Iddo Tzameret Tel Aviv University The Strength of Multilinear Proofs (Joint work with Ran Raz)
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Transcript of Iddo Tzameret Tel Aviv University The Strength of Multilinear Proofs (Joint work with Ran Raz)
![Page 1: Iddo Tzameret Tel Aviv University The Strength of Multilinear Proofs (Joint work with Ran Raz)](https://reader030.fdocuments.in/reader030/viewer/2022032802/56649e155503460f94aff0dd/html5/thumbnails/1.jpg)
Iddo TzameretTel Aviv University
The Strength of Multilinear
Proofs(Joint work with Ran Raz)
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Introduction:Algebraic Proof
Systems
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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
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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
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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)
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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
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–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
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•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:
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(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?
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•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
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Algebraic Proofs over
(General) Arithmetic Formulas
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• 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
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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
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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
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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
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Algebraic Proofs over
Multilinear Arithmetic Formulas
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• 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
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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
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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
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• 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
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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
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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
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PCR over Zp
PC over Zp
Frege systems
Bounded-depth Frege Modp
Resolution
Multilinear proofs
Depth 3-Multilinear proofs
Bounded-
depth Frege
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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:
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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
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•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
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Multilinear Proofs\Circuit
Correspondence
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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
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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
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The Functional Pigeonhole Principle
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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)
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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
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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:
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
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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
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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:
*
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Thank You!