Pebble Games and Complexity - Video Lectures · Pebble Games and Complexity Siu Man Chan Princeton...
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Pebble Games and Complexity Siu Man Chan Princeton CCI 21 Jan, 2014 @ IAS
Transcript of Pebble Games and Complexity - Video Lectures · Pebble Games and Complexity Siu Man Chan Princeton...
Pebble Games and Complexity
Siu Man Chan
Princeton CCI
21 Jan, 2014 @ IAS
Circuit Evaluation Problem
∧
∨ ∧
∧ ∨ ∧
x2 x1 x2 x1
I Input:circuit C
instance x
I Output:Result of evaluating C on x
Circuit Evaluation Problem
∧
∨ ∧
∧ ∨ ∧
x2 x1 x2 x1
I Input:circuit C
instance x
I Output:Result of evaluating C on x
Circuit Evaluation Problem
∧
∨ ∧
∧ ∨ ∧
x2 x1 x2 x1
I Input:circuit C
instance x
I Output:Result of evaluating C on x
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Circuit Evaluation Problem
I Complete for P
I “Complete” for NCi for restricted C of depth ≤ O(logi n)
NCi = size nO(1), depth O(logi n)
≈ processors nO(1), parallel time O(logi n)
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
I Concerns space and parallel complexity
I Applications: Database query algorithms, Data flow models,Big Data computation, etc.
Black Pebble Game
∧
∨ ∧
∧ ∨ ∧
x2 x1 x2 x1
⇒
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
. . . . . . . . .
⇒. . . . . . . . .
⇒
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
. . . . . . . . .
⇒. . . . . . . . .
⇒
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
. . . . . . . . .
⇒. . . . . . . . .
⇒
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
. . . . . . . . .
⇒. . . . . . . . .
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
. . . . . . . . .
⇒. . . . . . . . .
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
. . . . . . . . .
⇒. . . . . . . . .
. . . . . . . . .
⇒. . . . . . . . .
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
. . . . . . . . .
⇒. . . . . . . . .
. . . . . . . . .
⇒. . . . . . . . .
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
Time[t] ⊆ Space[t/ log t]
∀G#Pebbles(G) ≤ O
(|G |/ log|G |
)[Hopcroft–Paul–Valiant ’77]
Space needed ≤ O(#Pebbles)
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
Time[t] ⊆ Space[t/ log t]
∀G#Pebbles(G) ≤ O
(|G |/ log|G |
)[Hopcroft–Paul–Valiant ’77]
Space needed ≤ O(#Pebbles)
Black Pebble Game
Algorithms for Circuit Evaluation
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble atany time
I Pebbled means value storedin memory
Time[t] ⊆ Space[t/ log t]
∀G#Pebbles(G) ≤ O
(|G |/ log|G |
)[Hopcroft–Paul–Valiant ’77]
Space needed ≤ O(#Pebbles)
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99] Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99] Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99] Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99]
Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99]
Raz–McKenzie pebble game
m-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99]
Raz–McKenzie pebble game
m-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10]
Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99]
Raz–McKenzie pebble game
m-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10]
Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]
Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99]
Raz–McKenzie pebble game
m-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10]
Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]
Reversible pebble game
m-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99]
Raz–McKenzie pebble game
m-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10]
Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]
Reversible pebble game
m-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble game
sem-NC ( sem-P
Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99] Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99] Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1
[Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-NL m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-NLm-circuits [Raz–McKenzie ’99] Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1 [Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Lower Bounds by Pebble Games
NC1 ⊆ L ⊆ NL ⊆ NC2 ⊆ NC3 ⊆ · · · ⊆ NC ⊆ P
m-NC1 ( m-
N
L m-circuits [Karchmer–Wigderson ’90]
m-NC1 ( m-
N
Lm-circuits [Raz–McKenzie ’99] Raz–McKenzie pebble gamem-NCi ( m-NCi+1
m-NC ( m-P
m-L ( m-NL m-switching-networks [Potechin ’10] Reversible pebble game
m-L ( m-NLm-switching-networks
[C.–Potechin ’12]Reversible pebble gamem-NCi ( m-NCi+1 [Filmus–Pitassi–Robere–Cook ’13]
m-NC ( m-P
sem-NCi ( sem-NCi+1sem-circuits [C. ’13]
Dymond–Tompa pebble gamesem-NC ( sem-P Raz–McKenzie pebble game
Theorem ([C. ’13])
∀G Simulation of strategies amongRaz–McKenzie game, reversible game, and Dymond–Tompa game.
