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

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

Jul 09, 2023 203 likes •792 views

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

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SLIDE 1

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition Open Problems

Hardness of Function Composition for Semantic Read once Branching Programs

Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi June 23, 2018

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SLIDE 2

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition Open Problems

P and L

P: Polynomial time computable functions.

* * * *

L : Functions computable in logarithmic space. L

?

⊂ P

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SLIDE 3

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Branching Programs

f (x1, x2, .., xn) → {0, 1} xi ∈ {0, 1}, ∀i ∈ [n]

Definition

Deterministic Branching program DAG with a source node and two sinks, 1-sink (for accept) and 0-sink (for reject). Each non-sink node is labeled by some xi, outdegree 2 with an edge each for xi = 0 and xi = 1. x5 x2 x3 x4 x5 start, x1 x2 x3 x4 1

1 1 1 1 1 1 1 1 1

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SLIDE 4

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Non-det Branching Programs

Definition

Non-deterministic Branching program (NBP) allow unlabelled guessing nodes and arbitrary

  • ut-degree.

1

Start x12 x11 x13 x14 x22 x21 x23 x24 x32 x31 x33 x34 x42 x41 x43 x44

The size of a NBP= number of labelled nodes.

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SLIDE 5

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

NBP computing f : {0, 1}n → {0, 1}

1

Start x12 x11 x13 x14 x22 x21 x23 x24 x32 x31 x33 x34 x42 x41 x43 x44

f (u) = 1 ⇐ ⇒ ∃ a path from source to accept node that is consistent with input u.

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SLIDE 6

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Program Size and Space Complexity of computing f

BP(fn) = min

B∈ BP computing fn size (B)

S(fn) = min

T∈ non-uniform TMs computing fn space complexity (T)

log(BP(fn)) ≈ S(fn) [Cobham ‘66]

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SLIDE 7

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Big Picture

It is easy to show functions with high BP(fn) exist.

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SLIDE 8

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Big Picture

It is easy to show functions with high BP(fn) exist. Can we show that some function in P requires exponential size BP ?

amounts to showing L ⊂ P.

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SLIDE 9

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

BPs and other Computation Models

Formulas BranchingPrograms Circuits L(f ) ≥ BP(f ) ≥

1 3C(f )

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SLIDE 10

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

BPs and other Computation Models

Formulas BranchingPrograms Circuits L(f ) ≥ BP(f ) ≥

1 3C(f )

Ω

  • n3

Ω

  • n2

log2 n

  • Ω (n)

Random Restrictions

Nechiporuk

Gate Elimination

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SLIDE 11

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Restricted Branching Programs

Bounded Width: same as NC1, Barrington’s characterization.

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SLIDE 12

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Restricted Branching Programs

Bounded Width: same as NC1, Barrington’s characterization. Length Restricted: give Time-Space tradeoffs.

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SLIDE 13

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Time-Space tradeoffs

t ≤ cn =

⇒ s = 2Ω(n)

Jukna’09

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SLIDE 14

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Time-Space tradeoffs

t ≤ cn =

⇒ s = 2Ω(n)

Jukna’09 culmination results by Ajtai ‘99 and Beame,Jayram, Saks ‘01

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SLIDE 15

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Time-Space tradeoffs

t ≤ cn =

⇒ s = 2Ω(n)

Jukna’09 culmination results by Ajtai ‘99 and Beame,Jayram, Saks ‘01 We look at: time-space tradeoffs for iterated function composition.

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SLIDE 16

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Read Once

Syntactic read once: Along any path from source to sink any variable appears atmost once.

Start

1

x11 x12 x13 x21 x22 x23 x31 x32 x33

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SLIDE 17

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Read Once

Syntactic read once: Along any path from source to sink any variable appears atmost once.

Start

1

x11 x12 x13 x21 x22 x23 x31 x32 x33 Start

1

x11 x12 x13 x21 x22 x23 x31 x32 x33 ¯ x11 ¯ x21 ¯ x21 ¯ x31 ¯ x31 ¯ x11 ¯ x12 ¯ x22 ¯ x22 ¯ x32 ¯ x32 ¯ x12 ¯ x13 ¯ x23 ¯ x23 ¯ x33 ¯ x33 ¯ x13

Semantic read once: Along any consistent path from source to sink no variable is read more than once.

