Iddo TzameretTel Aviv University

The Strength of Multilinear

Proofs(Joint work with Ran Raz)

Introduction:Algebraic Proof

Systems

Algebraic Proofs

Example:x1-x1x2=0, x2-x2x3=0, 1-x1=0, x3=0

xi2 – xi=0 for every i

•Fix a field

•Demonstrate a collection of polynomial-equations has no 0/1 solutions over

Algebraic Proofs

x1-x1x2

x3

x2-x2x31-x1

x1x3-x1x2x3x1x2-x1x2x3

x3x1-x1x2

x1-x1x3

1-x1x3

1

x1x3

+

+

+

+

=0

=0 =

0

=0

=0

=0

=0

=0

=0

=0

=0

Defn: A Polynomial Calculus (PC) refutation of p1, ... pk is a sequence of polynomials terminating with 1generated as follows (CEI96) :

i

fx f

f gf g

Axioms: pi , xi2-xi

Inference rules:

The Polynomial Calculus

This enables completeness (the initial collection of polynomials is unsatisfiable over 0/1 values)

We can consider algebraic proof systems as proof systems for CNF formulas:

A k-CNF:

1 1 2 2 3 3x x x x x x

becomes a system of degree k monomials:

Translation of CNF Formulas

1 1 2 2 3 3, , ,x x x x x x Where we add the following axioms

(PCR): 1i ix x

–Degree lower bounds imply many monomials: –Linear degree lower bound means exponential number of monomials in proofs (Impagliazzo+Pudlák+Sgall ‘99)

Measuring the size of algebraic proofs:

Total number of monomials

Complexity Measures of Algebraic Proofs

≈size of total depth 2 arithmetic formulas

•A low-degree version of the Functional Pigeonhole Principle (Razb98, IPS99) – linear in the number of holes (n/2+1); EPHP (AR01)

•Tseitin’s graph tautologies (BGIP99, BSI99) – linear degree lower bounds

•Random k-CNF’s (BSI99, AR01) – linear degree lower bounds

•Pseudorandom Generators tautologies (ABSRW00, Razb03)

Known degree lower bounds:

(Informal) correspondence between circuit-based complexity classes and proof systems based on these circuits:

Proof/Circuit correspondence:

proof lines consist of circuits from the prescribed class

Examples: AC0-Frege = bounded-depth FregeNC1-Frege = FregeP/poly-Frege = Extended-Frege

Does showing lower bounds on proofs is at least as hard as showing lower bounds on circuits?

•Formulate an algebraic proof system stronger than PC, Resolution and PCR•But not “too strong”:Proof system based on a circuit class with known lower bounds•Illustrate the proof/circuit correspondence

Motivation

Algebraic Proofs over

(General) Arithmetic Formulas

• Field: • Variables: X1,...,Xn

• Gates:

• Every gate in the formula computes a polynomial in

• Example: (X1 · X1) ·(X2 + 1)

F

1[ ,..., ]F[ nx x

Arithmetic Formulas

Syntactic approach: • Each proof line is an arithmetic formula• Should verify efficiently formulas

conform to inference rules

“Semantic” approach:• Each proof line is an arithmetic formula• Don’t care to verify efficiently formulas

deduced from previous ones

Example:

Algebraic Proofs over Formulas

Ψ1 Ψ2

Ψ1+Ψ2

Ψ1 Ψ2

ΨSyntactic:

Semantic:

Any Ψ identical as a polynomial to Ψ1+Ψ2

Syntactic approach: •Proofs are deterministically

polynomial-time verifiable (Cook-Reckhow systems)

Semantic approach:•Proofs are probabilistically

polynomial-time verifiable (polynomial identity testing in BPP)

Algebraic Proofs over Formulas

In P? Open problem

In both semantic and syntactic approaches considering general arithmetic formulas make algebraic proofs considerably strong:

1.Polynomially simulate entire Frege system (BIKPRS97, Pit97, GH03)

(Super-polynomial lower bounds for Frege proofs: fundamental open problem)

2.No super-polynomial lower bounds are known for general arithmetic formulas

Algebraic Proofs over Formulas

Algebraic Proofs over

Multilinear Arithmetic Formulas

• Every gate in the formula computes a multilinear polynomial

• Example: (X1·X2) + (X2·X3)

• (No high powers of variables)• Unbounded fan-in gates(we shall consider bounded-

depth formulas)

Multilinear Formulas

Super-polynomial lower

bounds on multilinear arithmetic formulas for the Determinant and Permanent functions (Raz04), and also for other polynomials (Raz04b), were recently proved

Multilinear Formulas

We take the SEMANTIC approach: Defn. A formula Multilinear Calculus ( ) refutation of p1,...,pk is a sequence of multilinear polynomials

represented as multilinear formulas terminating with 1generated as follows:

Size = total size of multilinear formulas in the refutation

i ix xjp

fg f

f g

f g

1i ix x Axioms:

Inference rules:

Multilinear Proofs-Definition

g·f is multiline

ar

fMC

equivalent to multiplying by a single variable

• Are multilinear proofs strong “enough”: – What can multilinear proof systems

prove efficiently?– Which systems can multilinear

proofs polynomially simulate?• What about bounded-depth

multilinear proofs?• Connections to multilinear circuit

complexity?

