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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

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Hardness Escalation in Proof Complexity via Composition

Marc Vinyals

KTH Royal Institute of Technology Stockholm, Sweden

China Theory Week July 18 2017, Shanghai, China

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Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y}

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} x ∨ y

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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SLIDE 4

Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} x ∨ y x ∨ y

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} x ∨ y x ∨ y y

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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SLIDE 6

Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} x ∨ y x ∨ y y

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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SLIDE 7

Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} x ∨ y y

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} y y

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} y y ⊥

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 1 / 21

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Proof Complexity Composition Examples Applications

Proof Complexity

Setup Prove CNF formula unsatisfiable. 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 ⊥

F = {x ∨ y, x ∨ y, y} y y ⊥

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

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Proof Complexity Composition Examples Applications

Complexity Measures

Length x ∨ y x ∨ y x ∨ y x ∨ y x ∨ y y x ∨ y x ∨ y 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(2N), matching exp(Ω(N)).

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 2 / 21

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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 x ∨ y y x ∨ y x ∨ y 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

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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

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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

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Proof Complexity Composition Examples Applications

Proof Systems

Resolution

◮ Logic reasoning ◮ Very well understood

Polynomial calculus

◮ Algebraic reasoning ◮ Reasonably understood

Cutting planes

◮ Pseudoboolean reasoning ◮ Not well understood

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 3 / 21

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Proof Complexity Composition Examples Applications

Proof Systems

Resolution

◮ Logic reasoning ◮ Very well understood

Polynomial calculus

◮ Algebraic reasoning ◮ Reasonably understood

Cutting planes

◮ Pseudoboolean reasoning ◮ Not well understood

Sums of squares

◮ Semidefinite programming ◮ Not well understood

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 3 / 21

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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

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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

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Proof Complexity Composition Examples Applications

Composition in Proof Complexity

Have formula F with variables x1, . . . , xn. Replace variable xi with gadget g(x1

i , . . . , xk i ).

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 5 / 21

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Proof Complexity Composition Examples Applications

Composition in Proof Complexity

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}

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 5 / 21

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Proof Complexity Composition Examples Applications

Composition in Proof Complexity

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

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 5 / 21

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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

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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¨

  • m ’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

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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

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Proof Complexity Composition Examples Applications

Polynomial Calculus Space

Problem: Formulas with small degree but large space.

Theorem [Filmus, Lauria, Mikˇ

sa, Nordstr¨

  • m, 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

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Proof Complexity Composition Examples Applications

Sum of Squares Length

Problem: Formulas that require large length.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 8 / 21

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Proof Complexity Composition Examples Applications

Sum of Squares Length

Problem: Formulas that require large length.

Theorem [G¨

  • ¨
  • s, Pitassi ’14]

Rank(F ◦ ver) = Ω(Depth(F))

Strong measure: Tree-like length. Weak measure: Queries in decision tree. Composition: Versatile gadget.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 8 / 21

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Proof Complexity Composition Examples Applications

Falsified Clause Search Problem

◮ Given: CNF formula F ◮ Input: Assignment to variables α: x → {0, 1}n ◮ Task: Find clause C ∈ F falsified by assignment α

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 9 / 21

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Proof Complexity Composition Examples Applications

Sum of Squares Length

Problem: Formulas that require large length.

Theorem [G¨

  • ¨
  • s, Pitassi ’14]

Rank(F ◦ ver) = Ω(Depth(F))

Strong measure: Tree-like length. Weak measure: Queries in decision tree. Composition: Versatile gadget.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 10 / 21

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Proof Complexity Composition Examples Applications

Cutting Planes Trade-Offs

Problem: Formulas with small length and space, but not at the same time.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 11 / 21

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Proof Complexity Composition Examples Applications

Cutting Planes Trade-Offs

N1/40 N1/20 N2/5 N 2N1/40

Space Length Can do length N1+o(1), space N1/2+o(1). But space N1/2−ǫ requires size exp(Nǫ−o(1)).

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 12 / 21

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Proof Complexity Composition Examples Applications

Cutting Planes Trade-Offs

Problem: Formulas with small length and space, but not at the same time.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 13 / 21

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Proof Complexity Composition Examples Applications

Cutting Planes Trade-Offs

Problem: Formulas with small length and space, but not at the same time.

Theorem [de Rezende, Nordstr¨

  • m, V ’16]

F needs q queries with r rounds. Then F ◦ ind needs space q/r with length 2r.

Strong measure: (Length, Line Space). Weak measure: (Queries, Rounds). Composition: Indexing.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 13 / 21

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Proof Complexity Composition Examples Applications

Supercritical Trade-Offs in Tree-like Resolution

Problem: Trade-offs outside worst-case region

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 14 / 21

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Proof Complexity Composition Examples Applications

Supercritical Trade-Offs

N1/3 N2/3−ǫ N 2N 22Ω(N1/3)

Width Length Can do length 2N, width N.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 15 / 21

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Proof Complexity Composition Examples Applications

Supercritical Trade-Offs

N1/3 N2/3−ǫ N 2N 22Ω(N1/3)

Width Length Can do length 2N, width N1/3. But width N2/3−ǫ requires length 22Ω(N1/3).

