Linear lower bound on degrees of Positivstellensatz calculus proofs - - PowerPoint PPT Presentation

Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity (Dima Grigoriev)

talk by Anastasia Sofronova

Seminar “Modern Methods in CS”

April 21, 2020

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Proof systems

ϕ — an unsatisfiable formula in CNF. Propositional proof system — a formal way to show that ϕ is unsatisfiable.

• P(ϕ, Π) — an algorithm that checks whether Π is a proof of unsatisfiability of ϕ

in poly(|ϕ| + |Π|)

• how small can |Π| be?
• NP = coNP iff for every proof system there is a hard example

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Algebraic proof systems (equalities)

System of polynomial equalities: f1 = 0, . . . , fm = 0 Boolean axioms: x2

1 − x1 = 0, . . . , x2 n − xn = 0

• Nullstellensatz (NS) — static
• proof of unsatisfiability:

i gifi + j hj(x2 j − xj) = 1

• complexity measure: degree
• Polynomial Calculus (PC) — dynamic
• derivation using the rules: p, q ⊢ αp + βq, p ⊢ xp
• complexity measure: degree, size

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Algebraic proof systems (equalities)

System of polynomial equalities: f1 = 0, . . . , fm = 0 Boolean axioms: x2

1 − x1 = 0, . . . , x2 n − xn = 0

• Nullstellensatz (NS) — static
• proof of unsatisfiability:

i gifi + j hj(x2 j − xj) = 1

• complexity measure: degree
• Polynomial Calculus (PC) — dynamic
• derivation using the rules: p, q ⊢ αp + βq, p ⊢ xp
• complexity measure: degree, size

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Algebraic proof systems (equalities)

System of polynomial equalities: f1 = 0, . . . , fm = 0 Boolean axioms: x2

1 − x1 = 0, . . . , x2 n − xn = 0

• Nullstellensatz (NS) — static
• proof of unsatisfiability:

i gifi + j hj(x2 j − xj) = 1

• complexity measure: degree
• Polynomial Calculus (PC) — dynamic
• derivation using the rules: p, q ⊢ αp + βq, p ⊢ xp
• complexity measure: degree, size

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Semi-algebraic proof systems (inequalities)

Definition

The cone c(h1, . . . , hm) generated by polynomials h1, . . . , hm ∈ R[X1, . . . , Xn] is the smallest family of polynomials containing h1, . . . , hm and satisfying the following rules:

• e2 ∈ c(h1, . . . , hm) for any e ∈ R[X1, . . . , Xn];
• if a, b ∈ c(h1, . . . , hm), then a + b ∈ c(h1, . . . , hm);
• analogously ab ∈ c(h1, . . . , hm).

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Semi-algebraic proof systems (inequalities)

System of polynomial equalities and inequalities: f1 = 0, . . . , fm = 0, h1 ≥ 0, . . . , hk ≥ 0

• Positivstellensatz — static
• proof: f + h = −1, f ∈ I[f1, . . . , fm], h ∈ c(h1, . . . , hk)
• complexity measure: degree
• Positivstellensatz Calculus (PC>) — dynamic
• proof: we derive f from f1, . . . , fm using PC rules and h from h1, . . . , hk using rules

for cone c(h1, . . . , hk)

• complexity measure: degree, size
• note: if polynomials h1, . . . , hl are absent, h is just a sum of squares

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Semi-algebraic proof systems (inequalities)

System of polynomial equalities and inequalities: f1 = 0, . . . , fm = 0, h1 ≥ 0, . . . , hk ≥ 0

• Positivstellensatz — static
• proof: f + h = −1, f ∈ I[f1, . . . , fm], h ∈ c(h1, . . . , hk)
• complexity measure: degree
• Positivstellensatz Calculus (PC>) — dynamic
• proof: we derive f from f1, . . . , fm using PC rules and h from h1, . . . , hk using rules

for cone c(h1, . . . , hk)

• complexity measure: degree, size
• note: if polynomials h1, . . . , hl are absent, h is just a sum of squares

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Semi-algebraic proof systems (inequalities)

System of polynomial equalities and inequalities: f1 = 0, . . . , fm = 0, h1 ≥ 0, . . . , hk ≥ 0

• Positivstellensatz — static
• proof: f + h = −1, f ∈ I[f1, . . . , fm], h ∈ c(h1, . . . , hk)
• complexity measure: degree
• Positivstellensatz Calculus (PC>) — dynamic
• proof: we derive f from f1, . . . , fm using PC rules and h from h1, . . . , hk using rules

for cone c(h1, . . . , hk)

• complexity measure: degree, size
• note: if polynomials h1, . . . , hl are absent, h is just a sum of squares

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Laurent proofs for Boolean Thue systems

Definition (Grigoriev; Buss et al.)

