proof complexity lower bounds from algebraic circuit ...shpilka/talks/fstw.pdf · algebraic...

Proof Complexity Lower Bounds from

Algebraic Circuit Complexity

Michael A. ForbesPrinceton University

Iddo TzameretRoyal Holloway, University of London

Amir ShpilkaTel Aviv University

Avi WigdersonInstitute for Advanced Study

June 8, 2016

1 / 27

Talk overview

Very short introduction to proof complexity

Nullstellensatz proof system

Models of algebraic computations

Ideal Proof System (IPS)

Our results

Some proofs

2 / 27

Refuting CNFs

Question (3SAT)

Given a 3CNF φ = C1 ∧ C2 ∧ · · · ∧ Cm prove there is no

satisfying assignment to φ.

Question is coNP-hard

NP 6= coNP =⇒ any proof must be long

holy grail: prove unconditional lower bounds on lengths of proofs

in every proof system.

note: potentially harder than proving P 6= NP

goal: prove unconditional lower bounds on lengths of proofs in

strong proof systems

3 / 27

Frege Proof System

Example (Hilbert Propositional Calculus)

Axioms:

φ→ (ψ → φ)

((φ→ (ψ → ζ))→ ((φ→ ψ)→ (φ→ ζ)))

(φ→ ψ)→ (¬ψ → ¬φ)

Derivation rules:

Modus ponens:φ, (φ→ ψ)

ψ

Frege proof systems capture the way we write proofs

goal: lower bounds on lengths of proofs in Frege proof system

known: lower bounds for restricted versions of Frege

this talk: algebraic proof systems

4 / 27

Subset Sum

Question (Subset Sum)

Given a1, . . . , an, b ∈ C, prove there is no subset S ⊆ [n] such

that∑i∈S ai = b.

Equivalently, prove there are no solutions to

x21 − x1 = · · · = x2

n − xn = 0,

a1x1 + · · ·+ anxn − b = 0 .

Question is coNP-hard

NP 6= coNP =⇒ any proof must be long

goal: prove unconditional lower bounds on lengths of proofs in

strong algebraic proof systems.

5 / 27

Nullstellensatz Proofs (i)

Let f := (f1, . . . , fm) be a system of polynomials in C[x1, . . . , xn].

Theorem (Hilbert’s Nullstellensatz)

The system f1(x) = · · · = fm(x) = 0 has no solution iff there are

g1, . . . , gm ∈ C[x ] such that

g1(x) · f1(x) + · · ·+ gm(x) · fm(x) = 1 .

Gives a sound and complete proof system for unsatisfiability.

complexity: simple unsatisfiable f can require deg g ≥ exp(m).

but: coNP-statements concern x ∈ 0, 1n — polynomials over

0, 1n are degree ≤ n.

6 / 27

Nullstellensatz Proofs (ii)

f := (f1, . . . , fm), x2 − x := (x21 − x1, . . . , x

2n − xn).

Theorem (Boolean Nullstellensatz)

The system f = 0 has no solution over x ∈ 0, 1n⇔ the system f , x2 − x is unsatisfiable

⇔ there are g1, . . . , gm, h1, . . . , hn ∈ C[x ] such that∑j

gj(x) · fj(x) +∑i

hi(x) · (x2i − xi) = 1 .

complexity: deg g, h ≤ O(n), g, h have ≤ 2O(n) monomials

goal: prove lower bound on the complexity of g, h.

prior work ([BIK+96a, CEI96, BIK+96b, Raz98, Gri98, IPS99,

BGIP01, AR01,. . .]): exhibit simple f where

deg g, h ≥ Ω(n)

g, h require 2Ω(n) monomials7 / 27

Algebraic Complexity Measures

Degree and number of monomials are weak measures for

complexity

More interesting measure: size of representation in interesting

algebraic models

this talk: study Ideal Proof System (IPS) that looks at more

interesting measures of complexity

next: Algebraic models and Grochow-Pitassi IPS proof system

8 / 27

Algebraic Formulas

Algebraic formulas are a succinct model of computation for

polynomials, e.g. det

(a b

c d

)= ad − bc can be given by

+

×

a d

×

b c

−1

size: number of edges

depth: length of longest input-output path

known: [Kalorkoti85] Ω(n2) lower bound for algebraic formulas

9 / 27

The Ideal Proof System (IPS) (i)

f := (f1, . . . , fm), x2 − x := (x21 − x1, . . . , x

2n − xn).

