On the PPA-completeness ofthe Combinatorial Nullstellensatz and

the Chevalley-Warning Theorem

Miklos Santha

CNRS, Universite Paris Diderot, France

andCentre for Quantum Technologies, NUS, Singapore

joint work with

Alexander Belov Gabor Ivanyos Youming Qiao Siyi YangU. of Latvia, Riga SZTAKI, Budapest U. Tech., Sydney CQT, Singapore

1/44

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 andPPA-Circuit CNSS

5 PPA-hardness and PPA-easiness

2/44

The class PPA

3/44

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: xOutput: A solution y such that R(x , y) if there is any, or “failure”

ΠR is reducible to ΠS if there exist polynomial time computablefunctions f and g such that, for every positive x ,

S(f (x), y) =⇒ R(x , g(x , y)).

4/44

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 PLSExamples: Local optima, pure equilibrium in potential games

• Polynomial Pigeonhole Principle PPPExamples: Pigeonhole SubsetSum, Discrete Logarithm

• Polynomial Parity Argument classes PPA,PPAD.

5/44

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

LeafInput: (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

6/44

PPA with edge recognition and pairing

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

• Edge recognition: {v , v ′} ∈ Ez ⇔ ε(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 ofε and π is in PPA.

Proof: Let G ′z = (V ′z ,E′z) be defined as

• V ′z = Ez

• {{v ,w}, {v ,w ′}} ∈ E ′zif φ(w) = w ′.

7/44

Examples of problems in PPA

Few complete problems are known, all discretizations orcombinatorial 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 Mobius 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 hamsandwich [P’94]

• Algebra: Explicit Chevalley [P’94]

• Number theory: Square root and Factoring [J’16]

8/44

Combinatorial Nullstellensatz and

Chevalley-Warning Theorem

9/44

Combinatorial Nullstellensatz

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

• deg(P) =∑n

i=1 di ,

• the coefficient of xd11 . . . xdnn 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) 6= 0.

Consequences in algebra, graph theory, combinatorics, additivenumber theory ...

10/44

Chevalley-Warning Theorem

Theorem [Chevalley’36, Warning’36]: Let F be a field ofcharacteristic p, and let P1, . . . ,Pk ∈ F[x1, . . . , xn] be non-zeropolynomials.

If∑k

i=1 deg(Pi ) < n, then the number of common zeros ofP1, . . . ,Pk is divisible by p.

In particular, if the polynomials have a common root, they alsohave another one.

11/44

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 multilinearpolynomial 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 suchthat mdeg(P) = n.Then there exists a ∈ Fn

2 such that P(a) = 1.

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

2 such that P(a) = 0.Then there exists b 6= a such that P(b) = 0.

Theorem: mdeg(P) < n ⇐⇒ the number of zeros is even12/44

How to make them search problems?

Theorem[P’94]: The following problem is in PPA.

Explicit ChevalleyInput: Explicitly given polynomials P1, . . . ,Pk over F2 such that∑k

i=1 deg(Pi ) < n,

and a common root a ∈ Fn2.

Output: Another common root a′ 6= 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

13/44

Arithmetic circuits and parse subcircuits

14/44

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 + x22 + x2x4 + x23 + x3x4

15/44

Lagrange-circuits

Circuits computing the Lagrange basis polynomials La(x)

La(x) = 1 ⇐⇒ x = a

×

+ +x1

x2 1 x3

Lagrange-circuit L100

16/44

Degrees in a circuit

There are 3 types of degree

×3

×2

+

1

x1 x1 +

1

1

x2 x2

×2

×2

+

0

x1 x1 +

0

1

x2 x2

×1

×1

+

0

x1 x1 +

0

1

x2 x2

Formal degree = 3 Polynomial degree = 2 Multilinear degree =12 = 0 x2 = x

easy to compute ??

We are interested in the multilinear degree

17/44

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 ofMdeg = {C : mdeg(C (x)) = n}?