A Boolean (multiplicative) Thue system over field F in variables X1, . . . , Xn is a family T = {(a1m1, a2m2)} of pairs of terms such that (X 2

i , 1) ∈ T for any 1 ≤ i ≤ n.

Laurent monomial: l = X i1

1 . . . X in n . deg(l) = ij>0 ij − ij<0 ij.

Definition (Buss et al.)

For any natural number d we construct recursively a subset Ld ⊂ L of the terms of degrees at most d. Base: (a1m1, a2, m2) ∈ T ⇒ a1a−1

2 m1m−1 2

∈ Ld provided that its degree does not exceed d. Recursive step: l1, l2 ∈ Ld ⇒ l1l2 in Ld if deg(l1l2) ≤ d. l ∈ Ld ⇒ l−1 ∈ Ld.

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Laurent proofs for Boolean Thue systems

Definition (Grigoriev; Buss et al.)

A Boolean (multiplicative) Thue system over field F in variables X1, . . . , Xn is a family T = {(a1m1, a2m2)} of pairs of terms such that (X 2

i , 1) ∈ T for any 1 ≤ i ≤ n.

Laurent monomial: l = X i1

1 . . . X in n . deg(l) = ij>0 ij − ij<0 ij.

Definition (Buss et al.)

For any natural number d we construct recursively a subset Ld ⊂ L of the terms of degrees at most d. Base: (a1m1, a2, m2) ∈ T ⇒ a1a−1

2 m1m−1 2

∈ Ld provided that its degree does not exceed d. Recursive step: l1, l2 ∈ Ld ⇒ l1l2 in Ld if deg(l1l2) ≤ d. l ∈ Ld ⇒ l−1 ∈ Ld.

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Laurent proofs for Boolean Thue systems

Definition (Buss et al.)

Two terms t1, t2 are d-equivalent if t1 = lt2 for a certain l ∈ Ld.

Lemma (Buss et al.)

• If t1 is d-equivalent to t2 then t1Xj is d-equivalent to t2Xj, 1 ≤ j ≤ n.
• d-equivalence is a relation of equivalence on any subset of the set of all the terms
• f degree at most d.

Definition (Buss et al.)

The refutation degree D = D(T) is the minimal d such that Ld contains some a ∈ F ∗, a = 1.

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Laurent proofs for Boolean Thue systems

Support of a class of d-equivalence of terms = set of its monomials.

Lemma (Buss et al.)

Let d < D. The supports of two classes of d-equivalence either coincide or are disjoint. Two classes with the same support are obtained from one another by simultaneous multiplication of all the terms by an appropriate factor b ∈ F ∗. Thus, any class could be represented by a vector {cm}m where cm ∈ F and m runs over the support. Moreover, two classes with the same support have collinear corresponding vectors.

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Laurent proofs for Boolean Thue systems

PT — a binomial ideal, generated by the binomials a1m1 − a2m2

Lemma (Buss et al.)

Let d < D. A polynomial f can be expressed as a F-linear combination of binomials t1 − t2 where t1 = b1m3, t2 = b2m4 are d-equivalent and deg(t1), deg(t2) ≤ d. Then such a linear combination could be chosen in such a way that both monomials m3, m4

• ccur in f (this holds for all occurring binomials t1 − t2).

Lemma (Buss et al.)

If a polynomial f is deduced from PT in the fragment of the polynomial calculus of a degree at most d < D, then f can be expressed as F-linear combination of binomials t1 − t2 for d-equivalent t1, t2.

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Laurent proofs for Boolean Thue systems

PT — a binomial ideal, generated by the binomials a1m1 − a2m2

Lemma (Buss et al.)

Let d < D. A polynomial f can be expressed as a F-linear combination of binomials t1 − t2 where t1 = b1m3, t2 = b2m4 are d-equivalent and deg(t1), deg(t2) ≤ d. Then such a linear combination could be chosen in such a way that both monomials m3, m4

• ccur in f (this holds for all occurring binomials t1 − t2).

Lemma (Buss et al.)

If a polynomial f is deduced from PT in the fragment of the polynomial calculus of a degree at most d < D, then f can be expressed as F-linear combination of binomials t1 − t2 for d-equivalent t1, t2.

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Laurent proofs for Boolean Thue systems

All previous lemmas hold for arbitrary Thue system, from now on we take into account that T is just a Boolean Thue system.

Lemma

Let d < D

2 and a Laurent term al ∈ Ld. Then a ∈ {−1, +1}.

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PC> proofs for Boolean binomial systems

PC> refutation: 1 + h2

i = figi where fi ∈ PT

Theorem

The degree of any PC> refutation of a Boolean binomial ideal PT (over a real field) is greater than or equal to D

2 .

Proof.

Let d0 < D

2 be a degree of PC> refutation.

