Lifting Applied to Proof Complexity Marc Vinyals Technion Haifa, - - PowerPoint PPT Presentation
Lifting Applied to Proof Complexity Marc Vinyals Technion Haifa, Israel FSTTCS Workshop on Extension Complexity and Lifting Theorems Supported by ERC project HARMONIC Proof Complexity Lifting Examples Wishlist SAT Is this formula
Marc Vinyals
Technion Haifa, Israel
FSTTCS Workshop on Extension Complexity and Lifting Theorems Supported by ERC project “HARMONIC”
Proof Complexity Lifting Examples Wishlist
Is this formula satisfiable?
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32
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Proof Complexity Lifting Examples Wishlist
Is this formula satisfiable?
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32
Yes
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Proof Complexity Lifting Examples Wishlist
Is this formula satisfiable?
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32
Yes
x11 = 1, x12 = 0, x21 = 0, x22 = 1, x31 = 0, x32 = 1.
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Proof Complexity Lifting Examples Wishlist
Is this formula satisfiable?
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32
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Proof Complexity Lifting Examples Wishlist
Is this formula satisfiable?
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32
No
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Proof Complexity Lifting Examples Wishlist
Is this formula satisfiable?
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32
No I promise
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Proof Complexity Lifting Examples Wishlist
Is this formula satisfiable?
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32
No I promise Enumerate all 26 assignments
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32 x31 ∨ x22
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32 x31 ∨ x22 x31 ∨ x21
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32 x31 ∨ x22 x31 ∨ x21 x11 ∨ x21
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32 x31 ∨ x22 x31 ∨ x21 x11 ∨ x21 x11
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32 x31 ∨ x22 x31 ∨ x21 x11 ∨ x21 x11 ··· x11
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32 x31 ∨ x22 x31 ∨ x21 x11 ∨ x21 x11 ··· x11 ⊥
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Proof Complexity Lifting Examples Wishlist
x11 ∨ x12 x21 ∨ x22 x31 ∨ x32 x11 ∨ x21 x11 ∨ x31 x21 ∨ x31 x12 ∨ x22 x12 ∨ x32 x22 ∨ x32
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 x11 + x21 ≥ 1 x11 + x31 ≥ 1 x21 + x31 ≥ 1 x12 + x22 ≥ 1 x12 + x32 ≥ 1 x22 + x32 ≥ 1
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 1 − x11 + 1 − x21 ≥ 1 1 − x11 + 1 − x31 ≥ 1 1 − x21 + 1 − x31 ≥ 1 1 − x12 + 1 − x22 ≥ 1 1 − x12 + 1 − x32 ≥ 1 1 − x22 + 1 − x32 ≥ 1
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1 −2x11 − 2x21 − 2x31 ≥ −3
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1 −2x11 − 2x21 − 2x31 ≥ −3 − x11 − x21 − x31 ≥ −3/2
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1 −2x11 − 2x21 − 2x31 ≥ −3 − x11 − x21 − x31 ≥ −3/2 − x11 − x21 − x31 ≥ −1
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1 −2x11 − 2x21 − 2x31 ≥ −3 − x11 − x21 − x31 ≥ −3/2 − x11 − x21 − x31 ≥ −1 − x12 − x22 − x32 ≥ −1
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1 −2x11 − 2x21 − 2x31 ≥ −3 − x11 − x21 − x31 ≥ −3/2 − x11 − x21 − x31 ≥ −1 − x12 − x22 − x32 ≥ −1 − x11 − x21 − x31 − x12 − x22 − x32 ≥ −2
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1 −2x11 − 2x21 − 2x31 ≥ −3 − x11 − x21 − x31 ≥ −3/2 − x11 − x21 − x31 ≥ −1 − x12 − x22 − x32 ≥ −1 − x11 − x21 − x31 − x12 − x22 − x32 ≥ −2 x11 + x21 + x31 + x12 + x22 + x32 ≥ 3
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Proof Complexity Lifting Examples Wishlist
x11 + x12 ≥ 1 x21 + x22 ≥ 1 x31 + x32 ≥ 1 −x11 − x21 ≥ −1 −x11 − x31 ≥ −1 −x21 − x31 ≥ −1 −x12 − x22 ≥ −1 −x12 − x32 ≥ −1 −x22 − x32 ≥ −1 −2x11 − 2x21 − 2x31 ≥ −3 − x11 − x21 − x31 ≥ −3/2 − x11 − x21 − x31 ≥ −1 − x12 − x22 − x32 ≥ −1 − x11 − x21 − x31 − x12 − x22 − x32 ≥ −2 x11 + x21 + x31 + x12 + x22 + x32 ≥ 3 0 ≥ 1
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Proof Complexity Lifting Examples Wishlist
Resolution Lines are clauses Polynomial Calculus Lines are polynomials Cutting Planes Lines are linear inequalities Bounded Depth Frege Lines are AC0 circuits
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Proof Complexity Lifting Examples Wishlist
Resolution AC0-Frege PC CP A B: A simulates B (with only polynomial loss) A B: B cannot simulate A (separation) A B: simulation+separation A B: incomparable
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Proof Complexity Lifting Examples Wishlist
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Proof Complexity Lifting Examples Wishlist
◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds.