Semantic Circuit Bounds
Theorem (Karchmer–Wigderson)
Circuit Depth = Communication Complexity
NC1 vs NC2 Universal composition relation[Edmonds–Impagliazzo–Rudich–Sgall ’01]
[Gavinsky–Meir–Weinstein–Wigderson ’13]
NC1 vs NC2 Universal composition relation [Hastad–Wigderson ’97]
NCi vs NCi+1
Iterated indexing [C. ’13]
NC vs P
In the bounds for Iterated indexing:
I Upper bound by Dymond–Tompa game
I Lower bound by Raz–McKenzie game
Semantic Circuit Bounds
Theorem (Karchmer–Wigderson)
Circuit Depth = Communication Complexity
NC1 vs NC2 Universal composition relation[Edmonds–Impagliazzo–Rudich–Sgall ’01]
[Gavinsky–Meir–Weinstein–Wigderson ’13]
NC1 vs NC2 Universal composition relation [Hastad–Wigderson ’97]
NCi vs NCi+1
Iterated indexing [C. ’13]
NC vs P
In the bounds for Iterated indexing:
I Upper bound by Dymond–Tompa game
I Lower bound by Raz–McKenzie game
Semantic Circuit Bounds
Theorem (Karchmer–Wigderson)
Circuit Depth = Communication Complexity
NC1 vs NC2 Universal composition relation[Edmonds–Impagliazzo–Rudich–Sgall ’01]
[Gavinsky–Meir–Weinstein–Wigderson ’13]
NC1 vs NC2 Universal composition relation [Hastad–Wigderson ’97]
NCi vs NCi+1
Iterated indexing [C. ’13]
NC vs P
In the bounds for Iterated indexing:
I Upper bound by Dymond–Tompa game
I Lower bound by Raz–McKenzie game
Semantic Circuit Bounds
Theorem (Karchmer–Wigderson)
Circuit Depth = Communication Complexity
NC1 vs NC2 Universal composition relation[Edmonds–Impagliazzo–Rudich–Sgall ’01]
[Gavinsky–Meir–Weinstein–Wigderson ’13]
NC1 vs NC2 Universal composition relation [Hastad–Wigderson ’97]
NCi vs NCi+1
Iterated indexing [C. ’13]NC vs P
In the bounds for Iterated indexing:
I Upper bound by Dymond–Tompa game
I Lower bound by Raz–McKenzie game
Semantic Circuit Bounds
Theorem (Karchmer–Wigderson)
Circuit Depth = Communication Complexity
NC1 vs NC2 Universal composition relation[Edmonds–Impagliazzo–Rudich–Sgall ’01]
[Gavinsky–Meir–Weinstein–Wigderson ’13]
NC1 vs NC2 Universal composition relation [Hastad–Wigderson ’97]
NCi vs NCi+1
Iterated indexing [C. ’13]NC vs P
In the bounds for Iterated indexing:
I Upper bound by Dymond–Tompa game
I Lower bound by Raz–McKenzie game
Semantic Circuit Bounds
Theorem (Karchmer–Wigderson)
Circuit Depth = Communication Complexity
NC1 vs NC2 Universal composition relation[Edmonds–Impagliazzo–Rudich–Sgall ’01]
[Gavinsky–Meir–Weinstein–Wigderson ’13]
NC1 vs NC2 Universal composition relation [Hastad–Wigderson ’97]
NCi vs NCi+1
Iterated indexing [C. ’13]NC vs P
In the bounds for Iterated indexing:
I Upper bound by Dymond–Tompa game
I Lower bound by Raz–McKenzie game
My audience
I Parallel Complexity
I Space Complexity
I Randomised Complexity
I Communication Complexity
I Decision Tree Complexity (Certificate Complexity)
I Proof Complexity
I Algebraic Complexity
Reversible game
=
Dymond–Tompa game
=Raz–McKenzie game
Reversible Pebble Game
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble if allimmediate predecessors of vare pebbled
. . . . . . . . .
⇒. . . . . . . . .
Reversible Pebble Game
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble if allimmediate predecessors of vare pebbled
. . . . . . . . .
⇒. . . . . . . . .
Reversible Pebble Game
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble if allimmediate predecessors of vare pebbled
. . . . . . . . .
⇐. . . . . . . . .
Reversible Pebble Game
I Rule 1: add pebble to v ifall immediate predecessorsof v are pebbled
I Rule 2: remove pebble if allimmediate predecessors of vare pebbled
. . . . . . . . .
⇔. . . . . . . . .
