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 1 / 17

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 2 / 17

Algebraic proof systems (equalities) System of polynomial equalities: f 1 = 0 , . . . , f m = 0 Boolean axioms: x 2 1 − x 1 = 0 , . . . , x 2 n − x n = 0 • Nullstellensatz (NS) — static • proof of unsatisfiability: � j h j ( x 2 i g i f i + � j − x j ) = 1 • complexity measure: degree • Polynomial Calculus (PC) — dynamic • derivation using the rules: p , q ⊢ α p + β q , p ⊢ xp • complexity measure: degree, size 3 / 17

Algebraic proof systems (equalities) System of polynomial equalities: f 1 = 0 , . . . , f m = 0 Boolean axioms: x 2 1 − x 1 = 0 , . . . , x 2 n − x n = 0 • Nullstellensatz (NS) — static • proof of unsatisfiability: � j h j ( x 2 i g i f i + � j − x j ) = 1 • complexity measure: degree • Polynomial Calculus (PC) — dynamic • derivation using the rules: p , q ⊢ α p + β q , p ⊢ xp • complexity measure: degree, size 3 / 17

Algebraic proof systems (equalities) System of polynomial equalities: f 1 = 0 , . . . , f m = 0 Boolean axioms: x 2 1 − x 1 = 0 , . . . , x 2 n − x n = 0 • Nullstellensatz (NS) — static • proof of unsatisfiability: � j h j ( x 2 i g i f i + � j − x j ) = 1 • complexity measure: degree • Polynomial Calculus (PC) — dynamic • derivation using the rules: p , q ⊢ α p + β q , p ⊢ xp • complexity measure: degree, size 3 / 17

Semi-algebraic proof systems (inequalities) Definition The cone c ( h 1 , . . . , h m ) generated by polynomials h 1 , . . . , h m ∈ R [ X 1 , . . . , X n ] is the smallest family of polynomials containing h 1 , . . . , h m and satisfying the following rules: • e 2 ∈ c ( h 1 , . . . , h m ) for any e ∈ R [ X 1 , . . . , X n ]; • if a , b ∈ c ( h 1 , . . . , h m ), then a + b ∈ c ( h 1 , . . . , h m ); • analogously ab ∈ c ( h 1 , . . . , h m ). 4 / 17

Semi-algebraic proof systems (inequalities) System of polynomial equalities and inequalities: f 1 = 0 , . . . , f m = 0 , h 1 ≥ 0 , . . . , h k ≥ 0 • Positivstellensatz — static • proof: f + h = − 1, f ∈ I [ f 1 , . . . , f m ], h ∈ c ( h 1 , . . . , h k ) • complexity measure: degree • Positivstellensatz Calculus (PC > ) — dynamic • proof: we derive f from f 1 , . . . , f m using PC rules and h from h 1 , . . . , h k using rules for cone c ( h 1 , . . . , h k ) • complexity measure: degree, size • note: if polynomials h 1 , . . . , h l are absent, h is just a sum of squares 5 / 17

Semi-algebraic proof systems (inequalities) System of polynomial equalities and inequalities: f 1 = 0 , . . . , f m = 0 , h 1 ≥ 0 , . . . , h k ≥ 0 • Positivstellensatz — static • proof: f + h = − 1, f ∈ I [ f 1 , . . . , f m ], h ∈ c ( h 1 , . . . , h k ) • complexity measure: degree • Positivstellensatz Calculus (PC > ) — dynamic • proof: we derive f from f 1 , . . . , f m using PC rules and h from h 1 , . . . , h k using rules for cone c ( h 1 , . . . , h k ) • complexity measure: degree, size • note: if polynomials h 1 , . . . , h l are absent, h is just a sum of squares 5 / 17

Semi-algebraic proof systems (inequalities) System of polynomial equalities and inequalities: f 1 = 0 , . . . , f m = 0 , h 1 ≥ 0 , . . . , h k ≥ 0 • Positivstellensatz — static • proof: f + h = − 1, f ∈ I [ f 1 , . . . , f m ], h ∈ c ( h 1 , . . . , h k ) • complexity measure: degree • Positivstellensatz Calculus (PC > ) — dynamic • proof: we derive f from f 1 , . . . , f m using PC rules and h from h 1 , . . . , h k using rules for cone c ( h 1 , . . . , h k ) • complexity measure: degree, size • note: if polynomials h 1 , . . . , h l are absent, h is just a sum of squares 5 / 17

