ:

• Edge recognition: {v, v′} ∈ Ez ⇔ ǫ(v, v′) = 1
• Pairing: For every vertex v,
• if deg(v) is even the function φ(v, ·) is a pairing between the

vertices adjacent to v.

• if deg(v) is odd then

there exists exactly one neighbor w such that φ(v, w) = w, and on the remaining neighbors φ(v, ·) is a pairing.

. . . . . .

Fact: A problem defined in terms of ǫ and π is in PPA. Proof: Let G ′

z = (V ′ z, E ′ z) be defined as

• V ′

z = Ez

• {{v, w}, {v, w′}} ∈ E ′

z

if φ(w) = w′.

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Examples of problems in PPA

Few complete problems are known, all discretizations or combinatorial analogues of topological fixed point theorems:

• 3-D Sperner in some non-orientable space [G’01]
• Locally 2-D Sperner [FISV’06]
• Sperner and Tucker on the M¨
• bius band [DEFLQX16]
• 2-D Tucker in the Euclidean space [ABB’15]

Many problems of various origins are in PPA:

• Graph theory: Smith, Hamiltonian decomp. [P’94]
• Combinatorics: Necklace splitting and Discrete ham

sandwich [P’94]

• Algebra: Explicit Chevalley [P’94]
• Number theory: Square root and Factoring [J’16]

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Combinatorial Nullstellensatz and Chevalley-Warning Theorem

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Combinatorial Nullstellensatz

Theorem [Alon’99]: Let F be a field, let d1, . . . , dn be non-negative integers, and let P ∈ F[x1, . . . , xn] be a polynomial. Suppose that

• deg(P) = n

i=1 di,

• the coefficient of xd1

1 . . . xdn n

is non-zero. Then for every subsets S1, . . . , Sn of F with |Si| > di, there exists (s1, . . . sn) ∈ S1 × · · · × Sn such that P(s1, . . . , sn) = 0. Consequences in algebra, graph theory, combinatorics, additive number theory ...

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Chevalley-Warning Theorem

Theorem [Chevalley’36, Warning’36]: Let F be a field of characteristic p, and let P1, . . . , Pk ∈ F[x1, . . . , xn] be non-zero polynomials. If k

i=1 deg(Pi) < n, then the number of common zeros of

P1, . . . , Pk is divisible by p. In particular, if the polynomials have a common root, they also have another one.

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The theorems over F2

Definition A multilinear polynomial over F2 is M(x1, . . . , xn) =

• T⊆{1,...,n}

cT

• i∈T

xi, where cT ∈ F2 Fact: For every P over F2, there exists a unique multilinear polynomial MP such that P and MP compute the same function. Definition: The multilinear degree of P is mdeg(P) = deg(MP). Theorem [Combinatorial Nullstellensatz over F2]: Let P be such that mdeg(P) = n. Then there exists a ∈ Fn

2 such that P(a) = 1.

Theorem[Chevalley-Warning over F2]: Let P such that mdeg(P) < n, and let a ∈ Fn

2 such that P(a) = 0.

Then there exists b = a such that P(b) = 0. Theorem: mdeg(P) < n ⇐ ⇒ the number of zeros is even

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How to make them search problems?

Theorem[P’94]: The following problem is in PPA. Explicit Chevalley Input: Explicitly given polynomials P1, . . . , Pk over F2 such that k

i=1 deg(Pi) < n,

and a common root a ∈ Fn

2.

Output: Another common root a′ = a. Remark: a is common root ⇔ P(a) = 0 where P = 1 + k

i=1(Pi + 1)

Could this be PPA-hard? Probably not. Two restrictions:

• P is given by an arithmetic circuit of specific form
• even the degree of P is less than n

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Arithmetic circuits and parse subcircuits

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Arithmetic circuits

× + + + x1 x2 x3 x4

C is a labeled, directed, acyclic graph. Labels = {+, ×}, G + = sum gates, G × = product gates. Computational gates have indegree 2: left and right child Polynomial computed by C C(x) = (x1 + x2 + x3) × (x2 + x3 + x4) = x1x2 + x1x3 + x1x4 + x2

2 + x2x4 + x2 3 + x3x4

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Lagrange-circuits

Circuits computing the Lagrange basis polynomials La(x) La(x) = 1 ⇐ ⇒ x = a × + + x1 x2 1 x3 Lagrange-circuit L100

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Degrees in a circuit

There are 3 types of degree

× 3 × 2 + 1 x1 x1 + 1 1 x2 x2 × 2 × 2 + x1 x1 + 1 x2 x2 × 1 × 1 + x1 x1 + 1 x2 x2