Reversible Pebble Game
I Reversible Computation [Bennett ’73]:
I May reduce energy dissipationI Observation-free quantum computation is reversibleI Reversible simulation of irreversible computation [Bennett ’89]
[Li–Vitanyi ’96, ’97] [Kral’ovic ’01]
I Monotone space lower bounds [Potechin ’10] [C.–Potechin ’12]:
I Determinism equals reversibility/symmetry [Lange–McKenzie–Tapp ’00]
[Reingold ’08]
Irreversible Program Reversible Program
Reversible Pebble Game
I Reversible Computation [Bennett ’73]:I May reduce energy dissipation
I Observation-free quantum computation is reversibleI Reversible simulation of irreversible computation [Bennett ’89]
[Li–Vitanyi ’96, ’97] [Kral’ovic ’01]
I Monotone space lower bounds [Potechin ’10] [C.–Potechin ’12]:
I Determinism equals reversibility/symmetry [Lange–McKenzie–Tapp ’00]
[Reingold ’08]
Irreversible Program Reversible Program
Reversible Pebble Game
I Reversible Computation [Bennett ’73]:I May reduce energy dissipationI Observation-free quantum computation is reversible
I Reversible simulation of irreversible computation [Bennett ’89]
[Li–Vitanyi ’96, ’97] [Kral’ovic ’01]
I Monotone space lower bounds [Potechin ’10] [C.–Potechin ’12]:
I Determinism equals reversibility/symmetry [Lange–McKenzie–Tapp ’00]
[Reingold ’08]
Irreversible Program Reversible Program
Reversible Pebble Game
I Reversible Computation [Bennett ’73]:I May reduce energy dissipationI Observation-free quantum computation is reversibleI Reversible simulation of irreversible computation [Bennett ’89]
[Li–Vitanyi ’96, ’97] [Kral’ovic ’01]
I Monotone space lower bounds [Potechin ’10] [C.–Potechin ’12]:
I Determinism equals reversibility/symmetry [Lange–McKenzie–Tapp ’00]
[Reingold ’08]
Irreversible Program Reversible Program
Reversible Pebble Game
I Reversible Computation [Bennett ’73]:I May reduce energy dissipationI Observation-free quantum computation is reversibleI Reversible simulation of irreversible computation [Bennett ’89]
[Li–Vitanyi ’96, ’97] [Kral’ovic ’01]
I Monotone space lower bounds [Potechin ’10] [C.–Potechin ’12]:
I Determinism equals reversibility/symmetry [Lange–McKenzie–Tapp ’00]
[Reingold ’08]
Irreversible Program Reversible Program
Reversible Pebble Game
I Reversible Computation [Bennett ’73]:I May reduce energy dissipationI Observation-free quantum computation is reversibleI Reversible simulation of irreversible computation [Bennett ’89]
[Li–Vitanyi ’96, ’97] [Kral’ovic ’01]
I Monotone space lower bounds [Potechin ’10] [C.–Potechin ’12]:I Determinism equals reversibility/symmetry [Lange–McKenzie–Tapp ’00]
[Reingold ’08]
Irreversible Program Reversible Program
Dymond–Tompa Pebble Game
I To design parallel algorithms [Dymond–Tompa ’85, Gal–Jang ’11]
Give parallel speed-ups (when #processors is unbounded).
I Capture complexity classes and inclusions [Venkateswaran–Tompa ’89]
I (#Pebble used) characterizes parallelism in NCi ,NC,P, etc;I Simulates the inclusion of NL ⊆ NC2.
Dymond–Tompa Pebble Game
I To design parallel algorithms [Dymond–Tompa ’85, Gal–Jang ’11]
Give parallel speed-ups (when #processors is unbounded).
I Capture complexity classes and inclusions [Venkateswaran–Tompa ’89]
I (#Pebble used) characterizes parallelism in NCi ,NC,P, etc;I Simulates the inclusion of NL ⊆ NC2.
Dymond–Tompa Pebble Game
I To design parallel algorithms [Dymond–Tompa ’85, Gal–Jang ’11]
Give parallel speed-ups (when #processors is unbounded).
I Capture complexity classes and inclusions [Venkateswaran–Tompa ’89]
I (#Pebble used) characterizes parallelism in NCi ,NC,P, etc;I Simulates the inclusion of NL ⊆ NC2.
Dymond–Tompa Pebble Game
I To design parallel algorithms [Dymond–Tompa ’85, Gal–Jang ’11]
Give parallel speed-ups (when #processors is unbounded).
I Capture complexity classes and inclusions [Venkateswaran–Tompa ’89]
I (#Pebble used) characterizes parallelism in NCi ,NC,P, etc;I Simulates the inclusion of NL ⊆ NC2.
Dymond–Tompa Pebble Game
I To design parallel algorithms [Dymond–Tompa ’85, Gal–Jang ’11]
Give parallel speed-ups (when #processors is unbounded).
I Capture complexity classes and inclusions [Venkateswaran–Tompa ’89]
I (#Pebble used) characterizes parallelism in NCi ,NC,P, etc;
I Simulates the inclusion of NL ⊆ NC2.
Dymond–Tompa Pebble Game
I To design parallel algorithms [Dymond–Tompa ’85, Gal–Jang ’11]
Give parallel speed-ups (when #processors is unbounded).
I Capture complexity classes and inclusions [Venkateswaran–Tompa ’89]
I (#Pebble used) characterizes parallelism in NCi ,NC,P, etc;I Simulates the inclusion of NL ⊆ NC2.
