Hardness of Function Composition Toniann Pitassi for Semantic Read - PowerPoint PPT Presentation

Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Hardness of Function Composition Toniann Pitassi for Semantic Read once Branching Motivation Programs Branching Programs Lower Bounds against Function Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Composition Open Problems June 23, 2018

P and L Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation P : Polynomial time computable functions. Branching Programs Lower Bounds against Function L : Functions computable in logarithmic space. * * * * Composition Open Problems ? L ⊂ P

Branching Programs Hardness of Function Composition for Semantic Read once Branching Programs f ( x 1 , x 2 , .., x n ) → { 0 , 1 } x i ∈ { 0 , 1 } , ∀ i ∈ [ n ] Jeff Edmonds, Venkatesh Medabalimi, Definition Toniann Pitassi Deterministic Branching program Motivation DAG with a source node and two sinks, Branching Programs 1-sink (for accept) and 0-sink (for reject). Motivation Restricted: Read-Once Each non-sink node is labeled by some x i , outdegree 2 with Rectangles an edge each for x i = 0 and x i = 1. Lower Bounds against Function Composition Open Problems x 2 x 3 x 4 x 5 0 0 0 0 0 0 1 1 1 1 start , x 1 1 1 1 1 1 1 x 2 x 3 x 4 x 5 0 0 0 0

Non-det Branching Programs Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Definition Medabalimi, Toniann Pitassi Non-deterministic Branching program (NBP) Motivation allow unlabelled guessing nodes and arbitrary Branching out-degree. Programs Motivation Restricted: x 11 x 21 x 31 x 41 Read-Once Rectangles Lower Bounds x 12 x 22 x 32 x 42 against Function 1 Start Composition x 13 x 23 x 33 x 43 Open Problems x 14 x 24 x 34 x 44 The size of a NBP= number of labelled nodes.

NBP computing f : { 0 , 1 } n → { 0 , 1 } Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, x 11 x 21 x 31 x 41 Toniann Pitassi Motivation x 12 x 22 x 32 x 42 Branching 1 Start Programs x 13 x 23 x 33 x 43 Motivation Restricted: Read-Once Rectangles x 14 x 24 x 34 x 44 Lower Bounds against Function Composition f ( u ) = 1 ⇐ ⇒ ∃ a path from source to accept node Open Problems that is consistent with input u .

Program Size and Space Complexity of Hardness of Function Composition for computing f Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation BP ( f n ) = B ∈ BP computing f n size ( B ) min Branching Programs Motivation Restricted: Read-Once Rectangles S ( f n ) = T ∈ non-uniform TMs computing f n space complexity ( T ) min Lower Bounds against Function Composition Open Problems log( BP ( f n )) ≈ S ( f n ) [Cobham ‘66]

Big Picture Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching It is easy to show functions with high BP ( f n ) exist. Programs Motivation Restricted: Read-Once Rectangles Lower Bounds against Function Composition Open Problems

Big Picture Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching It is easy to show functions with high BP ( f n ) exist. Programs Motivation Can we show that some function in P requires Restricted: Read-Once exponential size BP ? Rectangles amounts to showing L ⊂ P . Lower Bounds against Function Composition Open Problems

BPs and other Computation Models Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Formulas BranchingPrograms Circuits Motivation Branching 1 L ( f ) ≥ BP ( f ) ≥ 3 C ( f ) Programs Motivation Restricted: Read-Once Rectangles Lower Bounds against Function Composition Open Problems

BPs and other Computation Models Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Formulas BranchingPrograms Circuits Motivation Branching 1 L ( f ) ≥ BP ( f ) ≥ 3 C ( f ) Programs Motivation Restricted: Read-Once � � n 2 n 3 � Rectangles � Ω Ω Ω ( n ) log 2 n Lower Bounds against Function Composition Nechiporuk Random Restrictions Gate Elimination Open Problems

Restricted Branching Programs Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Bounded Width: same as NC 1 , Barrington’s Programs Motivation characterization. Restricted: Read-Once Rectangles Lower Bounds against Function Composition Open Problems