Laurent proofs for Boolean Thue systems Definition (Grigoriev; Buss et al.) A Boolean (multiplicative) Thue system over field F in variables X 1 , . . . , X n is a family T = { ( a 1 m 1 , a 2 m 2 ) } of pairs of terms such that ( X 2 i , 1) ∈ T for any 1 ≤ i ≤ n . Laurent monomial: l = X i 1 1 . . . X i n n . deg ( l ) = � i j > 0 i j − � i j < 0 i j . Definition (Buss et al.) For any natural number d we construct recursively a subset L d ⊂ L of the terms of degrees at most d . Base: ( a 1 m 1 , a 2 , m 2 ) ∈ T ⇒ a 1 a − 1 2 m 1 m − 1 ∈ L d provided that its degree does not 2 exceed d . Recursive step: l 1 , l 2 ∈ L d ⇒ l 1 l 2 in L d if deg ( l 1 l 2 ) ≤ d . l ∈ L d ⇒ l − 1 ∈ L d . 6 / 17

Laurent proofs for Boolean Thue systems Definition (Grigoriev; Buss et al.) A Boolean (multiplicative) Thue system over field F in variables X 1 , . . . , X n is a family T = { ( a 1 m 1 , a 2 m 2 ) } of pairs of terms such that ( X 2 i , 1) ∈ T for any 1 ≤ i ≤ n . Laurent monomial: l = X i 1 1 . . . X i n n . deg ( l ) = � i j > 0 i j − � i j < 0 i j . Definition (Buss et al.) For any natural number d we construct recursively a subset L d ⊂ L of the terms of degrees at most d . Base: ( a 1 m 1 , a 2 , m 2 ) ∈ T ⇒ a 1 a − 1 2 m 1 m − 1 ∈ L d provided that its degree does not 2 exceed d . Recursive step: l 1 , l 2 ∈ L d ⇒ l 1 l 2 in L d if deg ( l 1 l 2 ) ≤ d . l ∈ L d ⇒ l − 1 ∈ L d . 6 / 17

Laurent proofs for Boolean Thue systems Definition (Buss et al.) Two terms t 1 , t 2 are d -equivalent if t 1 = lt 2 for a certain l ∈ L d . Lemma (Buss et al.) • If t 1 is d-equivalent to t 2 then t 1 X j is d-equivalent to t 2 X j , 1 ≤ j ≤ n. • d-equivalence is a relation of equivalence on any subset of the set of all the terms of degree at most d. Definition (Buss et al.) The refutation degree D = D ( T ) is the minimal d such that L d contains some a ∈ F ∗ , a � = 1. 7 / 17

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 { c m } m where c m ∈ F and m runs over the support. Moreover, two classes with the same support have collinear corresponding vectors. 8 / 17

Laurent proofs for Boolean Thue systems P T — a binomial ideal, generated by the binomials a 1 m 1 − a 2 m 2 Lemma (Buss et al.) Let d < D. A polynomial f can be expressed as a F-linear combination of binomials t 1 − t 2 where t 1 = b 1 m 3 , t 2 = b 2 m 4 are d-equivalent and deg ( t 1 ) , deg ( t 2 ) ≤ d. Then such a linear combination could be chosen in such a way that both monomials m 3 , m 4 occur in f (this holds for all occurring binomials t 1 − t 2 ). Lemma (Buss et al.) If a polynomial f is deduced from P T 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 t 1 − t 2 for d-equivalent t 1 , t 2 . 9 / 17

Laurent proofs for Boolean Thue systems P T — a binomial ideal, generated by the binomials a 1 m 1 − a 2 m 2 Lemma (Buss et al.) Let d < D. A polynomial f can be expressed as a F-linear combination of binomials t 1 − t 2 where t 1 = b 1 m 3 , t 2 = b 2 m 4 are d-equivalent and deg ( t 1 ) , deg ( t 2 ) ≤ d. Then such a linear combination could be chosen in such a way that both monomials m 3 , m 4 occur in f (this holds for all occurring binomials t 1 − t 2 ). Lemma (Buss et al.) If a polynomial f is deduced from P T 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 t 1 − t 2 for d-equivalent t 1 , t 2 . 9 / 17

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 ∈ L d . Then a ∈ {− 1 , +1 } . 10 / 17

PC > proofs for Boolean binomial systems PC > refutation: 1 + � h 2 i = � f i g i where f i ∈ P T Theorem The degree of any PC > refutation of a Boolean binomial ideal P T (over a real field) is greater than or equal to D 2 . Proof. Let d 0 < D 2 be a degree of PC > refutation. j ) ≤ deg ( � f i g i ) • deg ( h 2 • � f i g i = � ( b i 1 m i 1 − b i 2 m i 2 ) • linear mapping ϕ : if m is d 0 -equivalent to b ∈ F ∗ , ϕ ( m ) = b , otherwise ϕ ( m ) = 0 • ϕ ( � f i g i ) = 0 • ϕ (1 + � h 2 i ) ≥ 1 (to be proven) 11 / 17