Parallel Evaluation, Recursively
To compute the value at a
1. Pick a node, say c
2. In parallel, do
I Compute the value at cI For each possible value vc of c , assume
vc is correct and compute the value at a
3. Recurse!
4. Combine the results in Step 2 in constanttime
a
b
c
d
Parallel Evaluation, Recursively
To compute the value at a
1. Pick a node, say c
2. In parallel, do
I Compute the value at cI For each possible value vc of c , assume
vc is correct and compute the value at a
3. Recurse!
4. Combine the results in Step 2 in constanttime
a
b
c
d
Parallel Evaluation, Recursively
To compute the value at a
1. Pick a node, say c
2. In parallel, do
I Compute the value at cI For each possible value vc of c , assume
vc is correct and compute the value at a
3. Recurse!
4. Combine the results in Step 2 in constanttime
a
b
c
d
Parallel Evaluation, Recursively
To compute the value at a
1. Pick a node, say c
2. In parallel, doI Compute the value at c
I For each possible value vc of c , assumevc is correct and compute the value at a
3. Recurse!
4. Combine the results in Step 2 in constanttime
a
b
c
d
Parallel Evaluation, Recursively
To compute the value at a
1. Pick a node, say c
2. In parallel, doI Compute the value at cI For each possible value vc of c , assume
vc is correct and compute the value at a
3. Recurse!
4. Combine the results in Step 2 in constanttime
a
b
c
d
Parallel Evaluation, Recursively
To compute the value at a
1. Pick a node, say c
2. In parallel, doI Compute the value at cI For each possible value vc of c , assume
vc is correct and compute the value at a
3. Recurse!
4. Combine the results in Step 2 in constanttime
a
b
c
d
Parallel Evaluation, Recursively
To compute the value at a
1. Pick a node, say c
2. In parallel, doI Compute the value at cI For each possible value vc of c , assume
vc is correct and compute the value at a
3. Recurse!
4. Combine the results in Step 2 in constanttime
a
b
c
d
Parallel Evaluation, Recursively
a
b
c
d
Parallel Evaluation, Recursively
a
b
c
d
Parallel Evaluation, Recursively
a
b
c
d
Parallel Evaluation, Recursively
a
b
c
d
Parallel Evaluation, Recursively
a
b
c
d
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves first
I Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sink
I Challenger challenges sink (exactly one pebbled node ischallenged any time)
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒
. . . . . . . . .
⇒. . . . . . . . .
. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .
. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:I Pebbler chooses a node to pebble
I Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .
. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .
. . . . . . . . .
⇒
. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Dymond–Tompa Game
I Two players: Pebbler and Challenger, competitive
I Alternate to move, Pebbler moves firstI Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink (exactly one pebbled node is
challenged any time)
I Each round:I Pebbler chooses a node to pebbleI Challenger chooses to stay or jump
I Pebbler wins if, before she moves, the challenged node has allimmediate predecessors pebbled
I Challenger aims to delay the inevitable
. . . . . . . . .
⇒. . . . . . . . .
⇒. . . . . . . . .. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Give depth lower bounds to monotone circuits [Raz–McKenzie ’99]
Motivated by decision tree complexity of search problems[Lovasz–Naor–Newman–Wigderson ’95]
I Applications to Proof Complexity:
I Inspired pebbling contradictions (next slide)I Separation and Trade-off Results:
I Cutting plane refutations [Bonet–Esteban–Galesi–Galesi ’98]
I Treelike resolution refutations [Ben-Sasson–Impagliazzo–Wigderson ’04]
[Urquhart ’11]
I Regular resolution refutations [Alekhnovich–Johannsen–Pitassi–Urquhart ’07]
I Clause learning algorithms [Beame–Impagliazzo–Pitassi–Segerlind ’10]
I Nullstellenzatz and Polynomial Calculus[Buresh-Oppenheim–Clegg–Impagliazzo–Pitassi ’02]
I k-DNF resolution refutation [Esteban–Galesi–Messner ’04]
I Depth of resolution refutation [C. ’13]
Raz–McKenzie Pebble Game
I Give depth lower bounds to monotone circuits [Raz–McKenzie ’99]
Motivated by decision tree complexity of search problems[Lovasz–Naor–Newman–Wigderson ’95]
I Applications to Proof Complexity:
I Inspired pebbling contradictions (next slide)I Separation and Trade-off Results:
I Cutting plane refutations [Bonet–Esteban–Galesi–Galesi ’98]
I Treelike resolution refutations [Ben-Sasson–Impagliazzo–Wigderson ’04]
[Urquhart ’11]
I Regular resolution refutations [Alekhnovich–Johannsen–Pitassi–Urquhart ’07]
I Clause learning algorithms [Beame–Impagliazzo–Pitassi–Segerlind ’10]
I Nullstellenzatz and Polynomial Calculus[Buresh-Oppenheim–Clegg–Impagliazzo–Pitassi ’02]
I k-DNF resolution refutation [Esteban–Galesi–Messner ’04]
I Depth of resolution refutation [C. ’13]
Raz–McKenzie Pebble Game
I Give depth lower bounds to monotone circuits [Raz–McKenzie ’99]
