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

2. Introduction: Algebraic Proof Systems

3. Algebraic Proofs Fix a field
Demonstrate a collection of polynomial-equations has no 0/1 solutions over
Example:
x1-x1x2=0, x2-x2x3=0, 1-x1=0, x3=0
xi2 – xi=0 for every i

4. Algebraic Proofs   +  + + + x1-x1x2 x2-x2x3 1-x1 x3 x1x2-x1x2x3=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 + =0 1

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 (Impagliazzo+Pudlák+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/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)

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. (General) Arithmetic Formulas
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)
divisions are not needed, size = number of vertices, depth = maximal distance from leaf to root, fanin: not limited, arithmetic circuits

13. Algebraic Proofs over Formulas
Syntactic approach:
Each proof line is an arithmetic formula
Should verify efficiently formulas conform to inference rules
“Semantic” approach:
Don’t care to verify efficiently formulas deduced from previous ones
Example:
Any Ψ identical as a polynomial to Ψ1+Ψ2
Ψ1 Ψ2
Ψ1 Ψ2
Syntactic:
Semantic: Ψ
Ψ1+Ψ2

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. Multilinear Arithmetic Formulas
Algebraic Proofs
over
Multilinear Arithmetic Formulas

17. Multilinear 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)

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:
g·f 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]
Tseitin’s graph tautologies [BGIP99, BSI99, ABSRW00]
Bounded-depth Frege:
Functional PHP [PBI93, KPW95]
Tseitin mod 2 [BS02]

23. Bounded-depth Frege Modp Depth 3-Multilinear proofs
Frege systems
Bounded-depth Frege Modp
Multilinear proofs
Depth 3-Multilinear proofs
Bounded-
depth Frege
PCR over Zp
Resolution
PC over Zp

24. General simulation result:
Defn.(multilinearization of p) For a polynomial p, M[p] is the unique multilinear polynomial equal to p modulo
Example:
Q = unsatisfiable set of multilinear polynomials
(p1,...,pm) = sequence of polynomials that
forms a PCR refutation of Q
For all im, 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:

25. 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

26. No such lower bound is known
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. by the general simulation result
Proof.
arithmetic circuits
(C1,...,Cm):
cPCR * Q
and cMC * Q
(p1,...,pm) (pi is the polynomial Ci computes)
(M[p1],...,M[pm])
(φ1,...,φm) (φ1 computes M[pi])
multilinear circuits
by the general simulation result
If i=1|φi|=poly(n) then m
cMC * Q
Thus i=1|φi|>poly(n), and so m
i=1zi·M[pi] 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: yk:=x1k+…+xmk
Gn:=y1+...+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 π of Gn·(Gn-1)·…·(Gn-n) with polynomial # of steps
Show that the multilinearization of each polynomial in π 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:
Gi+1·(Gi-1)·…·(Gi-i)·(yi+1-1)
x11 x12 … x1i x1(i+1) … x1n
x21 x22 … x2i x2(i+1) … x2n
...
xm1 xm2 … xmi xm(i+1) … xmn
Gi+1 symmetric over
(Gi−1) · · · (Gi−i) symmetric over
(yi+1−1) 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!