Iddo Tzameret Tel Aviv University The Strength of Multilinear Proofs (Joint work with Ran Raz)
Dec 26, 2015
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!