Motivated by decision tree complexity of search problems[Lovasz–Naor–Newman–Wigderson ’95]
I Applications to Proof Complexity:
I Inspired pebbling contradictions (next slide)I Separation and Trade-off Results:
I Cutting plane refutations [Bonet–Esteban–Galesi–Galesi ’98]
I Treelike resolution refutations [Ben-Sasson–Impagliazzo–Wigderson ’04]
[Urquhart ’11]
I Regular resolution refutations [Alekhnovich–Johannsen–Pitassi–Urquhart ’07]
I Clause learning algorithms [Beame–Impagliazzo–Pitassi–Segerlind ’10]
I Nullstellenzatz and Polynomial Calculus[Buresh-Oppenheim–Clegg–Impagliazzo–Pitassi ’02]
I k-DNF resolution refutation [Esteban–Galesi–Messner ’04]
I Depth of resolution refutation [C. ’13]
Raz–McKenzie Pebble Game
I Give depth lower bounds to monotone circuits [Raz–McKenzie ’99]
Motivated by decision tree complexity of search problems[Lovasz–Naor–Newman–Wigderson ’95]
I Applications to Proof Complexity:I Inspired pebbling contradictions (next slide)
I Separation and Trade-off Results:I Cutting plane refutations [Bonet–Esteban–Galesi–Galesi ’98]
I Treelike resolution refutations [Ben-Sasson–Impagliazzo–Wigderson ’04]
[Urquhart ’11]
I Regular resolution refutations [Alekhnovich–Johannsen–Pitassi–Urquhart ’07]
I Clause learning algorithms [Beame–Impagliazzo–Pitassi–Segerlind ’10]
I Nullstellenzatz and Polynomial Calculus[Buresh-Oppenheim–Clegg–Impagliazzo–Pitassi ’02]
I k-DNF resolution refutation [Esteban–Galesi–Messner ’04]
I Depth of resolution refutation [C. ’13]
Raz–McKenzie Pebble Game
I Give depth lower bounds to monotone circuits [Raz–McKenzie ’99]
Motivated by decision tree complexity of search problems[Lovasz–Naor–Newman–Wigderson ’95]
I Applications to Proof Complexity:I Inspired pebbling contradictions (next slide)I Separation and Trade-off Results:
I Cutting plane refutations [Bonet–Esteban–Galesi–Galesi ’98]
I Treelike resolution refutations [Ben-Sasson–Impagliazzo–Wigderson ’04]
[Urquhart ’11]
I Regular resolution refutations [Alekhnovich–Johannsen–Pitassi–Urquhart ’07]
I Clause learning algorithms [Beame–Impagliazzo–Pitassi–Segerlind ’10]
I Nullstellenzatz and Polynomial Calculus[Buresh-Oppenheim–Clegg–Impagliazzo–Pitassi ’02]
I k-DNF resolution refutation [Esteban–Galesi–Messner ’04]
I Depth of resolution refutation [C. ’13]
Raz–McKenzie Pebble Game
I Give depth lower bounds to monotone circuits [Raz–McKenzie ’99]
Motivated by decision tree complexity of search problems[Lovasz–Naor–Newman–Wigderson ’95]
I Applications to Proof Complexity:I Inspired pebbling contradictions (next slide)I Separation and Trade-off Results:
I Cutting plane refutations [Bonet–Esteban–Galesi–Galesi ’98]
I Treelike resolution refutations [Ben-Sasson–Impagliazzo–Wigderson ’04]
[Urquhart ’11]
I Regular resolution refutations [Alekhnovich–Johannsen–Pitassi–Urquhart ’07]
I Clause learning algorithms [Beame–Impagliazzo–Pitassi–Segerlind ’10]
I Nullstellenzatz and Polynomial Calculus[Buresh-Oppenheim–Clegg–Impagliazzo–Pitassi ’02]
I k-DNF resolution refutation [Esteban–Galesi–Messner ’04]
I Depth of resolution refutation [C. ’13]
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
d
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per node
I Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
d
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,
Implication Truth propagates through the graph,Sink The sink variable is false.
Example:
d
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
d
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
d
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
d
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
d
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
de
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
de
fa
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
de
f
d ∨ e ∨ bd ∧ e ⇒ b a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
de
f
d ∨ e ∨ b
e ∨ f ∨ c
d ∧ e ⇒ b
e ∧ f ⇒ c
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
de
f
d ∨ e ∨ b
e ∨ f ∨ c
b ∨ c ∨ a
d ∧ e ⇒ b
e ∧ f ⇒ c
b ∧ c ⇒ a
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Example:
de
f
d ∨ e ∨ b
e ∨ f ∨ c
b ∨ c ∨ aa
d ∧ e ⇒ b
e ∧ f ⇒ c
b ∧ c ⇒ a
a
b c
d e f
Pebbling Contradictions
I Given G , construct an unsatisfiable CNF ΣG :
I One variable per nodeI Add the following clauses:
Source All source variables are True,Implication Truth propagates through the graph,
Sink The sink variable is false.
Resolution refutation of minimum depth for ΣG .
Resolution Refutation
Resolution Refutation
Resolution Step:A ∨ x B ∨ x
A ∨ B
Resolution Refutation
Resolution Step:A ∨ x B ∨ x
A ∨ B
∅
b
bc
. . . . . .
bc
bce
. . . . . .
bce
. . . . . .
b
. . . . . .
Resolution Refutation
Resolution Step:A ∨ x B ∨ x
A ∨ B
∅
b
bc
. . . . . .
bc
bce
. . . . . .
bce
. . . . . .
b
. . . . . .
Resolution Refutation
Resolution Step:A ∨ x B ∨ x
A ∨ B
∅
b
bc
. . . . . .
bc
bce
. . . . . .
bce
. . . . . .
b
. . . . . .
Resolution Refutation
Resolution Step:A ∨ x B ∨ x
A ∨ B
∅
b
bc
. . . . . .
bc
bce
. . . . . .
bce
. . . . . .
b
. . . . . .