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SLIDE 18

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

syntactic weaker than semantic

The Exact Perfect matching function(EPMn): accept a matrix iff it is a permutation matrix. Jukna and Razborov ‘98 showed

Theorem

Every syntactic read once NBP computing EPMn must have size 2Ω(n).

slide-19
SLIDE 19

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

EPMn ∈ semantic read once

Theorem (Jukna)

EPMn can be solved by a semantic read once NBP of size O(n3).   0 0 1 1 0 0 0 1 0  

slide-20
SLIDE 20

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

EPMn ∈ semantic read once

Theorem (Jukna)

EPMn can be solved by a semantic read once NBP of size O(n3).   0 0 1 1 0 0 0 1 0  

Start x11 x12 x13 x21 x22 x23 x31 x32 x33

slide-21
SLIDE 21

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

EPMn ∈ semantic read once

Theorem (Jukna)

EPMn can be solved by a semantic read once NBP of size O(n3).   0 0 1 1 0 0 0 1 0  

Start

1

x11 x12 x13 x21 x22 x23 x31 x32 x33 ¯ x11 ¯ x21 ¯ x21 ¯ x31 ¯ x31 ¯ x11 ¯ x12 ¯ x22 ¯ x22 ¯ x32 ¯ x32 ¯ x12 ¯ x13 ¯ x23 ¯ x23 ¯ x33 ¯ x33 ¯ x13

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SLIDE 22

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

EPMn ∈ semantic read once

Theorem (Jukna)

EPMn can be solved by a semantic read once NBP of size O(n3).   0 0 1 1 0 0 0 1 0  

Start

1

x11 x12 x13 x21 x22 x23 x31 x32 x33 ¯ x11 ¯ x21 ¯ x21 ¯ x31 ¯ x31 ¯ x11 ¯ x12 ¯ x22 ¯ x22 ¯ x32 ¯ x32 ¯ x12 ¯ x13 ¯ x23 ¯ x23 ¯ x33 ¯ x33 ¯ x13

Sees only 1s

slide-23
SLIDE 23

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

EPMn ∈ semantic read once

Theorem (Jukna)

EPMn can be solved by a semantic read once NBP of size O(n3).   0 0 1 1 0 0 0 1 0  

Start

1

x11 x12 x13 x21 x22 x23 x31 x32 x33 ¯ x11 ¯ x21 ¯ x21 ¯ x31 ¯ x31 ¯ x11 ¯ x12 ¯ x22 ¯ x22 ¯ x32 ¯ x32 ¯ x12 ¯ x13 ¯ x23 ¯ x23 ¯ x33 ¯ x33 ¯ x13

Sees only 1s Sees only 0s

slide-24
SLIDE 24

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

EPMn ∈ semantic read once

Theorem (Jukna)

EPMn can be solved by a semantic read once NBP of size O(n3).   0 0 1 1 0 0 0 1 0  

Start

1

x11 x12 x13 x21 x22 x23 x31 x32 x33 ¯ x11 ¯ x21 ¯ x21 ¯ x31 ¯ x31 ¯ x11 ¯ x12 ¯ x22 ¯ x22 ¯ x32 ¯ x32 ¯ x12 ¯ x13 ¯ x23 ¯ x23 ¯ x33 ¯ x33 ¯ x13

Sees only 1s Sees only 0s

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SLIDE 25

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Embedded Rectangles Cred × {w} × Cblue ⊆ [D]n

A ⊂ [n]

Cred ⊆ DA

B ⊂ [n]

Cblue ⊆ DB w

w ∈ D[n]−A−B

     012210101 012010201 101012202      × {10210001} ×          012001012 201101002 110020120 210200120         

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SLIDE 26

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Embedded Rectangles Cred × {w} × Cblue ⊆ [D]n

A ⊂ [n]

Cred ⊆ DA

B ⊂ [n]

Cblue ⊆ DB w

w ∈ D[n]−A−B

     012210101 012010201 101012202      × {102101} ×          012001012 201101002 110020120 210200120         

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SLIDE 27

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

KRW Conjecture

an approach to separating NC1 from NC2. KRW conjecture that for every random f and ∀ g, D(fog) ≥ ǫD(f ) + D(g). KRW conjecture on formula size of a composed function fog. L(fog) ≈ L(f )L(g) ?

