Post on 31-May-2020
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
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)
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
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
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 =
depth-3 multilinear formulas
read-once oblivious algebraic branching programs (roABPs)
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
depth-3 multilinear formulas
read-once oblivious algebraic branching programs (roABPs)
lower bounds for proving unsatisfiability of∑i<j zi ,jxixj + 1, x2 − x , z 2 − z
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
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