Definition ([GrochowPitassi14])

A formula-IPS proof of f , x2 − x is a pair (g, h) such that∑gj fj +

∑hi · (x2

i − xi) = 1

the size of the proof is the formula size of gj , hi

proof verification: via Polynomial Identity Testing, only

randomized algorithms known in general.

goal: prove lower bounds for interesting f

How realistic is this goal?

10 / 27

The Ideal Proof System (IPS) (ii)

Theorem ([GrochowPitassi14])

Let CNF C = C1 ∧C2 ∧ · · · ∧Cm be unsatisfiable and encoded by

the equations f1, . . . , fm, x2 − x. Then there are g, h such that∑j

gj · fj +∑i

hi · (x2i − xi) = 1, where

If there is a size-s Frege proof that C is unsatisfiable, then

there are g, h with poly(n,m, s)-size algebraic formulas.

There are g, h in VNP ≈ explicit polynomials.

Corollary

formula-IPS as powerful as Frege

lower bounds for CNFs =⇒ lower bounds for the permanent

goal: prove lower bounds for restricted, yet interesting and

powerful, algebraic models11 / 27

The Ideal Proof System (IPS) (iii)

f := (f1, . . . , fm), x2 − x := (x21 − x1, . . . , x

2n − xn).

Definition ([GrochowPitassi14])

For an algebraic model C, a C-IPS proof of f , x2 − x is a pair

(g, h) such that∑gj fj +

∑hi · (x2

i − xi) = 1

the size of the proof is the maximum size of the gj , hi as

C-computations.

goal: prove lower bounds for C-IPS for interesting C

12 / 27

Multilinear Formulas

Definition

A polynomial f ∈ C[x1, . . . , xn] is multilinear if the individual

degree of each variable xi is at most 1, that is

f (x) =∑S⊆[n]

αS∏i∈S

xi

A formula is multilinear if each gate is multilinear

multilinear polynomials uniquely determined by evaluations

over 0, 1n

known: [Raz04,RY09]: permanent and determinant require

nΩ(lg n)-size multilinear formulas, 2nΩ(1)

-size constant-depth

multilinear formulas

goal: prove lower bounds for multilinear-formula-IPS

13 / 27

Algebraic Branching Programs

s...

x1 − 1

x7 + 2

x3 + 9

...

x2 + 1

3x1 · · · ...t

2xn − 2

xk

x3 − 4

Width

each s → t path computes multiplication of edge labels

program computes the sum of those over all s → t paths

output is the (1, 1) entry of the matrix product∏ni=1 Mi(x)

(Mi is adjacency matrix of graph on layers i ,i + 1)

goal: prove lower bounds for ABP-IPS

problem: ABPs more expresfull than formulas

14 / 27

Read-Once Oblivious ABPs (i)

read-once oblivious ABP (roABP): every variable appears in

one layer

s...

x1 − 1

x1 + 2

x1 + 9

...

x2 + 1

3x2

...t

2xn − 2

xn

xn − 4

Width

[Nisan91]: exponential lower bounds for roABPs

[RS05]: polynomial time PIT for roABPs

[FS13,AGKS14,FSS14] quasi-poly black-box PIT for roABPs

goal: prove lower bounds for roABP-IPS15 / 27

Recap

Nullstellensatz proof system: proof of unsatisfiability of

f , x2 − x is g, h such that∑j

gj(x) · fj(x) +∑i

hi(x) · (x2i − xi) = 1

IPS: each gj , hi has “small” C-computation

Algebraic classes:

multilinear formulas

read-once oblivious Algebraic Branching Programs (roABPs)

Next: our results and some proofs

16 / 27

Our Results: Upper Bounds

x1 + · · ·+ xn + 1, x2 − x , is unsatisfiable subset-sum instance

Theorem ([ImpagliazzoPudlakSgall99])

x1 + · · ·+ xn + 1, x2 − x requires Nullstellensatz refutations of

degree ≥ Ω(n).