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Proof Complexity Lifting Examples Wishlist
◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds.
Plan
1
Prove formula F hard in weak model/measure.
2 Lift to F ◦ g. 3 Prove generic lifting theorem. 4 Lifted problem hard in strong model/measure.
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Proof Complexity Lifting Examples Wishlist
◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds.
Plan
1
Prove formula F hard in weak model/measure.
2 Lift to F ◦ g. 3 Prove generic lifting theorem. 4 Lifted problem hard in strong model/measure.
◮ Many results in proof complexity follow this pattern.
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Proof Complexity Lifting Examples Wishlist
◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds.
Plan
1
Prove formula F hard in weak model/measure.
2 Lift to F ◦ g. 3 Prove generic lifting theorem. 4 Lifted problem hard in communication complexity. 5 Lifted problem has no short proofs.
◮ Many results in proof complexity follow this pattern. ◮ This talk: communication complexity techniques.
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Proof Complexity Lifting Examples Wishlist
◮ Have formula F with variables x1,...,xn. ◮ Replace variable xi with gadget g(x1
i ,...,xk i ).
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Proof Complexity Lifting Examples Wishlist
◮ Have formula F with variables x1,...,xn. ◮ Replace variable xi with gadget g(x1
i ,...,xk i ).
Example
F = {x ∨ y, x ∨ y, y} F ◦ ⊕ = {x1 ⊕ x2 ∨ y1 ⊕ y2, x1 ⊕ x2 ∨ y1 ⊕ y2, y1 ⊕ y2}
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Proof Complexity Lifting Examples Wishlist
◮ Have formula F with variables x1,...,xn. ◮ Replace variable xi with gadget g(x1
i ,...,xk i ).
Example
F = {x ∨ y, x ∨ y, y} F ◦ ⊕ = {x1 ⊕ x2 ∨ y1 ⊕ y2, x1 ⊕ x2 ∨ y1 ⊕ y2, y1 ⊕ y2} = x1 ∨ x2 ∨ y1 ∨ y2, x1 ∨ x2 ∨ y1 ∨ y2, x1 ∨ x2 ∨ y1 ∨ y2, x1 ∨ x2 ∨ y1 ∨ y2, ··· y1 ∨ y2, y1 ∨ y2
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Proof Complexity Lifting Examples Wishlist
Given CNF formula F Input Assignment to variables α: x → {0,1}n Output Clause C ∈ F falsified by assignment α
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Proof Complexity Lifting Examples Wishlist
Given CNF formula F Input Assignment to variables α: x → {0,1}n Output Clause C ∈ F falsified by assignment α Example Given F = {x ∨ y, x ∨ y, y} Input x = 0, y = 1 Output y
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Proof Complexity Lifting Examples Wishlist
x ∨ y x ∨ y y y ⊥ x ∨ y x ∨ y y x? y?
1 1
◮ Small proof =⇒ small decision tree.
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Proof Complexity Lifting Examples Wishlist
x ∨ y x ∨ y y y ⊥ x ∨ y x ∨ y y x? y?
1 1
◮ Small proof =⇒ small decision tree. ◮ But proofs cannot be balanced, we only get depth lower bounds. ◮ Use communication complexity.
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Proof Complexity Lifting Examples Wishlist
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Proof Complexity Lifting Examples Wishlist
[Bonet, Esteban, Galesi, Johannsen ’98]
Theorem
There exists a formula family Fn such that
◮ Fn has resolution proofs of length poly(n) ◮ But every tree-like CP proof must have length exp(Ω(n))
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Proof Complexity Lifting Examples Wishlist
Resolution AC0-Frege PC CP TreeCP A B: A simulates B (with only polynomial loss) A B: B cannot simulate A (separation) A B: simulation+separation A B: incomparable
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Proof Complexity Lifting Examples Wishlist
x ∨ y x ∨ y y y ⊥ x ∨ y x ∨ y y x? y?