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SLIDE 28

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Understanding Composition

How does space compose ?

slide-29
SLIDE 29

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

TEPh

d

Cook et.al 2012

f : [k]2 → [k] ∈ [k]

Figure: TEP4

2 that is height 4, degree 2.

slide-30
SLIDE 30

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

TEPh

d

Cook et.al 2012

f : [k]2 → [k] ∈ [k]

Figure: TEP4

2 that is height 4, degree 2.

Is BP(TEPh

2 ) = Ω(kh) ??

slide-31
SLIDE 31

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

TEPh

d

Cook et.al 2012

f : [k]2 → [k] ∈ [k]

Figure: TEP4

2 that is height 4, degree 2.

Is BP(TEPh

2 ) = Ω(kh) ?? = ⇒ L⊂P

slide-32
SLIDE 32

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs

Motivation Restricted: Read-Once Rectangles

Lower Bounds against Function Composition Open Problems

Boolean TEP

K 1−ǫ

K −ǫ density

Tree

F,ǫ(·)

slide-33
SLIDE 33

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Lower Bound for composition

Theorem

For any h, and k sufficiently large, there exists ǫ and F such that any k-ary nondeterministic semantic read-once branching program for ternary Tree

F,ǫ requires size at least

  • k

log k

h .

slide-34
SLIDE 34

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Black White pebbling Upperbound

slide-35
SLIDE 35

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

1 2(d − 1)h + 1 pebbles at this moment

slide-36
SLIDE 36

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Guess the remaining siblings

slide-37
SLIDE 37

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Infer the root

slide-38
SLIDE 38

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Unpebble blacks

slide-39
SLIDE 39

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Verify Guesses

slide-40
SLIDE 40

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

lower bound uses Invertible functions

Definition

A Latin Cube is a function f : [k]3 → [k] such that f is invertible in each of its coordinates. Equivalently, every element of [k] appears exactly once along every row,column and leg in the cube [k]3.

slide-41
SLIDE 41

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

4-invertible function, f : [k]3 → [k]

Definition

Any element in [k] appears at most 4 times along any row,column or leg in the cube [k]3.

slide-42
SLIDE 42

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Proof Overview

A small BP for a

= ⇒

Tree

F,ǫ accepts a large

Tree

F,ǫ

rectangle of inputs

  • ver its leaves
slide-43
SLIDE 43

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Proof Overview

A small BP for a

= ⇒

Tree

F,ǫ accepts a large

Tree

F,ǫ

rectangle of inputs

  • ver its leaves

A large rectangle

= ⇒

∃ a special node v* in the tree

  • ver leaves

whose Fv∗ can be described in few bits.

slide-44
SLIDE 44

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Proof Overview

A small BP for a

= ⇒

Tree

F,ǫ accepts a large

Tree

F,ǫ

rectangle of inputs

  • ver its leaves

A large rectangle

= ⇒

∃ a special node v* in the tree

  • ver leaves

whose Fv∗ can be described in few bits. Show that the distribution on F is rich or sufficiently random looking that one cannot save these bits.

slide-45
SLIDE 45

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

for every accepting input ∃ a special query state:

Bh(x, y) = Fh(Ah(x), Bh−1(x, y)Ch(y)) Ah(x) Ch−1(y) Ah−2(x) vi Bi(x, y) = Fi(Ai(x), Bi−1(x, y)Ci(y)) Ci(y) A3(x) A2(x) ? A1(x) C1(y) C2(y) C3(y) Ai(x) Ch−2(y) Ah−1(x) Ch(y)

i − 1 i − 1

A red subtree has at least a fraction of leaf node which are red h − 2 A white subtree has at least a fraction of leaf node which are white h − 2

slide-46
SLIDE 46

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

few special states = ⇒ ∃ a large embedded rectangle over leaves

q Most popular start 1 v v v × Choose a popular labelled path down the tree. Choose a popular red variable for the first red-subtree. Prune the input set. Continue to choose h red variables

  • ne for each red-subtree. Similarly for each

blue-subtree. Fix the remaining variables in [n]-Red-Blue to the most popular projection ‘w’.

slide-47
SLIDE 47

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

∃ a large embedded rectangle over the leaves.