Resolution Refutation
Resolution Step:A ∨ x B ∨ x
A ∨ B
∅
b
bc
. . . . . .
bc
bce
. . . . . .
bce
. . . . . .
b
. . . . . .
Partial Assignment on Resolution Refutation
I When a branch grows to a clause of ΣG , this partialassignment falsifies the clause
I If this partial assignment does not falsify any clause of ΣG ,then the branch must grow deeper!
I To falsify a clause from ΣG :
Source d set d to FalseImplication b ∨ c ∨ a set b, c to True, a to False
Sink a set a to True
I Adversary Argument:
I When a variable is queried, answer True or FalseI Try to avoid falsifying a clause from ΣG (as above)I Number of answers before falsifying ≤ depth of resolution
refutation
Partial Assignment on Resolution Refutation
I When a branch grows to a clause of ΣG , this partialassignment falsifies the clause
I If this partial assignment does not falsify any clause of ΣG ,then the branch must grow deeper!
I To falsify a clause from ΣG :
Source d set d to FalseImplication b ∨ c ∨ a set b, c to True, a to False
Sink a set a to True
I Adversary Argument:
I When a variable is queried, answer True or FalseI Try to avoid falsifying a clause from ΣG (as above)I Number of answers before falsifying ≤ depth of resolution
refutation
Partial Assignment on Resolution Refutation
I When a branch grows to a clause of ΣG , this partialassignment falsifies the clause
I If this partial assignment does not falsify any clause of ΣG ,then the branch must grow deeper!
I To falsify a clause from ΣG :
Source d set d to FalseImplication b ∨ c ∨ a set b, c to True, a to False
Sink a set a to True
I Adversary Argument:
I When a variable is queried, answer True or FalseI Try to avoid falsifying a clause from ΣG (as above)I Number of answers before falsifying ≤ depth of resolution
refutation
Partial Assignment on Resolution Refutation
I When a branch grows to a clause of ΣG , this partialassignment falsifies the clause
I If this partial assignment does not falsify any clause of ΣG ,then the branch must grow deeper!
I To falsify a clause from ΣG :
Source d set d to FalseImplication b ∨ c ∨ a set b, c to True, a to False
Sink a set a to True
I Adversary Argument:I When a variable is queried, answer True or False
I Try to avoid falsifying a clause from ΣG (as above)I Number of answers before falsifying ≤ depth of resolution
refutation
Partial Assignment on Resolution Refutation
I When a branch grows to a clause of ΣG , this partialassignment falsifies the clause
I If this partial assignment does not falsify any clause of ΣG ,then the branch must grow deeper!
I To falsify a clause from ΣG :
Source d set d to FalseImplication b ∨ c ∨ a set b, c to True, a to False
Sink a set a to True
I Adversary Argument:I When a variable is queried, answer True or FalseI Try to avoid falsifying a clause from ΣG (as above)
I Number of answers before falsifying ≤ depth of resolutionrefutation
Partial Assignment on Resolution Refutation
I When a branch grows to a clause of ΣG , this partialassignment falsifies the clause
I If this partial assignment does not falsify any clause of ΣG ,then the branch must grow deeper!
I To falsify a clause from ΣG :
Source d set d to FalseImplication b ∨ c ∨ a set b, c to True, a to False
Sink a set a to True
I Adversary Argument:I When a variable is queried, answer True or FalseI Try to avoid falsifying a clause from ΣG (as above)I Number of answers before falsifying ≤ depth of resolution
refutation
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves first
I Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound
. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebble
I Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound
. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound
. . . . . . . . .
⇒
. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Is exactly the depth of resolution refutation for ΣG [C. ’13]
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound
. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie Pebble Game
I Two players: Pebbler and Colourer, competitive
I Alternate to move, Pebbler moves firstI Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it True or False
I Pebbler wins if, before she moves, some False node has allimmediate predecessors True (source and sink are treatedanalogously)
I Colourer aims to delay the inevitable
I Add an initial set-up to make it more like Dymond–Tompagame.
I Colourer strategy gives lower bound
I Pebbler strategy gives upper bound
. . . . . . . . .
⇒. . . . . . . . .
⇒
⇒
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
Raz–McKenzie pebble game
I Two players: Pebbler and Colourer,competitive
I Alternate to move, Pebbler moves first
I Initial Set-up:
I Pebbler pebbles sinkI Colourer colours sink False
I Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it
True or False
I Pebbler wins if, before she moves,some False node has all immediatepredecessors True
I Colourer aims to delay the inevitable
Dymond–Tompa pebble game
I Two players: Pebbler and Challenger,competitive
I Alternate to move, Pebbler moves first
I Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or
jump
I Pebbler wins if, before she moves, thechallenged node has all immediatepredecessors pebbled
I Challenger aims to delay the inevitable
Raz–McKenzie pebble game
I Two players: Pebbler and Colourer,competitive
I Alternate to move, Pebbler moves first
I Initial Set-up:
I Pebbler pebbles sinkI Colourer colours sink False
I Each round:
I Pebbler chooses a node to pebbleI Colourer chooses to colour it
True or False
I Pebbler wins if, before she moves,some False node has all immediatepredecessors True
I Colourer aims to delay the inevitable
Dymond–Tompa pebble game
I Two players: Pebbler and Challenger,competitive
I Alternate to move, Pebbler moves first
I Initial Set-up:
I Pebbler pebbles sinkI Challenger challenges sink
I Each round:
I Pebbler chooses a node to pebbleI Challenger chooses to stay or
jump
I Pebbler wins if, before she moves, thechallenged node has all immediatepredecessors pebbled
I Challenger aims to delay the inevitable
Dymond–Tompa Game = Raz–McKenzie Game
I Simulation argument (reduction in combinatorial game):
