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

Introduction: Algebraic Proof Systems

Algebraic Proofs Example: x 1 -x 1 x 2 =0, x 2 -x 2 x 3 =0, 1-x 1 =0, x 3 =0 x i 2 – x i =0 for every i Fix a field Demonstrate a collection of polynomial- equations has no 0/1 solutions over

Algebraic Proofs x 1 -x 1 x 2 x3x3 x 2 -x 2 x 3 1-x 1 x 1 x 3 -x 1 x 2 x 3 x 1 x 2 -x 1 x 2 x 3 x 3 x 1 -x 1 x 2 x 1 -x 1 x 3 1-x 1 x 3 1 x1x3x1x3    =0

Defn : A Polynomial Calculus (PC) refutation of p 1,... p k is a sequence of polynomials terminating with 1generated as follows (CEI96) : Axioms: p i, x i 2 -x i 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: becomes a system of degree k monomials: Translation of CNF Formulas Where we add the following axioms (PCR):

–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/ Ci rcuit correspondence: proof lines consist of circuits from the prescribed class Examples: AC 0 -Frege = bounded-depth Frege NC 1 -Frege = Frege P/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: X 1,...,X n Gates: Every gate in the formula computes a polynomial in Example: (X 1 · X 1 ) ·(X 2 + 1) 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 Ψ 1 Ψ 2 Ψ1+Ψ2Ψ1+Ψ2 Ψ Syntactic: Semantic: Any Ψ identical as a polynomial to Ψ 1 +Ψ 2

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: (X 1 ·X 2 ) + (X 2 ·X 3 ) (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 p 1,...,p k is a sequence of multilinear polynomials represented as multilinear formulas terminating with 1 generated as follows: Size = total size of multilinear formulas in the refutation Axioms: Inference rules: Multilinear Proofs-Definition g·f is multilinear 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

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

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 Z p PC over Z p Frege systems Bounded-depth Frege Mod p 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 (p 1,...,p m ) = sequence of polynomials that forms a PCR refutation of Q For all i  m, Ψ i is a multilinear formula for M[p i ] S:=  |Ψ i | and d:=Max(depth(Ψ i )) Theorem : Depth d-fMC has a polynomial-size (in S) refutation of Q m (Proof. ) Consider (M[p 1 ],…,M[p m ]). Let U:=( Ψ 1,…, Ψ m ); Does U constitute a legitimate fMC proof? pjpj xi·pjxi·pj M[p j ] M[x i ·p j ] NOTE: If x i occurs in p j then M[x i ·p j ]  x i ·M[p j ] 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 * Q then there is an explicit polynomial with NO p-size multilinear circuit

cPCR * Qand cMC * Q(C 1,...,C m ): (p 1,...,p m ) (p i is the polynomial C i computes) (M[p 1 ],...,M[p m ]) (φ 1,...,φ m ) (φ 1 computes M[p i ]) If  i=1 |φ i |=poly(n) then m cMC * Q by the general simulation result Thus  i=1 |φ i |>poly(n), and so  i=1 z i ·M[p i ] has no p-size multilinear circuit. m m Proof. z i - new variables arithmetic circuits multilinear circuits

The Functional Pigeonhole Principle

Functional Pigeonhole Principle (¬FPHP): m pigeons and n holes Abbreviate: y k :=x 1k +…+x mk G n :=y y n ; roughly a sum of n Boolean variables (by the Holes axioms)

A depth 3-fMC refutation of ¬FPHP Roughly can be reduced in PCR to proving: G n · (G n -1) · … · (G n -n) By the general simulation result suffices: 1)Show a PCR proof of π of G n · (G n -1) · … · (G n -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 (x ij )

Example: A typical PCR proof line from the previous refutation: G i+1 ·(G i -1)·…·(G i -i)·(y i+1 -1) G i+1 symmetric over (G i −1) · · · (G i −i) symmetric over (y i+1 −1) is symmetric over x 11 x 12 … x 1i x 1(i+1) … x 1n x 21 x 22 … x 2i x 2(i+1) … x 2n... x m1 x m2 … x mi x m(i+1) … x mn

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 (x ij ), 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 separation ii) 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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