1 1
◮ Small proof =⇒ small decision tree. ◮ But proofs cannot be balanced, we only get depth lower bounds. ◮ Use communication complexity.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 ◮ Alice sends sum of her variables; Bob evaluates inequality.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 −1 ◮ Alice sends sum of her variables; Bob evaluates inequality.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 ◮ Alice sends sum of her variables; Bob evaluates inequality.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 ◮ Alice sends sum of her variables; Bob evaluates inequality.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 ◮ Alice sends sum of her variables; Bob evaluates inequality.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 1 ◮ Alice sends sum of her variables; Bob evaluates inequality.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 ◮ Alice sends sum of her variables; Bob evaluates inequality.
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Proof Complexity Lifting Examples Wishlist
0 ≥ 1 −x11 − x21 − x31 −x12 − x22 − x32 ≥ −2 x11 + x21 + x31 +x12 + x22 + x32 ≥ 3 −x11 − x21 −x31 ≥ −1 −x12 − x22 −x32 ≥ −1
Alice Bob
x11 = 0, x22 = 1, x31 = 0 x12 = 1, x21 = 0, x32 = 1 ◮ Alice sends sum of her variables; Bob evaluates inequality. ◮ Ok if small coefficients, in general solve GT.
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Proof Complexity Lifting Examples Wishlist
◮ Want a lifting theorem for a model of communication where GT is easy. ◮ e.g. Randomized ◮ or Deterministic with a GT oracle.
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Proof Complexity Lifting Examples Wishlist
◮ Want a lifting theorem for a model of communication where GT is easy. ◮ e.g. Randomized ◮ or Deterministic with a GT oracle.
Alice Bob Oracle
x y ◮ Send f(x), g(y) to oracle ◮ Both parties see answer ◮ Cost number of calls
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Proof Complexity Lifting Examples Wishlist
◮ Want a lifting theorem for a model of communication where GT is easy. ◮ e.g. Randomized ◮ or Deterministic with a GT oracle.
Alice Bob Oracle
x y eπ πe ◮ Send f(x), g(y) to oracle ◮ Both parties see answer ◮ Cost number of calls
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Proof Complexity Lifting Examples Wishlist
◮ Want a lifting theorem for a model of communication where GT is easy. ◮ e.g. Randomized ◮ or Deterministic with a GT oracle.
Alice Bob Oracle
x y eπ πe 1 1 ◮ Send f(x), g(y) to oracle ◮ Both parties see answer ◮ Cost number of calls
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Proof Complexity Lifting Examples Wishlist
Theorem
PGT
cc (f ◦ IND) = Ω(Pdt(f) · logn)
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Proof Complexity Lifting Examples Wishlist
Theorem
PGT
cc (f ◦ IND) = Ω(Pdt(f) · logn)
Proof
◮ Essentially like Arkadev’s talk.
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Proof Complexity Lifting Examples Wishlist
Theorem
PGT
cc (f ◦ IND) = Ω(Pdt(f) · logn)
Proof
◮ Essentially like Arkadev’s talk. Recall: ◮ One bit of communication partitions inputs into two rectangles. ◮ At least one is large.
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Proof Complexity Lifting Examples Wishlist
Theorem
PGT
cc (f ◦ IND) = Ω(Pdt(f) · logn)
Proof
◮ Essentially like Arkadev’s talk. Recall: ◮ One bit of communication partitions inputs into two rectangles. ◮ At least one is large. ◮ Now partition inputs into two triangles. ◮ At least one contains a large rectangle.
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Proof Complexity Lifting Examples Wishlist
[Garg, Göös, Kamath, Sokolov ’18; Göös, Kamath, Robere, Sokolov ’19]
Theorem
There exists a formula family Fn such that
◮ Fn has polynomial calculus proof of length poly(n) ◮ But every CP proof must have length exp(Ω(n))
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Proof Complexity Lifting Examples Wishlist
[Garg, Göös, Kamath, Sokolov ’18; Göös, Kamath, Robere, Sokolov ’19]
Theorem
There exists a formula family Fn such that
◮ Fn has polynomial calculus proof of length poly(n) ◮ But every CP proof must have length exp(Ω(n)) ◮ Uses “DAG-like” lifting ◮ More after tea!
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Proof Complexity Lifting Examples Wishlist
Resolution AC0-Frege PC CP TreeCP A B: A simulates B (with only polynomial loss) A B: B cannot simulate A (separation) A B: simulation+separation A B: incomparable
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Proof Complexity Lifting Examples Wishlist
◮ Every Boolean function that can be represented with a linear inequality
has a representation with coefficients of size O(n!).
[Muroga, Toda, Takasu ’61]
◮ And this is tight.
[Håstad ’94]
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Proof Complexity Lifting Examples Wishlist
◮ Every Boolean function that can be represented with a linear inequality
has a representation with coefficients of size O(n!).