2Ω(n)-monomials.

Related to Pigeonhole Principle, well-known “hard” principle

Theorem (Upper Bounds for Subset-Sum)

x1 + · · ·+ xn + 1, x2 − x has a poly(n)-size C-IPS proof for C =

depth-3 multilinear formulas

read-once oblivious algebraic branching programs (roABPs)

Strengthens related upper bounds of [GH03,RT08]

17 / 27

Our Results: Lower Bounds

Theorem (Lower Bounds for Subset-Sum Variants)∑i<j zi ,jxixj + 1, x2 − x , z 2 − z requires

multilinear-formula-IPS proofs of nΩ(lg n)-size

constant-depth-multilinear-formula-IPS proofs of 2nΩ(1)

-size

roABP-IPS proofs of 2Ω(n)-size (in every order)

First such lower bounds, matches much of the frontier of lower

bounds in algebraic complexity theory

Proven via functional lower bounds.

18 / 27

Functional Lower Bounds

circuit complexity: single polynomial requires large formulas

proof complexity: every proof requires large formulas

idea: if “unique” proof then only study single polynomial

Consider an unsatisfiable system f (x), x2 − x , with proof

g(x) · f (x) +∑i

hi(x) · (x2i − xi) = 1

This implies

g(x) · f (x) = 1, x ∈ 0, 1n

g(x) = 1/f (x), x ∈ 0, 1n

=⇒ g unique as a function over 0, 1n or as multilinear

polynomial

goal: find easy f (x) so any g with g|0,1n = 1f |0,1n is hard

need: proof techniques that works in the functional setting and

not just for syntactic polynomials (e.g. partial derivative method)19 / 27

Structure of Proof




-size

roABP-IPS proofs of 2Ω(n)-size (in every order)

Proof.

prove functional lower bound for degree

“lift” to functional lower bound for evaluation dimension

conclude functional lower bound for circuit classes via known

relations to evaluation dimension

conclude IPS lower bounds

20 / 27

Functional Lower Bound — Degree

x1 + · · ·+ xn + 1, x2 − x

Proposition

Let g ∈ C[x ] such that

g(x) =1

x1 + · · ·+ xn + 1, x ∈ 0, 1n

Then deg g ≥ n

Tight. Strengthens prior deg g ≥ n/2 [IPS99].

Proof.

multilinearize: g 7→ ml(g) with g|0,1n = ml(g)|0,1n , and

deg g ≥ deg ml(g)

ml(g) uniquely determined, compute it explicitly,

deg ml(g) = n

21 / 27

Evaluation Dimension

g ∈ C[x1, . . . , xn, y1, . . . , yn]. Study interaction between x and y

Definition ([Nis91,Sap12,FS13b])

The set of evaluations of g is

Evalx |y (g) := g(x , b)b∈0,1n ⊆ C[x ]

The evaluation dimension of g is dimC Evalx |y (g)

Well-studied complexity measure, (implicitly) used for many

lower bounds:

multilinear formulas [Raz04,RY09,. . .]

non-commutative ABPs, roABPs [Nisan91,FS13,. . .]

depth-3 powering formulas [Saxena08,FS13,. . .]

22 / 27

Functional Lower Bound — Evaluation Dimension∑ni=1 xiyi + 1, x2 − x , y 2 − y

Proposition

g(x , y ) = 1∑ixiyi+1

for x , y ∈ 0, 1n, then dimEvalx |y (g) ≥ 2n

Proof.

