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Iddo Tzameret Tel Aviv University

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Presentation on theme: "Iddo Tzameret Tel Aviv University"— Presentation transcript:

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!


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