• deg(h2

j ) ≤ deg( figi)

• figi = (bi1mi1 − bi2mi2)
• linear mapping ϕ: if m is d0-equivalent to b ∈ F ∗, ϕ(m) = b, otherwise ϕ(m) = 0
• ϕ( figi) = 0
• ϕ(1 + h2

i ) ≥ 1 (to be proven)

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PC> proofs for Boolean binomial systems

Lemma

ϕ(1 + h2

i ) ≥ 1

Proof.

• fix h = hj =

I aIX I, deg(XI) ≤ D 4

• graph Q with vertices corresponding to monomials of h,

(I, J) ∈ Q ⇔ bX IX J ∈ Ld0 for some b ∈ F ∗ (in particular we have all loops)

• apart from terms (aIxI)2 only terms 2aIaJX IX J where (I, J) ∈ Q contribute to
• ur sum
• Q is a disjoint union of cliques, so let us prove that for a fixed clique C its

contribution is non-negative

• partition C onto V1 and V2 according to signs of edges

I∈V1∪V2(aI)2 + 2 I∈V1,J∈V1 aIaJ + 2 I∈V2,J∈V2 aIaJ − 2 I∈V1,J∈V1 aIaJ =

(

I∈V1 aI)2 + ( I∈V2 aI)2 − 2( I∈V1 aI)( I∈V2 aI) ≥ 0

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Lower bounds on PC> for Tseitin and parity

Tseitin formulas TSk(2):

• undirected graph G
• charge uv ∈ {−1, 1} for each vertex v,

v uv = −1

• each edge e has a corresponding variable Xe
• Thue system: {(X(v) = uv
• e contains v Xe, 1} ∪ {(X 2

e , 1)}

We assume that G = Gk is an r-regular expander with k nodes: for any S ∈ V (G) the number of adjacent to S nodes is ≥ (1 + ε(1 − |S|

k ))|S| for some constant ε.

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Lower bounds on PC> for Tseitin and parity

Any Laurent monomial in {Xe}e could be reduced to multilinear using Boolean axioms. Pseudo-degree of the monomial — the number of variables in such reduction. Any Laurent monomial in {X(v)}v, {X 2

e }e could be reduced to multilinear in {X(v)}v

using Boolean axioms. Weight of the monomial — the number of X(v).

Lemma

• Any reduced monomial in {X 2

e }e, {X(v)}v which is equal to an element of the

form am2 where a ∈ F ∗ and m is a monomial, is either 1 or

v X(v) = −1

• For any 1

2 ≥ ε1 > 0 there exists ε0 > 0 such that any reduced monomial in

{X 2

e }e, {X(v)}v with the weight between ε1k and (1 − ε1)k has the

pseudo-degree at least ε0k.

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Lower bounds on PC> for Tseitin and parity

Lemma

The refutation degree D = D(TSk(2)) is greater than Ω(k).

Proof.

• a chain of Laurent monomials l1, . . . , lN in {X 2

e }e, {X(v)}v such that lN = −1

• each lj is either one of {X 2

e }e, {X(v)}v, either l−1 j1

• r lj1lj2 for some j1, j2 < j
• the degree of each lj does not exceed D
• w(lN) = k, w(X(v)) = 1, w(lj) ≤ w(lj1) + w(lj2)
• there exists j0 such that 1

3k ≤ w(lj0) ≤ 2 3k

• pseudo-degree of lj0 is Ω(k)

Corollary

The degree of any PC> refutation of the Boolean binomial system PTSk(2) is greater than Ω(k).

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Lower bounds on PC> for Tseitin and parity

MODn

2:

• n

2

• variables Xe, e ⊂ [n], |e| = 2
• X 2

e = Xe, XeXf = 0 for e = f , e ∩ f = ∅

i∈e Xe = 1 for each i ∈ [n]

• not a binomial system

Definition (Buss et al.)

Let P = P(x1, . . . , xn), Q = Q(y1, . . . , ym) be two sets of polynomials. Then P is (d1, d2)-reducible to Q if for every 1 ≤ i ≤ m there exists a polynomial si(x1, . . . , xn)

• f degree at most d1 such that there exists a degree d2 derivation in the PC of the

polynomials Q(s1, . . . , sm) from the polynomials P.

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Lower bounds on PC> for Tseitin and parity

Lemma (Buss et al.)

Suppose that P is (d1, d2)-reducible to Q. Then if there is a degree d3 PC> refutation

• f Q then there is a degree max{d2, d3d1} PC> refutation of P.

Lemma (Buss et al.)

For all k the Boolean binomial system PTSk(2) is (4r, 4r)-reducible to MODk(1+2r)

2

(where r denotes the valency of the expander Gk; one could take r = 6).

Corollary

The degree of any PC> refutation of MODk

2 is greater than Ω(k).

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