1. Turn a Colourer strategy (Raz–McKenzie game) into aChallenger strategy (Dymond–Tompa game).
2. If the Dymond–Tompa game is over, so is the Raz–McKenziegame.
3. Implies Dymond–Tompa #Pebble ≥ Raz–McKenzie #Pebble.
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:
I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
v
c
Dymond–Tompa Game = Raz–McKenzie Game
I Simulation argument (reduction in combinatorial game):
1. Turn a Colourer strategy (Raz–McKenzie game) into aChallenger strategy (Dymond–Tompa game).
2. If the Dymond–Tompa game is over, so is the Raz–McKenziegame.
3. Implies Dymond–Tompa #Pebble ≥ Raz–McKenzie #Pebble.
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:
I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
v
c
Dymond–Tompa Game = Raz–McKenzie Game
I Simulation argument (reduction in combinatorial game):
1. Turn a Colourer strategy (Raz–McKenzie game) into aChallenger strategy (Dymond–Tompa game).
2. If the Dymond–Tompa game is over, so is the Raz–McKenziegame.
3. Implies Dymond–Tompa #Pebble ≥ Raz–McKenzie #Pebble.
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:
I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
v
c
Dymond–Tompa Game = Raz–McKenzie Game
I Simulation argument (reduction in combinatorial game):
1. Turn a Colourer strategy (Raz–McKenzie game) into aChallenger strategy (Dymond–Tompa game).
2. If the Dymond–Tompa game is over, so is the Raz–McKenziegame.
3. Implies Dymond–Tompa #Pebble ≥ Raz–McKenzie #Pebble.
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:
I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
v
c
Dymond–Tompa Game = Raz–McKenzie Game
I Simulation argument (reduction in combinatorial game):
1. Turn a Colourer strategy (Raz–McKenzie game) into aChallenger strategy (Dymond–Tompa game).
2. If the Dymond–Tompa game is over, so is the Raz–McKenziegame.
3. Implies Dymond–Tompa #Pebble ≥ Raz–McKenzie #Pebble.
Colourer strategy ⇒ Challenger strategy:
I Assume c is challenged. If v is pebbled, see what a Colourerwould do, and Challenger:
I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
v
c
Dymond–Tompa Game = Raz–McKenzie Game
I Simulation argument (reduction in combinatorial game):
1. Turn a Colourer strategy (Raz–McKenzie game) into aChallenger strategy (Dymond–Tompa game).
2. If the Dymond–Tompa game is over, so is the Raz–McKenziegame.
3. Implies Dymond–Tompa #Pebble ≥ Raz–McKenzie #Pebble.
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:
I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
v
c
Dymond–Tompa Game = Raz–McKenzie Game
I Simulation argument (reduction in combinatorial game):
1. Turn a Colourer strategy (Raz–McKenzie game) into aChallenger strategy (Dymond–Tompa game).
2. If the Dymond–Tompa game is over, so is the Raz–McKenziegame.
3. Implies Dymond–Tompa #Pebble ≥ Raz–McKenzie #Pebble.
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:
I c is pebbled,I all immediate predecessors of c are pebbled,I Raz–McKenzie game is over.
v
c
⇒v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.
I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:
I c is pebbled,I all immediate predecessors of c are pebbled,I Raz–McKenzie game is over.
v
c
⇒v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:
I c is pebbled,I all immediate predecessors of c are pebbled,I Raz–McKenzie game is over.
v
c
⇒v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:
I c is pebbled,I all immediate predecessors of c are pebbled,I Raz–McKenzie game is over.
v
c
⇒v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:
I c is pebbled,I all immediate predecessors of c are pebbled,I Raz–McKenzie game is over.
v
c
⇒v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:I c is pebbled,I all immediate predecessors of c are pebbled,
I Raz–McKenzie game is over.v
c
⇒v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:I c is pebbled (False),I all immediate predecessors of c are pebbled (True),
I Raz–McKenzie game is over.v
c
⇒v
c
Simulation Argument
Colourer strategy ⇒ Challenger strategy:I Assume c is challenged. If v is pebbled, see what a Colourer
would do, and Challenger:I jump if v is a predecessor of c , and v is coloured FalseI stay otherwise
If Dymond–Tompa game is over, so is Raz–McKenzie game.I Invariant: challenged node c is the ‘earliest’ False node.
I Proof: by induction.
