Hardness Escalation in Proof Complexity via Composition Marc - PowerPoint PPT Presentation

Hardness Escalation in Proof Complexity via Composition Marc Vinyals KTH Royal Institute of Technology Stockholm, Sweden China Theory Week July 18 2017, Shanghai, China

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses ◮ Infer new clauses C ∨ x D ∨ x C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses x ∨ y ◮ Infer new clauses C ∨ x D ∨ x C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses x ∨ y ◮ Infer new clauses x ∨ y C ∨ x D ∨ x C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses x ∨ y ◮ Infer new clauses x ∨ y C ∨ x D ∨ x y C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses x ∨ y ◮ Infer new clauses x ∨ y C ∨ x D ∨ x y C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses x ∨ y ◮ Infer new clauses y C ∨ x D ∨ x C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses y ◮ Infer new clauses y C ∨ x D ∨ x C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses y ◮ Infer new clauses y C ∨ x D ∨ x ⊥ C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Proof Complexity Setup Prove CNF formula unsatisfiable. = { x ∨ y , x ∨ y , y } F Present proof on board. ◮ Write down axiom clauses y ◮ Infer new clauses y C ∨ x D ∨ x ⊥ C ∨ D ◮ Erase clauses to save space Goal: derive empty clause ⊥ Questions ◮ How much time will this take? (Length) ◮ How large is the blackboard? (Space) Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

Proof Complexity Composition Examples Applications Complexity Measures Length x ∨ y x ∨ y x ∨ y x ∨ y y y x ∨ y x ∨ y x ∨ y y y y y ⊥ 1 2 3 4 5 Length of a proof: # Lines Length of refuting a formula: Min over all proofs Worst case O ( 2 N ) , matching exp ( Ω ( N )) . Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 2 / 21

Proof Complexity Composition Examples Applications Complexity Measures Line Space [Esteban, Tor´ an ’99] [Alekhnovich, Ben Sasson, Razborov, Wigderson ’00] x ∨ y x ∨ y x ∨ y x ∨ y y y x ∨ y x ∨ y x ∨ y y y y y ⊥ 1 2 3 2 2 3 Line Space of a proof: Max lines in configuration (whiteboard) Line Space of refuting a formula: Min over all proofs Worst case N + O ( 1 ) , matching Ω ( N ) . Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 2 / 21

Proof Complexity Composition Examples Applications Proof Systems Resolution ◮ Logic reasoning ◮ Very well understood Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 3 / 21

Proof Complexity Composition Examples Applications Proof Systems Resolution ◮ Logic reasoning ◮ Very well understood Polynomial calculus ◮ Algebraic reasoning ◮ Reasonably understood Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 3 / 21

Proof Complexity Composition Examples Applications Proof Systems Resolution Cutting planes ◮ Logic reasoning ◮ Pseudoboolean reasoning ◮ Very well understood ◮ Not well understood Polynomial calculus ◮ Algebraic reasoning ◮ Reasonably understood Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 3 / 21

Proof Complexity Composition Examples Applications Proof Systems Resolution Cutting planes ◮ Logic reasoning ◮ Pseudoboolean reasoning ◮ Very well understood ◮ Not well understood Polynomial calculus Sums of squares ◮ Algebraic reasoning ◮ Semidefinite programming ◮ Reasonably understood ◮ Not well understood Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 3 / 21

Proof Complexity Composition Examples Applications Composition ◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds. Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 4 / 21

Proof Complexity Composition Examples Applications Composition ◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds. Would not it be nice if...? 1 Have original problem. 2 Prove hard in weak model/measure. 3 Compose. 4 Composed problem hard in strong model/measure. Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 4 / 21

Proof Complexity Composition Examples Applications Composition in Proof Complexity Have formula F with variables x 1 , . . . , x n . Replace variable x i with gadget g ( x 1 i , . . . , x k i ) . Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 5 / 21

Proof Complexity Composition Examples Applications Composition in Proof Complexity Have formula F with variables x 1 , . . . , x n . Replace variable x i with gadget g ( x 1 i , . . . , x k i ) . Example = { x ∨ y , x ∨ y , y } F { x 1 ⊕ x 2 ∨ y 1 ⊕ y 2 , x 1 ⊕ x 2 ∨ y 1 ⊕ y 2 , y 1 ⊕ y 2 } F ◦ ⊕ = Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 5 / 21

Proof Complexity Composition Examples Applications Composition in Proof Complexity Have formula F with variables x 1 , . . . , x n . Replace variable x i with gadget g ( x 1 i , . . . , x k i ) . Example = { x ∨ y , x ∨ y , y } F { x 1 ⊕ x 2 ∨ y 1 ⊕ y 2 , x 1 ⊕ x 2 ∨ y 1 ⊕ y 2 , y 1 ⊕ y 2 } F ◦ ⊕ = x 1 ∨ x 2 ∨ y 1 ∨ y 2 , x 1 ∨ x 2 ∨ y 1 ∨ y 2 , = x 1 ∨ x 2 ∨ y 1 ∨ y 2 , x 1 ∨ x 2 ∨ y 1 ∨ y 2 , · · · y 1 ∨ y 2 , y 1 ∨ y 2 Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 5 / 21

Proof Complexity Composition Examples Applications Resolution Space ◮ Width: Size of largest clause in proof. ◮ Theorem: Width ≤ Space. [Atserias, Dalmau ’02] Problem: Formulas with small width but large space. Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 6 / 21

Proof Complexity Composition Examples Applications Resolution Space ◮ Width: Size of largest clause in proof. ◮ Theorem: Width ≤ Space. [Atserias, Dalmau ’02] Problem: Formulas with small width but large space. Theorem [Ben Sasson, Nordstr¨ om ’11] LinSp ( F ◦ ⊕ ) = Ω ( VarSp ( F )) Strong measure: Line Space. Weak measure: Variable Space. Max variables in configuration. Composition: XOR Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 6 / 21

Proof Complexity Composition Examples Applications Polynomial Calculus Space Problem: Formulas with small degree but large space. Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 7 / 21

Proof Complexity Composition Examples Applications Polynomial Calculus Space Problem: Formulas with small degree but large space. Theorem [Filmus, Lauria, Mikˇ sa, Nordstr¨ om, V ’13] MonSp ( F ◦ ⊕ ) = Ω ( Width ( F )) Strong measure: Monomial Space. Max monomials in configuration. Weak measure: Width in resolution proof. Composition: XOR Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 7 / 21