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 15 / 21

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Proof Complexity Composition Examples Applications

Supercritical Trade-Offs in Tree-like Resolution

Problem: Trade-offs outside worst-case region

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 16 / 21

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Proof Complexity Composition Examples Applications

Supercritical Trade-Offs in Tree-like Resolution

Problem: Trade-offs outside worst-case region

Theorem [Razborov ’16] π proof of F ◦R ⊕. Then Len(π) = exp(Ω(Depth(F)/Width(π)))

Strong measure: (Length, Width). Weak measure: Depth. Composition: XOR with reusing.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 16 / 21

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Proof Complexity Composition Examples Applications

Cutting Planes Trade-Offs

Theorem [de Rezende, Nordstr¨

  • m, V ’16]

F needs q queries with r rounds. Then F ◦ ind needs space q/r with length 2r.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 17 / 21

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Proof Complexity Composition Examples Applications

Cutting Planes Trade-Offs

Theorem [de Rezende, Nordstr¨

  • m, V ’16]

F needs q queries with r rounds. Then F ◦ ind needs space q/r with length 2r.

Let us build F.

◮ Can be solved in few queries. ◮ Can be solved in few rounds. ◮ But not both.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 17 / 21

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Proof Complexity Composition Examples Applications

Pebbling Formulas

◮ Sources are true

u v w

◮ Truth propagates

(u ∧ v) → x (v ∧ w) → y (x ∧ y) → z

◮ Sink is false

z

u v w x y z

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 18 / 21

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Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink

Rounds 0 Pebbles 1

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink ◮ Each round:

◮ Pebbler adds some pebbles

Rounds 1 Pebbles 4

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink ◮ Each round:

◮ Pebbler adds some pebbles ◮ Challenger may challenge one new pebble

Rounds 1 Pebbles 4

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink ◮ Each round:

◮ Pebbler adds some pebbles ◮ Challenger may challenge one new pebble

Rounds 2 Pebbles 7

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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SLIDE 47

Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink ◮ Each round:

◮ Pebbler adds some pebbles ◮ Challenger may challenge one new pebble

Rounds 2 Pebbles 7

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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SLIDE 48

Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink ◮ Each round:

◮ Pebbler adds some pebbles ◮ Challenger may challenge one new pebble

Rounds 3 Pebbles 9

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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SLIDE 49

Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink ◮ Each round:

◮ Pebbler adds some pebbles ◮ Challenger may challenge one new pebble

◮ Ends when challenged pebble is surrounded

Rounds 3 Pebbles 9

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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SLIDE 50

Proof Complexity Composition Examples Applications

Dymond–Tompa Game

2-player pebble game on a DAG [Dymond, Tompa ’85]

◮ Start with a challenged pebble on the sink ◮ Each round:

◮ Pebbler adds some pebbles ◮ Challenger may challenge one new pebble

◮ Ends when challenged pebble is surrounded ◮ Equivalent to decision tree on pebbling formula

Rounds 3 Pebbles 9

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 19 / 21

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Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

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SLIDE 52

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

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SLIDE 53

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-54
SLIDE 54

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-55
SLIDE 55

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-56
SLIDE 56

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-57
SLIDE 57

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-58
SLIDE 58

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-59
SLIDE 59

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-60
SLIDE 60

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-61
SLIDE 61

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-62
SLIDE 62

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-63
SLIDE 63

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-64
SLIDE 64

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-65
SLIDE 65

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-66
SLIDE 66

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-67
SLIDE 67

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-68
SLIDE 68

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

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SLIDE 69

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

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SLIDE 70

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-71
SLIDE 71

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-72
SLIDE 72

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

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SLIDE 73

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds ◮ But r rounds require n/4 pebbles

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-74
SLIDE 74

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds ◮ But r rounds require n/4 pebbles

◮ Challenger does nothing... ◮ ...if reachable from n/2 vertices log n rows below.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-75
SLIDE 75

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds ◮ But r rounds require n/4 pebbles

◮ Challenger does nothing... ◮ ...if reachable from n/2 vertices log n rows below.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-76
SLIDE 76

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds ◮ But r rounds require n/4 pebbles

◮ Challenger does nothing... ◮ ...if reachable from n/2 vertices log n rows below.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-77
SLIDE 77

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds ◮ But r rounds require n/4 pebbles

◮ Challenger does nothing... ◮ ...if reachable from n/2 vertices log n rows below. ◮ Otherwise jump to vertex that ◮ can reach current vertex, and ◮ reachable from n/2 vertices log n rows below.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-78
SLIDE 78

Proof Complexity Composition Examples Applications

Trade-off for Dymond–Tompa

◮ Stack of r + 1 butterfly graphs ◮ Can do 2r log n pebbles in r log n rounds ◮ Or n log(r log n) pebbles in log(r log n) rounds ◮ But r rounds require n/4 pebbles

◮ Challenger does nothing... ◮ ...if reachable from n/2 vertices log n rows below. ◮ Otherwise jump to vertex that ◮ can reach current vertex, and ◮ reachable from n/2 vertices log n rows below.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 20 / 21

slide-79
SLIDE 79

Take Home

Remarks

◮ Composition is a very powerful technique

◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds.

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 21 / 21

slide-80
SLIDE 80

Take Home

Remarks

◮ Composition is a very powerful technique

◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds.

Open problems

◮ More applications ◮ Smaller gadgets

◮ Even identity?

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 21 / 21

slide-81
SLIDE 81

Take Home

Remarks

◮ Composition is a very powerful technique

◮ Proving lower bounds is hard. ◮ Let us prove easier lower bounds.

Open problems

◮ More applications ◮ Smaller gadgets

◮ Even identity?

Thanks!

Marc Vinyals (KTH) Hardness Escalation in Proof Complexity via Composition 21 / 21