Title: Iddo Tzameret
1The Strength of Multilinear Proofs (Joint work
with Ran Raz)
- Iddo Tzameret
- Tel Aviv University
2IntroductionAlgebraic Proof Systems
3Algebraic 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
4Algebraic 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
5The 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)
6Translation 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)
7Complexity 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)
8Known 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)
9Proof/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
10Motivation
- 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
11Algebraic Proofs over (General) Arithmetic
Formulas
12Arithmetic Formulas
- Field
- Variables X1,...,Xn
- Gates
- Every gate in the formula
- computes a polynomial in
- Example (X1 X1) (X2 1)
13Algebraic 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
14Algebraic 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
15Algebraic 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
16Algebraic Proofs over Multilinear Arithmetic
Formulas
17Multilinear 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)
18Multilinear 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
19Multilinear 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
20Multilinear 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?
21Results
- 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
22Corollary 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
23Frege systems
Bounded-depth Frege Modp
Multilinear proofs
Depth 3-Multilinear proofs
PCR over Zp
Bounded- depth Frege
PC over Zp
Resolution
24General simulation result
25General 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
26Results
- 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
27Multilinear 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
29Proof.
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
30The Functional Pigeonhole Principle
31Functional Pigeonhole Principle (FPHP)
Abbreviate ykx1kxmk
Gny1...yn roughly a sum of n Boolean
variables (by the Holes axioms)
32A 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
33Step 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)
34Example 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
35Proposition 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)
36Further 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)
37Thank You!