Iddo Tzameret Tel Aviv University

1. The Strength of Multilinear Proofs (Joint work withRan 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 =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) refutationof p1, ... pk is a sequence of polynomials terminating with 1generated as follows (CEI96) : Axioms:pi , xi2-xi Inference rules: This enables completeness (the initial collection of polynomials is unsatisfiable over0/1 values)

6. Translation of CNF Formulas We can consideralgebraic proof systemsas proof systems forCNFformulas: Ak-CNF: becomes a system of degree kmonomials: 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 theFunctionalPigeonhole Principle(Razb98, IPS99) – linear in the number of holes (n/2+1); EPHP(AR01) • Tseitin’sgraph tautologies(BGIP99, BSI99) – linear degree lower bounds • Random k-CNF’s(BSI99, AR01) – linear degree lower bounds • Pseudorandom Generatorstautologies (ABSRW00, Razb03)

9. Proof/Circuitcorrespondence: (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 thanPC, ResolutionandPCR • 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 Syntacticapproach: • 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: AnyΨidenticalas a polynomial toΨ1+Ψ2 Ψ1Ψ2 Ψ1Ψ2 Semantic: Syntactic: 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 testingin BPP) In P? Open problem

15. Algebraic Proofs over Formulas In both semantic and syntactic approachesconsideringgeneral arithmetic formulas make algebraic proofs considerably strong: • Polynomially simulate entireFregesystem(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:(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() refutationof p1,...,pk is a sequence of multilinear polynomials represented as multilinear formulas terminatingwith1 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 proofsstrong “enough”: • What can multilinear proof systemsproveefficiently? • Which systems can multilinear proofs polynomiallysimulate? • What aboutbounded-depthmultilinearproofs? • Connections to multilinear circuit complexity?

21. Results Polynomial Simulations: • Depth 2-fMCpolynomially simulates Resolution, PC (and PCR) Efficient proofs: • Depth 3-fMC(over characteristic 0)has polynomial-size refutations of theFunctional Pigeonhole Principle • Depth 3-fMChas polynomial-size refutations of the Tseitinmodp 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 thefunctional PHP[Razb98, IPS99], EPHP [AR01] • Tseitin’sgraph 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 Resolution PC over Zp

24. Defn.(multilinearization of p)For a polynomial p, M[p] is the unique multilinear polynomial equal to p modulo Q= unsatisfiable set of multilinear polynomials (p1,...,pm)= sequence of polynomials that forms a PCR refutation of Q For all im, Ψiis a multilinearformulafor M[pi] S:=|Ψi|andd:=Max(depth(Ψi)) Theorem: Depth d-fMC has a polynomial-size (in S) refutation of Q Example: • (Proof.) Consider(M[p1],…,M[pm]). • Let U:=(Ψ1,…,Ψm); Does U constitute a legitimate fMC proof? m NO: M[pj] M[xi·pj] pj NOTE: If xi occurs in pj then M[xi·pj]xi·M[pj] xi·pj General simulation result:

25. General Simulation Result Lemma:Letφbea depthdmultilinear formula computing M[p].Then there is adepthd-fMCproofofM[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. Results • Proof\Circuitcorrespondence: Theorem:An explicit separation between proofs manipulating general arithmetic circuitsand proofs manipulating multilinearcircuitsimplies 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 generalarithmetic circuits • cMC – extension of fMC to multilinear arithmeticcircuits Theorem:Let Q be an unsatisfiable set of multilinear polynomials. If Q cPCR Q and cMC then there is an explicit polynomial with NO p-size multilinear circuit

29. * * * Proof. arithmetic circuits (C1,...,Cm): Q cPCR Q andcMC (p1,...,pm) (piis the polynomialCi 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 Q cMC 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 FunctionalPigeonhole Principle

31. Functional Pigeonhole Principle (¬FPHP): mpigeons andnholes Abbreviate: yk:=x1k+…+xmk Gn:=y1+...+yn;roughly asumofnBoolean 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 aPCRproof ofπofGn·(Gn-1)·…·(Gn-n)withpolynomial#of steps • Show that themultilinearizationof each polynomial inπhas 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+1symmetric 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 char0) • Note: these are not symmetric polynomials in themselves Proof based on: Theorem (Ben-Or): Multilinearsymmetricpolynomials have p-sizedepth 3 multilinearformulas (over char 0)

36. * Further Research: 1)Weaker algebraic systems based on arithmetic formulas (susceptible to lower bounds?Nullstellensatzproofs) 2) Proof/circuit correspondence: one of the following is true: i) Extended-Frege/Frege separation impliesArithmetic circuit/formula separation ii) Frege “polynomial identity testing is in NP/poly” (note in preparation)

37. Thank You!