Formal degree = 3 Polynomial degree = 2 Multilinear degree =1 2 = 0 x2 = x easy to compute ?? We are interested in the multilinear degree

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Multilinear degree and monomials

+ (x1 + x2)20 x1 x2 × (x1 + x2)21 × (x1 + x2)22 . . . × (x1 + x2)2n How can we certify mdeg(C(x)) = n? What is the the complexity of Mdeg = {C : mdeg(C(x)) = n}? We wish Mdeg ∈ NP A monomial m computed by C is maximal if mdeg(m) = n Fact: mdeg(C(x)) = n ⇐ ⇒

• dd number of maximal monomials

Difficulty: the number of monomials computed by C can be doubly exponential in the size of C We can certainly say that Mdeg ∈ ⊕EXP

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Monomials in arithmetic formulae

Let F be an arithmetic formula Monomials are computed by parse subtrees defined by the marking

• f appropriate sum gates: S : G + → {ℓ, r, ∗}:

× + ℓ + r 1 x2 x2 x1

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Parse subcircuits

C arithmetic circuit. A parse subcircuit is a partial marking S : G + → {ℓ, r, ∗} such that marked vertices = accessible vertices × + r + ℓ + r x1 x2 x3 x4 × + ℓ + r + ∗ x1 x2 x3 x4 computes x2

3

computes x1x4

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Parse subcircuits witness monomials

S(C) = set of parse subcircuits of C, mS(x) = monomial computed by parse sub circuit S Theorem: Let F be a field of characteristic 2. Then C(x) =

• S∈S(C)

mS(x). × + U + W + g x1 x2 x3 x4 × + U ′ + W ′ + g x1 x2 x3 x4 mUmW = x2x3 mU′mW ′ = x2x3 Corollary: Mdeg ∈ ⊕P Proposition: Mdeg is ⊕P-hard.

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The problems PPA-Circuit Chevalley and PPA-Circuit CNSS

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Towards PPA-circuits

We would like to characterize PPA with arithmetic circuits Auxiliary circuits I and I ⋄ C:

× + · · · + x1 · · · xn y1 · · · yn 1 C x1 · · · xn 1 + · · · + × · · · · · ·

I(x1, . . . , xn, y1, . . . , yn) = n

i=1(xi + yi + 1)

I(x, y) = 1 ⇐ ⇒ x = y I ⋄ C(x) = 1 ⇐ ⇒ C(x) = x

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PPA-circuits

Definition: A PPA-circuit is the PPA-composition CD,F of two n-variable, n-output arithmetic circuits D and F over F2

+ x1 · · · xn I1 ⋄ D1 ◦ F1 · · · I2 ⋄ F2 ◦ D2 · · · I3 ⋄ D3 ◦ D4 · · · I4 ⋄ D5 · · · I5 ⋄ F3 ◦ F4 · · · I5 ⋄ F5 · · ·

CD,F PPA-Circuit Matching Lemma: If C is a PPA-circuit then in polynomial time a perfect matching µ can be computed between the maximal parse subcircuits of C.

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PPA-Circuit Matching Lemma

We want to define a polynomial time computable µ: perfect matching on the maximal parse subcircuits of CD,F

+ + C1 + C2 + C3 x1 · · · xn I1 ⋄ D1 ◦ F1 · · · I2 ⋄ F2 ◦ D2 · · · I3 ⋄ D3 ◦ D4 · · · I4 ⋄ D5 · · · I5 ⋄ F3 ◦ F4 · · · I5 ⋄ F5 · · ·

CD,F = C1 + C2 + C3 µ is defined inside C1, inside C2 and inside C3

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The matching µ inside C1

+ C1 x1 · · · xn I ⋄ D ◦ F · · · I′ ⋄ F ′ ◦ D′ · · ·

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The matching µ inside C1

x1 x2 x3 1 + h1 + h2 + h3 × D F f1 f2 f3 d1 d2 d3 Sin = {1, 2, 3} Smiddle = {1, 3} Sout = {1, 2}

i ∈ Sout if the edge from the di to hi belongs to S i ∈ Smiddle if there exists an edge in S from fi to a gate in D i ∈ Sin if there exists an edge in S from xi to a gate in F Claim: Sout ⊆ Sin

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The matching µ inside C1

Case 1: Sout ⊂ Sin Let i be the smallest index in Sin \ Sout

× + ℓ xi di 1 × + r xi di 1

S µ(S)