[Muroga, Toda, Takasu ’61]
◮ And this is tight.
[Håstad ’94]
◮ Every formula that has a CP proof of length L has a proof of similar
length and coefficients of size O(2L).
[Buss, Clote ’96]
◮ Is this needed?
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Proof Complexity Lifting Examples Wishlist
[de Rezende, Meir, Nordström, Pitassi, Robere, V]
Theorem
There exists a formula family Fn such that
◮ Fn has tree-like CP proofs of length L = poly(n) ◮ Fn has tree-like CP proofs with coefficient size c = O(1) ◮ But every tree-like CP proof must have logL · c = Ω(n)
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Proof Complexity Lifting Examples Wishlist
Resolution AC0-Frege PC CP TreeCP TreeCP* A B: A simulates B (with only polynomial loss) A B: B cannot simulate A (separation) A B: simulation+separation A B: incomparable
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Proof Complexity Lifting Examples Wishlist
How to encode x1 = y1,...,xn = yn?
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Proof Complexity Lifting Examples Wishlist
How to encode x1 = y1,...,xn = yn?
◮ 2n inequalities x1 ≥ y1 y1 ≥ x1 ... xn ≥ yn yn ≥ xn
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Proof Complexity Lifting Examples Wishlist
How to encode x1 = y1,...,xn = yn?
◮ 2n inequalities x1 ≥ y1 y1 ≥ x1 ... xn ≥ yn yn ≥ xn
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Proof Complexity Lifting Examples Wishlist
How to encode x1 = y1,...,xn = yn?
◮ 2n inequalities x1 ≥ y1 y1 ≥ x1 ... xn ≥ yn yn ≥ xn ◮ 1 equality x1 + 2x2 + ··· + 2n−1xn = y1 + 2y2 + ···2n−1yn
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Proof Complexity Lifting Examples Wishlist
How to encode x1 = y1,...,xn = yn?
◮ 2n inequalities x1 ≥ y1 y1 ≥ x1 ... xn ≥ yn yn ≥ xn ◮ 2 inequalities x1 + 2x2 + ··· + 2n−1xn ≥ y1 + 2y2 + ···2n−1yn y1 + 2y2 + ··· + 2n−1yn ≥ x1 + 2x2 + ···2n−1xn
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Proof Complexity Lifting Examples Wishlist
Theorem
Pcc(Search(F ◦ g)) ≥ degNss(F) for all gadgets g such that rank(g) ≥ n/degNss(F). Nullstellensatz degree degNss(F):
◮ Interpret F as polynomials {fi}. ◮ Pick polynomials gi such that
figi = 1
with minimal maxi deg(figi).
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Proof Complexity Lifting Examples Wishlist
◮ DAG-like lifting for intersections of triangles?
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Proof Complexity Lifting Examples Wishlist
◮ DAG-like lifting for intersections of triangles? ◮ Multi-party lifting?
Marc Vinyals (Technion) Lifting Applied to Proof Complexity 26 / 27
Proof Complexity Lifting Examples Wishlist
◮ DAG-like lifting for intersections of triangles? ◮ Multi-party lifting? ◮ Simulation version of lifting with equality gadget?
Marc Vinyals (Technion) Lifting Applied to Proof Complexity 26 / 27
Proof Complexity Lifting Examples Wishlist
◮ DAG-like lifting for intersections of triangles? ◮ Multi-party lifting? ◮ Simulation version of lifting with equality gadget?
◮ Round-aware lifting with equality? ◮ DAG-like lifting with equality?
Marc Vinyals (Technion) Lifting Applied to Proof Complexity 26 / 27
◮ Have a new lifting theorem? ◮ Chances are it implies something for proof complexity!
Marc Vinyals (Technion) Lifting Applied to Proof Complexity 27 / 27
◮ Have a new lifting theorem? ◮ Chances are it implies something for proof complexity!
Marc Vinyals (Technion) Lifting Applied to Proof Complexity 27 / 27
◮ Proof for F ◦ g =⇒ protocol for Search(F ◦ g). ◮ But lower bound for Search(F) ◦ g.
Marc Vinyals (Technion) Lifting Applied to Proof Complexity 28 / 27
◮ Proof for F ◦ g =⇒ protocol for Search(F ◦ g). ◮ But lower bound for Search(F) ◦ g. ◮ Not a problem:
protocol for Search(F ◦ g) =⇒ protocol for Search(F) ◦ g. ◮ On input (x,y) obtain clause D falsified by (x,y). ◮ D ∈ CNF(C ◦ g) with C ∈ F. ◮ Answer C falsified by z = g(x,y).
Marc Vinyals (Technion) Lifting Applied to Proof Complexity 28 / 27