Iddo Tzameret

1
The Strength of Multilinear Proofs (Joint work
with Ran Raz)

• Iddo Tzameret
• Tel Aviv University

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IntroductionAlgebraic Proof Systems
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Algebraic Proofs

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

• Example
• x1-x1x20, x2-x2x30, 1-x10, x30

xi2 xi0 for every i
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Algebraic Proofs
x1-x1x2
0
x2-x2x3
0
1-x1
0
?
?
x3
0
x1x2-x1x2x3
0
x1x3-x1x2x3
0
?

x3x1-x1x2
0
x1x3
0

x1-x1x3
0

1-x1x3
0

1
0
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The Polynomial Calculus
Defn A Polynomial Calculus (PC) refutation of
p1, ... pk is a sequence of polynomials
terminating with 1 generated as follows (CEI96)
Axioms pi , xi2-xi
Inference rules
This enables completeness (the initial collection
of polynomials is unsatisfiable over 0/1 values)
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Translation of CNF Formulas

• We can consider algebraic proof systems as proof
  systems for CNF formulas
• A k-CNF

becomes a system of degree k monomials
Where we add the following axioms (PCR)
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Complexity Measures of Algebraic Proofs

• Measuring the size of algebraic proofs
• Total number of monomials

size of total depth 2 arithmetic formulas

• Degree lower bounds imply many monomials
• Linear degree lower bound means exponential
  number of monomials in proofs (ImpagliazzoPudlák
  Sgall 99)

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Known degree lower bounds

• A low-degree version of the Functional Pigeonhole
  Principle (Razb98, IPS99) linear in the number
  of holes (n/21) EPHP (AR01)
• Tseitins graph tautologies (BGIP99, BSI99)
  linear degree lower bounds
• Random k-CNFs (BSI99, AR01) linear degree
  lower bounds
• Pseudorandom Generators tautologies (ABSRW00,
  Razb03)

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Proof/Circuit correspondence

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

Examples AC0-Frege bounded-depth
Frege NC1-Frege Frege P/poly-Frege
Extended-Frege Does showing lower bounds on
proofs is at least as hard as showing lower
bounds on circuits?
proof lines consist of circuits from the
prescribed class
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Motivation

• 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

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Algebraic Proofs over (General) Arithmetic
Formulas
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Arithmetic Formulas

• Field
• Variables X1,...,Xn
• Gates
• Every gate in the formula
• computes a polynomial in
• Example (X1 X1) (X2 1)

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Algebraic Proofs over 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
• Dont care to verify efficiently formulas deduced
  from previous ones
• Example

Any ? identical as a polynomial to ?1?2
?1 ?2
?1 ?2
Syntactic
Semantic
?
?1?2
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Algebraic Proofs over Formulas

• Syntactic approach
• Proofs are deterministically polynomial-time
  verifiable (Cook-Reckhow systems)
• Semantic approach
• Proofs are probabilistically polynomial-time
  verifiable (polynomial identity testing in BPP)

In P? Open problem
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Algebraic Proofs over Formulas

• In both semantic and syntactic approaches
  considering general arithmetic formulas make
  algebraic proofs considerably strong
• Polynomially simulate entire Frege system
  (BIKPRS97, Pit97, GH03)
• (Super-polynomial lower bounds for Frege proofs
  fundamental open problem)
• No super-polynomial lower bounds are known for
  general arithmetic formulas

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Algebraic Proofs over Multilinear Arithmetic
Formulas
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Multilinear Formulas

• Every gate in the formula computes a multilinear
  polynomial
• Example (X1X2) (X2X3)
• (No high powers of variables)
• Unbounded fan-in gates
• (we shall consider bounded-
• depth formulas)

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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

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Multilinear Proofs-Definition
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 1 generated as follows
fMC
Axioms
Inference rules
gf is multilinear
equivalent to multiplying by a single variable
Size total size of multilinear formulas in the
refutation
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Multilinear Proofs

• 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?

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Results

• Polynomial 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
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Corollary separation results

• 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
• Tseitins graph tautologies BGIP99, BSI99,
  ABSRW00
• Bounded-depth Frege
• Functional PHP PBI93, KPW95
• Tseitin mod 2 BS02

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Frege systems
Bounded-depth Frege Modp
Multilinear proofs
Depth 3-Multilinear proofs
PCR over Zp
Bounded- depth Frege
PC over Zp
Resolution
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General simulation result
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General Simulation Result
Lemma Let f be a depth d multilinear formula
computing Mp. Then there is a depth d-fMC proof
of Mxp from Mp of size O(f).
One should check that everything can be done
without increasing the size depth of formulas
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Results

• 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.

No such lower bound is known
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Multilinear Proofs\Circuit Correspondence
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• Defn.
• cPCR semantic algebraic proofs where
  polynomials are represented as general arithmetic
  circuits
• cMC extension of fMC to multilinear arithmetic
  circuits

Theorem Let Q be an unsatisfiable set of
multilinear polynomials. If
cPCR
Q
and cMC
Q
then there is an explicit polynomial with NO
p-size multilinear circuit
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Proof.
arithmetic circuits
cPCR
Q
and cMC
Q
(C1,...,Cm)
(p1,...,pm) (pi is the polynomial Ci computes)
(Mp1,...,Mpm)
(f1,...,fm) (f1 computes Mpi)
multilinear circuits
by the general simulation result
If ?i1fipoly(n) then
m
cMC
Q
Thus ?i1figtpoly(n), and so
m
?i1ziMpi has no p-size multilinear circuit.
m
zi - new variables
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The Functional Pigeonhole Principle
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Functional Pigeonhole Principle (FPHP)

• m pigeons and n holes

Abbreviate ykx1kxmk
Gny1...yn roughly a sum of n Boolean
variables (by the Holes axioms)
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A depth 3-fMC refutation of FPHP
Roughly can be reduced in PCR to
proving Gn(Gn-1)(Gn-n)

• By the general simulation result suffices
• Show a PCR proof of p of Gn(Gn-1)(Gn-n) with
  polynomial of steps
• Show that the multilinearization of each
  polynomial in p has p-size depth 3-multilinear
  formula

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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)
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Example A typical PCR proof line from the
previous refutation Gi1(Gi-1)(Gi-i)(yi1-1
)
x11 x12 x1i x1(i1) x1n x21 x22 x2i
x2(i1) x2n ... ... ... xm1 xm2 xmi
xm(i1) xmn
Gi1 symmetric over (Gi-1) (Gi-i)
symmetric over (yi1-1) is symmetric over
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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
Proof based on Theorem (Ben-Or) Multilinear
symmetric polynomials have p-size depth 3
multilinear formulas (over char 0)
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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
i) Extended-Frege/Frege separation implies
Arithmetic circuit/formula separation ii) Frege
polynomial identity testing is in NP/poly
(note in preparation)
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Thank You!