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The matching µ inside C1

Case 2: Sout = Sin

+ D F x3 x1 x2 + F ′ D′ x2 x3 x1

S µ(S)

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The matching µ inside C2

+ C2 x1 · · · xn I ⋄ D ◦ D′ · · · I∗ ⋄ D∗ · · ·

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The matching µ inside C2

x1 x2 x3 1 + h1 + h2 + h3 × D D′ d′

1

d′

2

d′

3

d1 d2 d3 Sin = {1, 2, 3} Smiddle = {1, 3} Sout = {1, 2} 31/44

The matching µ inside C2

Case 1: Sout ⊂ Sin Let i be the smallest index in Sin \ Sout

× + ℓ xi di 1 × + r xi di 1

S µ(S)

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The matching µ inside C2

Case 2: Sout = Sin and S(g) = S(g′) for some sum gate in D

+ D D′ x3 x1 x2 + D D′ x3 x1 x2

S µ(S)

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The matching µ inside C2

Case 3: Sout = Sin and S(g) = S(g′) for all sum gate in D

+ D D′ x3 x1 x2 + D∗ x3 x1 x2

S µ(S)

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The computational problems

PPA-Circuit Chevalley Input: (C, a), where C: an n-variable PPA-circuit over F2, a: a root of C. Output: Another root b = a of C. PPA-Circuit CNSS Input: (C ′, a), where C: an n-variable PPA-circuit over F2, a: an element of Fn

2.

Output: An element b ∈ Fn

2 satisfying C = C ′ ⊕ La.

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The result

Main Theorem: PPA-Circuit Chevalley and PPA-Circuit-CNSS are PPA-complete. The proof contains three parts: Proposition: PPA-Circuit Chevalley and PPA-Circuit CNSS are polynomially equivalent. Hardness Theorem: PPA-Circuit Chevalley is PPA-hard. Easiness Theorem: PPA-Circuit CNSS is in PPA.

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PPA-hardness and PPA-easiness

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PPA-hardness

Theorem: PPA-Circuit Chevalley is PPA-hard. Proof: Reduce Leaf to PPA-Circuit Chevalley Express the ≤ 2 neighbours M(u) of u via D(u) and F(u):

• Case 1:

u

then D(u) = F(u) = u,

• Case 2:

u v

then D(u) = v and F(u) = u,

• Case 3:

u v w

then D(u) = v and F(u) = w Claim: Parity of deg(u) = Parity of satisfied components of CD,F u

(a) Case 1

u v · · ·

(b) Case 2-a

u v · · ·

(c) Case 2-b

u v w · · · · · ·

(d) Case 3-a

u v w · · · · · ·

(e) Case 3-b

u v w · · · · · ·

(f) Case 3-c

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PPA-easiness

We prove something stronger Matched-Circuit CNSS Input: (C, T, µ), where C: an n-variable arithmetic circuit over F2, T: maximal parse subcircuit µ: polynomial time perfect matching for the maximal parse subcircuits in C but T. Output: An element b ∈ Fn

2 satisfying C.

Theorem: Matched-Circuit CNSS is in PPA

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An instance of Matched-Circuit CNSS

Input: N = (C, T, µ) Remark: C(x) = x1x2

× + ℓ + ℓ + r x1 x2 × + ℓ + r + ∗ x1 x2 × + r + r + ℓ x1 x2

µ matches ℓℓr and ℓr∗ unmatched T = rrℓ

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PPA-easiness

Theorem: Matched-Circuit CNSS is in PPA Proof: We reduce Matched-Circuit CNSS to Leaf.

00 01 10 11 ℓℓℓ : x2

1

ℓℓr : x1x2 ℓr∗ : x1x2 rℓℓ : x2

1

rℓr : x2

2

rrℓ : x1x2 T rrr : x2

2

µ

GN resulting from the Circuit-CNSS instance N = (C, µ, T)

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The pairing on the left hand side

Vertex 01 of even degree: For all parse subcircuit S, mS(a) = 1, ∃ sum gate g with Pg(a) = 0

× + 1 + + 1 x1 x2 1 × + 1 + + 1 x1 x2 1

rℓr rrr

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The pairing on the left hand side

Vertex 11 of odd degree: There exists a unique S, mS(a) = 1, such that Pg(a) = 1 for all sum gate g

× 1 + 1 + 1 + x1 1 x2 1 × 1 + 1 + 1 + x1 1 x2 1 × 1 + 1 + 1 + x1 1 x2 1

ℓℓℓ ℓℓr unique unmatched parse subcircuit

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Thank you

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