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Title: Iddo Tzameret


1
The Strength of Multilinear Proofs (Joint work
with Ran Raz)
  • Iddo Tzameret
  • Tel Aviv University

2
IntroductionAlgebraic Proof Systems
3
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
4
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
5
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)
6
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)
7
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)

8
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)

9
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
10
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

11
Algebraic Proofs over (General) Arithmetic
Formulas
12
Arithmetic Formulas
  • Field
  • Variables X1,...,Xn
  • Gates
  • Every gate in the formula
  • computes a polynomial in
  • Example (X1 X1) (X2 1)

13
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
14
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
15
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

16
Algebraic Proofs over Multilinear Arithmetic
Formulas
17
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)

18
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

19
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
20
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?

21
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
22
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

23
Frege systems
Bounded-depth Frege Modp
Multilinear proofs
Depth 3-Multilinear proofs
PCR over Zp
Bounded- depth Frege
PC over Zp
Resolution
24
General simulation result
25
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
26
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
27
Multilinear Proofs\Circuit Correspondence
28
  • 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
29
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
30
The Functional Pigeonhole Principle
31
Functional Pigeonhole Principle (FPHP)
  • m pigeons and n holes

Abbreviate ykx1kxmk
Gny1...yn roughly a sum of n Boolean
variables (by the Holes axioms)
32
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

33
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)
34
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
35
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)
36
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)
37
Thank You!
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