I When Dymond–Tompa game is over:I c is pebbled (False),I all immediate predecessors of c are pebbled (True),I Raz–McKenzie game is over.
v
c
⇒v
c
Summary of Results
I Equivalence of Pebble GamesI Reversible Pebble GameI Dymond–Tompa Pebble GameI Raz–McKenzie Pebble Game
I Relations to Computational ComplexityI Restricted lower boundsI Depth complexity of circuits
I Applications to Proof ComplexityI Depth of resolution refutationsI Size of Tree-Like resolution refutations
I Complexity of Pebble GamesI PSPACE-complete
Summary of Results
I Equivalence of Pebble GamesI Reversible Pebble GameI Dymond–Tompa Pebble GameI Raz–McKenzie Pebble Game
I Relations to Computational ComplexityI Restricted lower boundsI Depth complexity of circuits
I Applications to Proof ComplexityI Depth of resolution refutationsI Size of Tree-Like resolution refutations
I Complexity of Pebble GamesI PSPACE-complete
Summary of Results
I Equivalence of Pebble GamesI Reversible Pebble GameI Dymond–Tompa Pebble GameI Raz–McKenzie Pebble Game
I Relations to Computational ComplexityI Restricted lower boundsI Depth complexity of circuits
I Applications to Proof ComplexityI Depth of resolution refutationsI Size of Tree-Like resolution refutations
I Complexity of Pebble GamesI PSPACE-complete
Summary of Results
I Equivalence of Pebble GamesI Reversible Pebble GameI Dymond–Tompa Pebble GameI Raz–McKenzie Pebble Game
I Relations to Computational ComplexityI Restricted lower boundsI Depth complexity of circuits
I Applications to Proof ComplexityI Depth of resolution refutationsI Size of Tree-Like resolution refutations
I Complexity of Pebble GamesI PSPACE-complete
Summary of Results
I Equivalence of Pebble GamesI Reversible Pebble GameI Dymond–Tompa Pebble GameI Raz–McKenzie Pebble Game
I Relations to Computational ComplexityI Restricted lower boundsI Depth complexity of circuits
I Applications to Proof ComplexityI Depth of resolution refutationsI Size of Tree-Like resolution refutations
I Complexity of Pebble GamesI PSPACE-complete (bounded fan-in)
Other Approaches
Lower Bounds by Communication ComplexityMulti-party pointer jumping
[Chakrabarti ’07] [Brody–Chakrabarti ’08] [Viola–Wigderson ’09] ACC0 ?= P
Extensions of Karchmer–Wigderson framework
[Aaronson–Wigderson ’09] NL?= NP
[Kol–Raz ’13] NC?= P
Size and Depth of Circuits
[Allender–Koucky ’10] TC0 ?= NC1
[Lipton–Williams ’12] NC?= P
Geometric Complexity Theory
[Mulmuley–Sohoni ’01 ’08] VP?= VNP
Other Approaches
Lower Bounds by Communication ComplexityMulti-party pointer jumping
[Chakrabarti ’07] [Brody–Chakrabarti ’08] [Viola–Wigderson ’09] ACC0 ?= P
Extensions of Karchmer–Wigderson framework
[Aaronson–Wigderson ’09] NL?= NP
[Kol–Raz ’13] NC?= P
Size and Depth of Circuits
[Allender–Koucky ’10] TC0 ?= NC1
[Lipton–Williams ’12] NC?= P
Geometric Complexity Theory
[Mulmuley–Sohoni ’01 ’08] VP?= VNP
Other Approaches
Lower Bounds by Communication ComplexityMulti-party pointer jumping
[Chakrabarti ’07] [Brody–Chakrabarti ’08] [Viola–Wigderson ’09] ACC0 ?= P
Extensions of Karchmer–Wigderson framework
[Aaronson–Wigderson ’09] NL?= NP
[Kol–Raz ’13] NC?= P
Size and Depth of Circuits
[Allender–Koucky ’10] TC0 ?= NC1
[Lipton–Williams ’12] NC?= P
Geometric Complexity Theory
[Mulmuley–Sohoni ’01 ’08] VP?= VNP
Other Approaches
Lower Bounds by Communication ComplexityMulti-party pointer jumping
[Chakrabarti ’07] [Brody–Chakrabarti ’08] [Viola–Wigderson ’09] ACC0 ?= P
Extensions of Karchmer–Wigderson framework
[Aaronson–Wigderson ’09] NL?= NP
[Kol–Raz ’13] NC?= P
Size and Depth of Circuits
[Allender–Koucky ’10] TC0 ?= NC1
[Lipton–Williams ’12] NC?= P
Geometric Complexity Theory
[Mulmuley–Sohoni ’01 ’08] VP?= VNP
Other Approaches
Lower Bounds by Communication ComplexityMulti-party pointer jumping
[Chakrabarti ’07] [Brody–Chakrabarti ’08] [Viola–Wigderson ’09] ACC0 ?= P
Extensions of Karchmer–Wigderson framework
[Aaronson–Wigderson ’09] NL?= NP
[Kol–Raz ’13] NC?= P
Size and Depth of Circuits
[Allender–Koucky ’10] TC0 ?= NC1
[Lipton–Williams ’12] NC?= P
Geometric Complexity Theory
[Mulmuley–Sohoni ’01 ’08] VP?= VNP
Questions