Evalx |y (g) = g(x , b)b∈0,1n = g(x ,1S)S⊆0,1n

For x ∈ 0, 1n, g(x ,1S) = 1∑i∈S xi+1

ml(g(x ,1S)) has degree |S |ml(g(x ,1S)) =

∏i∈S xi + (lower terms)

ml(g(x ,1S)) linearly independent

dimEvalx |y (g)≥ dim ml(Evalx |y (g))= dimml(g(x ,1S))S⊆[n]

= dim∏i∈S

xi + (lower terms)S

= 2n

23 / 27

Lower Bounds for IPS




-size

roABP-IPS proofs of 2Ω(n)-size

Proof.

degree ≥ n functional lower bound for 1∑ixi+1

dimEvalx |y ≥ 2n functional lower bound for 1∑ixiyi+1

dimEvalv |u ≥ 2n/2 functional lower bounds for 1∑i<jzi,jxixj+1

,

for any equipartition x = v ∪ u

implies lower bounds for the mentioned circuit classes

24 / 27

Upper Bounds for IPS (i)

Theorem (Upper Bounds for Subset-Sum)

x1 + · · ·+ xn + 1, x2 − x has a poly(n)-size C-IPS proof for C =



Proof.

Let p(t) =∏ni=0(t − i)

Then, p(∑

xi) = 0 for x ∈ 0, 1n

Equivalently, p(∑

xi) = 0 modulo x2 − x

Note p(t)−p(−1)t−(−1) = q(t) for some polynomial q

q(∑

xi)(∑

xi + 1) = p(∑

xi)− p(−1) ≡ −p(−1) 6= 0

Are we really done?25 / 27

Upper Bounds for IPS (ii)

Showed that for p(t) =∏ni=0(t − i) and q(t) = p(t)−p(−1)

t−(−1) ,

q(∑

xi) (∑

xi + 1)

= p(∑

xi)− p(−1) ≡ −p(−1) 6= 0

To show poly(n)-size C-IPS proofs we need

q(x)(∑

xi + 1)

+n∑i=1

hi(x)(x2i − xi) = 1 , where

q(∑

xi) has short C-computations

can efficiently multilinearize q(x) (∑

xi + 1) in C(this gives h)

Proposition

q(∑

xi) =∑poly(n)i=1

∏nj=1(ai ,jxj + bi ,j)

Can express this as small roABP and depth-3 multilinear formula

and not too difficult to multilinearize26 / 27

Conclusions

this talk:

upper bounds for proving unsatisfiability of∑i xi + 1, x2− x



lower bounds for proving unsatisfiability of∑i<j zi ,jxixj + 1, x2 − x , z 2 − z



-size

roABP-IPS proofs of 2Ω(n)-size

open question:

lower bounds for unsatisfiability of f1, . . . , fm, x2 − x where

each fi depends on o(n) many variables ?

27 / 27

TOC

1 Title

2 Talk overview

3 Refuting CNFs

4 Frege Proof System

5 Subset Sum

6 Nullstellensatz Proofs (i)

7 Nullstellensatz Proofs (ii)8 Algebraic Complexity

Measures

9 Algebraic Formulas10 The Ideal Proof System

(IPS) (i)11 The Ideal Proof System

(IPS) (ii)12 The Ideal Proof System

(IPS) (iii)

13 Multilinear Formulas

14 Algebraic Branching

Programs

15 roABPs (i)

16 Recap

17 Our Results: Upper Bounds

18 Our Results: Lower Bounds

19 Functional Lower Bounds

20 Structure of Proof21 Functional Lower Bound —

Degree

22 Evaluation Dimension23 Functional Lower Bound —

Evaluation Dimension

24 Lower Bounds for IPS

25 Upper Bounds for IPS (i)

26 Upper Bounds for IPS (ii)

27 Conclusions27 / 27

proof complexity lower bounds from algebraic circuit ...shpilka/talks/fstw.pdf · algebraic...

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