Restricted Branching Programs Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Bounded Width: same as NC 1 , Barrington’s Programs Motivation characterization. Restricted: Read-Once Rectangles Length Restricted: give Time-Space tradeoffs. Lower Bounds against Function Composition Open Problems

Time-Space tradeoffs Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi ⇒ s = 2 Ω( n ) t ≤ cn = Jukna’09 Motivation Branching Programs Motivation Restricted: Read-Once Rectangles Lower Bounds against Function Composition Open Problems

Time-Space tradeoffs Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi ⇒ s = 2 Ω( n ) t ≤ cn = Jukna’09 Motivation culmination results by Ajtai ‘99 and Beame,Jayram, Branching Programs Saks ‘01 Motivation Restricted: Read-Once Rectangles Lower Bounds against Function Composition Open Problems

Time-Space tradeoffs Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi ⇒ s = 2 Ω( n ) t ≤ cn = Jukna’09 Motivation culmination results by Ajtai ‘99 and Beame,Jayram, Branching Programs Saks ‘01 Motivation Restricted: Read-Once We look at: Rectangles time-space tradeoffs Lower Bounds against for Function Composition iterated function composition . Open Problems

Read Once Hardness of Function Composition for Semantic Read once Branching Syntactic read once: Along any path from source to Programs sink any variable appears atmost once. Jeff Edmonds, Venkatesh x 11 x 21 x 31 Medabalimi, Toniann Pitassi Motivation 1 Start Branching x 12 x 22 x 32 Programs Motivation Restricted: Read-Once x 33 x 13 x 23 Rectangles Lower Bounds against Function Composition Open Problems

Read Once Hardness of Function Composition for Semantic Read once Branching Syntactic read once: Along any path from source to Programs sink any variable appears atmost once. Jeff Edmonds, Venkatesh x 11 x 21 x 31 Medabalimi, Toniann Pitassi Motivation 1 Start Branching x 12 x 22 x 32 Programs Motivation Restricted: Read-Once x 33 x 13 x 23 Rectangles x 11 x 21 x 31 Lower Bounds x 11 ¯ x 21 ¯ x 12 ¯ x 22 ¯ x 13 ¯ x 23 ¯ against Function Composition Open Problems 1 Start x 12 x 22 x 32 x 21 ¯ ¯ x 31 x 22 ¯ ¯ x 32 x 23 ¯ ¯ x 33 x 33 x 31 ¯ x 11 ¯ x 32 ¯ x 12 ¯ x 33 ¯ x 13 ¯ x 13 x 23 Semantic read once: Along any consistent path from source to sink no variable is read more than once.

syntactic weaker than semantic Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi The Exact Perfect matching function( EPM n ): accept a Motivation matrix iff it is a permutation matrix. Branching Programs Motivation Jukna and Razborov ‘98 showed Restricted: Read-Once Rectangles Theorem Lower Bounds against Every syntactic read once NBP computing EPM n must have Function Composition size 2 Ω( n ) . Open Problems

EPM n ∈ semantic read once Hardness of Function Composition for Semantic Read once Branching Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Theorem (Jukna) Motivation EPM n can be solved by a semantic read once NBP of size Branching O ( n 3 ) . Programs Motivation Restricted: Read-Once Rectangles   0 0 1 Lower Bounds 1 0 0 against   Function 0 1 0 Composition Open Problems

EPM n ∈ semantic read once Hardness of Function Composition for Semantic Read once Branching Programs Theorem (Jukna) Jeff Edmonds, Venkatesh EPM n can be solved by a semantic read once NBP of size Medabalimi, O ( n 3 ) . Toniann Pitassi Motivation Branching   0 0 1 Programs Motivation 1 0 0   Restricted: Read-Once 0 1 0 Rectangles Lower Bounds against x 11 x 21 x 31 Function Composition Open Problems Start x 12 x 22 x 32 x 13 x 23 x 33