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 satisfiable? x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 1 / 27

Proof Complexity Lifting Examples Wishlist SAT Is this formula satisfiable? x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 Yes Marc Vinyals (Technion) Lifting Applied to Proof Complexity 1 / 27

Proof Complexity Lifting Examples Wishlist SAT Is this formula satisfiable? x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 Yes x 11 = 1 , x 12 = 0 , x 21 = 0 , x 22 = 1 , x 31 = 0 , x 32 = 1 . Marc Vinyals (Technion) Lifting Applied to Proof Complexity 1 / 27

Proof Complexity Lifting Examples Wishlist SAT Is this formula satisfiable? x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 2 / 27

Proof Complexity Lifting Examples Wishlist SAT Is this formula satisfiable? x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 No Marc Vinyals (Technion) Lifting Applied to Proof Complexity 2 / 27

Proof Complexity Lifting Examples Wishlist SAT Is this formula satisfiable? x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 No I promise Marc Vinyals (Technion) Lifting Applied to Proof Complexity 2 / 27

Proof Complexity Lifting Examples Wishlist SAT Is this formula satisfiable? x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 No I promise Enumerate all 2 6 assignments Marc Vinyals (Technion) Lifting Applied to Proof Complexity 2 / 27

Proof Complexity Lifting Examples Wishlist Resolution x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 3 / 27

Proof Complexity Lifting Examples Wishlist Resolution x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 x 31 ∨ x 22 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 3 / 27

Proof Complexity Lifting Examples Wishlist Resolution x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 x 31 ∨ x 22 x 31 ∨ x 21 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 3 / 27

Proof Complexity Lifting Examples Wishlist Resolution x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 x 31 ∨ x 22 x 31 ∨ x 21 x 11 ∨ x 21 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 3 / 27

Proof Complexity Lifting Examples Wishlist Resolution x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 x 31 ∨ x 22 x 31 ∨ x 21 x 11 ∨ x 21 x 11 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 3 / 27

Proof Complexity Lifting Examples Wishlist Resolution x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 x 31 ∨ x 22 x 31 ∨ x 21 x 11 ∨ x 21 x 11 ··· x 11 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 3 / 27

Proof Complexity Lifting Examples Wishlist Resolution x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 x 31 ∨ x 22 x 31 ∨ x 21 x 11 ∨ x 21 x 11 ··· x 11 ⊥ Marc Vinyals (Technion) Lifting Applied to Proof Complexity 3 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 ∨ x 12 x 21 ∨ x 22 x 31 ∨ x 32 x 11 ∨ x 21 x 11 ∨ x 31 x 21 ∨ x 31 x 12 ∨ x 22 x 12 ∨ x 32 x 22 ∨ x 32 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 x 11 + x 21 ≥ 1 x 11 + x 31 ≥ 1 x 21 + x 31 ≥ 1 x 12 + x 22 ≥ 1 x 12 + x 32 ≥ 1 x 22 + x 32 ≥ 1 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 1 − x 11 + 1 − x 21 ≥ 1 1 − x 11 + 1 − x 31 ≥ 1 1 − x 21 + 1 − x 31 ≥ 1 1 − x 12 + 1 − x 22 ≥ 1 1 − x 12 + 1 − x 32 ≥ 1 1 − x 22 + 1 − x 32 ≥ 1 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 − 2 x 11 − 2 x 21 − 2 x 31 ≥ − 3 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 − 2 x 11 − 2 x 21 − 2 x 31 ≥ − 3 − x 11 − x 21 − x 31 ≥ − 3 / 2 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 − 2 x 11 − 2 x 21 − 2 x 31 ≥ − 3 − x 11 − x 21 − x 31 ≥ − 3 / 2 − x 11 − x 21 − x 31 ≥ − 1 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 − 2 x 11 − 2 x 21 − 2 x 31 ≥ − 3 − x 11 − x 21 − x 31 ≥ − 3 / 2 − x 11 − x 21 − x 31 ≥ − 1 − x 12 − x 22 − x 32 ≥ − 1 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 − 2 x 11 − 2 x 21 − 2 x 31 ≥ − 3 − x 11 − x 21 − x 31 ≥ − 3 / 2 − x 11 − x 21 − x 31 ≥ − 1 − x 12 − x 22 − x 32 ≥ − 1 − x 11 − x 21 − x 31 − x 12 − x 22 − x 32 ≥ − 2 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 − 2 x 11 − 2 x 21 − 2 x 31 ≥ − 3 − x 11 − x 21 − x 31 ≥ − 3 / 2 − x 11 − x 21 − x 31 ≥ − 1 − x 12 − x 22 − x 32 ≥ − 1 − x 11 − x 21 − x 31 − x 12 − x 22 − x 32 ≥ − 2 x 11 + x 21 + x 31 + x 12 + x 22 + x 32 ≥ 3 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist Cutting Planes x 11 + x 12 ≥ 1 x 21 + x 22 ≥ 1 x 31 + x 32 ≥ 1 − x 11 − x 21 ≥ − 1 − x 11 − x 31 ≥ − 1 − x 21 − x 31 ≥ − 1 − x 12 − x 22 ≥ − 1 − x 12 − x 32 ≥ − 1 − x 22 − x 32 ≥ − 1 − 2 x 11 − 2 x 21 − 2 x 31 ≥ − 3 − x 11 − x 21 − x 31 ≥ − 3 / 2 − x 11 − x 21 − x 31 ≥ − 1 − x 12 − x 22 − x 32 ≥ − 1 − x 11 − x 21 − x 31 − x 12 − x 22 − x 32 ≥ − 2 x 11 + x 21 + x 31 + x 12 + x 22 + x 32 ≥ 3 0 ≥ 1 Marc Vinyals (Technion) Lifting Applied to Proof Complexity 4 / 27

Proof Complexity Lifting Examples Wishlist A Few Proof Systems Resolution Lines are clauses Polynomial Calculus Lines are polynomials Cutting Planes Lines are linear inequalities Bounded Depth Frege Lines are AC 0 circuits Marc Vinyals (Technion) Lifting Applied to Proof Complexity 5 / 27