We wish Mdeg ∈ NP

A monomial m computed by C ismaximal if mdeg(m) = n

Fact: mdeg(C (x)) = n ⇐⇒odd number of maximal monomials

Difficulty: the number of monomials computed by C can be doublyexponential in the size of C

We can certainly say that Mdeg ∈ ⊕EXP18/44

Monomials in arithmetic formulae

Let F be an arithmetic formula

Monomials are computed by parse subtrees defined by the markingof appropriate sum gates: S : G+ → {`, r , ∗}:

×

+

ℓ

+

r

1 x2 x2 x1

19/44

Parse subcircuits

C arithmetic circuit. A parse subcircuit is a partial marking

S : G+ → {`, r , ∗}such that

marked vertices = accessible vertices

×

+

r

+

ℓ

+

r

x1

x2 x3

x4

×

+

ℓ

+

r

+

∗x1

x2 x3

x4

computes x23 computes x1x4

20/44

Parse subcircuits witness monomialsS(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.21/44

The problems

PPA-Circuit Chevalley

and PPA-Circuit CNSS

22/44

Towards PPA-circuitsWe 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) = x23/44

PPA-circuits

Definition: A PPA-circuit is the PPA-composition CD,F of twon-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 µcan be computed between the maximal parse subcircuits of C .

24/44

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

25/44

The matching µ inside C1

+C1

x1 · · · xn

I ⋄ D ◦ F

· · ·I ′ ⋄ F ′ ◦ D′

· · ·

26/44

The matching µ inside C1

x1 x2 x3

1

+h1 +h2 +h3

×

D

Ff1 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 Si ∈ Smiddle if there exists an edge in S from fi to a gate in Di ∈ Sin if there exists an edge in S from xi to a gate in F

Claim: Sout ⊆ Sin27/44

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)

28/44

The matching µ inside C1

Case 2: Sout = Sin

+

D

F

x3

x1 x2

+

F ′

D′

x2 x3

x1

S µ(S)

29/44

The matching µ inside C2

+C2

x1 · · · xn

I ⋄ D ◦ D′

· · ·I∗ ⋄ D∗

· · ·

30/44

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)

32/44

The matching µ inside C2

Case 2: Sout = Sin and S(g) 6= S(g ′) for some sum gate in D

+

D

D′

x3

x1 x2

+

D

D′

x3

x1 x2

S µ(S)

33/44

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)

34/44

The computational problems

PPA-Circuit Chevalley

Input: (C , a), whereC : an n-variable PPA-circuit over F2,a: a root of C .

Output: Another root b 6= a of C .

PPA-Circuit CNSS

Input: (C ′, a), whereC : an n-variable PPA-circuit over F2,a: an element of Fn

2.

Output: An element b ∈ Fn2 satisfying C = C ′ ⊕ La.

35/44

The result

Main Theorem: PPA-Circuit Chevalleyand PPA-Circuit-CNSS are PPA-complete.

The proof contains three parts:

Proposition: PPA-Circuit Chevalleyand PPA-Circuit CNSS are polynomially equivalent.

Hardness Theorem: PPA-Circuit Chevalley is PPA-hard.

Easiness Theorem: PPA-Circuit CNSS is in PPA.

36/44

PPA-hardness and PPA-easiness

37/44

PPA-hardnessTheorem: PPA-Circuit Chevalley is PPA-hard.

Proof: Reduce Leaf to PPA-Circuit ChevalleyExpress 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:uv 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

uv w· · · · · ·

(d) Case 3-a

uv w· · · · · ·

(e) Case 3-b

uv w· · · · · ·

(f) Case 3-c

38/44

PPA-easiness

We prove something stronger

Matched-Circuit CNSS

Input: (C ,T , µ), whereC : an n-variable arithmetic circuit over F2,T : maximal parse subcircuitµ: polynomial time perfect matching for the maximal parsesubcircuits in C but T .

Output: An element b ∈ Fn2 satisfying C .

Theorem: Matched-Circuit CNSS is in PPA

39/44

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 ``r and `r∗ unmatched T = rr`

40/44

PPA-easinessTheorem: Matched-Circuit CNSS is in PPA

Proof: We reduce Matched-Circuit CNSS to Leaf.

00

01

10

11

ℓℓℓ : x21

ℓℓr : x1x2

ℓr∗ : x1x2

rℓℓ : x21

rℓr : x22

rrℓ : x1x2T

rrr : x22

µ

GN resulting from the Circuit-CNSS instance N = (C , µ,T )41/44

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

×0

+1 + 0

+

1

x10 x2 1

×0

+1 + 0

+

1

x10 x2 1

r`r rrr

42/44

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 allsum gate g

×1

+1 + 1

+

0

x11 x2 1

×1

+1 + 1

+

0

x11 x2 1

×1

+1 + 1

+

0

x11 x2 1

``` ``r unique unmatchedparse subcircuit

43/44

Thank you

44/44