Multilinear Proofs

ResultsPolynomial Simulations:

• Depth 2-fMC polynomially simulates Resolution, PC (and PCR)

Efficient proofs:

• Depth 3-fMC (over characteristic 0) has polynomial-size refutations of the Functional Pigeonhole Principle

• Depth 3-fMC has polynomial-size refutations of the Tseitin mod p contradictions (over any characteristic)

depth 2 multilinear formulas

Known size lower bounds:

Resolution: – Functional PHP [Hak85]

– Tseitin [Urq87, BSW99]

PC (and PCR):– Low-degree version of the functional PHP

[Razb98, IPS99], EPHP [AR01]

– Tseitin’s graph tautologies [BGIP99, BSI99, ABSRW00]

Bounded-depth Frege: – Functional PHP [PBI93, KPW95]

– Tseitin mod 2 [BS02]

Corollary: separation results

PCR over Zp

PC over Zp

Frege systems

Bounded-depth Frege Modp

Resolution

Multilinear proofs

Depth 3-Multilinear proofs

Bounded-

depth Frege

Defn.(multilinearization of p) For a polynomial p, M[p] is the unique multilinear polynomial equal to p modulo

Example:

General simulation result:

Q = unsatisfiable set of multilinear polynomials(p1,...,pm) = sequence of polynomials that

forms a PCR refutation of QFor all im, Ψi is a multilinear formula for M[pi]

S:=|Ψi| and d:=Max(depth(Ψi))

Theorem: Depth d-fMC has a polynomial-size (in S) refutation of Q

m

(Proof.) Consider (M[p1],…,M[pm]).

Let U:=(Ψ1 ,…,Ψm ); Does U constitute a legitimate fMC proof?

pj

xi·pj

M[pj]M[xi·pj]

NOTE: If xi occurs in pj then

M[xi·pj] xi·M[pj]

NO:

General Simulation Result

Lemma: Let φ be a depth d multilinear formula computing M[p]. Then there is a depth d-fMC proof of M[x·p] from M[p] of size O(|φ|).

One should check that everything can be done without increasing the size & depth of formulas

•Proof\Circuit correspondence:Theorem: An explicit separation between proofs manipulating general arithmetic circuits and proofs manipulating multilinear circuits implies a lower bound on multilinear circuits for an explicit polynomial.

Results

No such lower bound is known

Multilinear Proofs\Circuit

Correspondence

cPCR

Theorem: Let Q be an unsatisfiable set of multilinear polynomials. If

Defn.

1. cPCR – semantic algebraic proofs where polynomials are represented as general arithmetic circuits

2. cMC – extension of fMC to multilinear arithmetic circuits

* Q and cMC * Qthen there is an explicit polynomial with NO p-size multilinear circuit

cPCR * Q and cMC * Q(C1,...,Cm):

(p1,...,pm) (pi is the polynomial Ci computes)(M[p1],...,M[pm])(φ1,...,φm) (φ1 computes M[pi])

If i=1|φi|=poly(n) then m

cMC * Q

by the general simulation

result

Thus i=1|φi|>poly(n), and so i=1zi·M[pi] has no p-size multilinear circuit.

m

m

Proof.

zi - new variables

arithmetic circuits

multilinear circuits

The Functional Pigeonhole Principle

Functional Pigeonhole Principle (¬FPHP):

m pigeons and n holes

1 [ ]

[ ]. [ ]

, [ ]. [ ]

i in

ik jk

ik il

Pigeons

Ho

x x

x x

i m

k n i j m

k

les

Functionx x n m al li

1 [ ]

[ ]. [ ]

, [ ]. [ ]

...i in

ik jk

ik il

i m

k n i j

Pige

m

k l n i m

x x

x x

x x

ons

Holes

Functional

Abbreviate: yk:=x1k+…+xmk

Gn:=y1+...+yn;

roughly a sum of n Boolean variables (by the Holes axioms)

A depth 3-fMC refutation of ¬FPHPRoughly can be reduced in PCR to

proving:

Gn·(Gn-1)·…·(Gn-n)By the general simulation result

suffices:

1)Show a PCR proof of π of Gn·(Gn-1)·…

·(Gn-n) with polynomial # of steps

2)Show that the multilinearization of each polynomial in π has p-size depth 3-multilinear formula

Step 2:

Observation: Each polynomial in the PCR refutation is a product of const number of symmetric polynomials, each over some (not necessarily disjoint) subset of basic variables (xij)

Example: A typical PCR proof line from the previous refutation:

Gi+1·(Gi-1)·…·(Gi-i)·(yi+1-1)

Gi+1 symmetric over

(Gi−1) · · · (Gi−i) symmetric over

(yi+1−1) is symmetric over

x11 x12 … x1i x1(i+1) … x1n

x21 x22 … x2i x2(i+1) … x2n

...

...

...

xm1 xm2 … xmi xm(i+1) … xmn

Proof based on:

Theorem (Ben-Or): Multilinear symmetric polynomials have p-size depth 3 multilinear formulas (over char 0)

Proposition: Multilinearization of product of const number of symmetric polynomials, each over some different (not necessarily disjoint) subset of basic variables (xij), has p-size depth 3 multilinear formulas (over char 0)Note: these are not symmetric

polynomials in themselves

i) Extended-Frege/Frege separation implies Arithmetic circuit/formula separationii) Frege “polynomial identity testing is in NP/poly”

(note in preparation)

Further Research:1) Weaker algebraic systems based on

arithmetic formulas (susceptible to lower bounds? Nullstellensatz proofs)

2) Proof/circuit correspondence: one of the following is true:

*

Thank You!