Bh(x, y) = Fh(Ah(x), Bh−1(x, y)Ch(y)) Ah(x) Ch−1(y) Ah−2(x) vi Bi(x, y) = Fi(Ai(x), Bi−1(x, y)Ci(y)) Ci(y) A3(x) A2(x) ? A1(x) C1(y) C2(y) C3(y) Ai(x) Ch−2(y) Ah−1(x) Ch(y)

i − 1 i − 1

slide-48
SLIDE 48

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

∃ a node v∗ at which leaves in both red and blue trees take a lot of values.

Bh(x, y) = Fh(Ah(x), Bh−1(x, y)Ch(y)) Ah(x) Ch−1(y) Ah−2(x) vi Bi(x, y) = Fi(Ai(x), Bi−1(x, y)Ci(y)) Ci(y) A3(x) A2(x) ? A1(x) C1(y) C2(y) C3(y) Ai(x) Ch−2(y) Ah−1(x) Ch(y)

i − 1 i − 1

slide-49
SLIDE 49

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

∃ a node v∗ at which both red and blue child take a lot of values.

Bh(x, y) = Fh(Ah(x), Bh−1(x, y)Ch(y)) Ah(x) Ch−1(y) Ah−2(x) vi Bi(x, y) = Fi(Ai(x), Bi−1(x, y)Ci(y)) Ci(y) A3(x) A2(x) ? A1(x) C1(y) C2(y) C3(y) Ai(x) Ch−2(y) Ah−1(x) Ch(y)

i − 1 i − 1

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SLIDE 50

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

∃ a node v∗ with low entropy on a Two-way product set

Bh(x, y) = Fh(Ah(x), Bh−1(x, y)Ch(y)) Ah(x) Ch−1(y) Ah−2(x) vi Bi(x, y) = Fi(Ai(x), Bi−1(x, y)Ci(y)) Ci(y) A3(x) A2(x) ? A1(x) C1(y) C2(y) C3(y) Ai(x) Ch−2(y) Ah−1(x) Ch(y)

i − 1 i − 1

slide-51
SLIDE 51

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Two-way Product Set at v∗, in Fv∗()

A C

A, C ⊂ [k] |A| = |C| = r << k

slide-52
SLIDE 52

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Two-way Product Set at v∗, in Fv∗()

A C

A, C ⊂ [k] |A| = |C| = r << k Sr = {(x, Q(x, y), y)|x ∈ A, y ∈ C} |S| = r 2

slide-53
SLIDE 53

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Enropy of spread on Two-way Product Set can’t be low

A C

A, C ⊂ [k] |A| = |C| = r << k Sr = {(x, B(x, y), y)|x ∈ A, y ∈ C} |S| = r 2 ∀ Two-way Product Sets Sr and target set Tǫ Pr

f ∼U(All 4-invertible cubes)[f (Sr) ⊂ Tǫ] ≤

1 kǫr 2

slide-54
SLIDE 54

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

Label

αh αh−1 αh−2 vi ∗ βi α2 α1 ? β1 β2 αi βh−2 βh−1 βh A white subtree has at least

  • ne white leaf node

h − 2 A red subtree has at least one red leaf node h − 2

Figure: This figure depicts a label L

F associated with a problem

instance Tree

F

{bottlenecktessellation

slide-55
SLIDE 55

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition

Proof OverView ∃ A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

Open Problems

∃ many F that remain unaccounted without such a special label.

< v∗, SA, SB.. >

  • F
slide-56
SLIDE 56

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition Open Problems

Open Problems

More general time space tradeoffs for composition. Exponential lower bound for boolean semantic NBPs for some problem in P. super-quadratic lower bound for BPs via understanding composition fog where g is element distinctness.

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SLIDE 57

Hardness of Function Composition for Semantic Read

  • nce Branching

Programs Jeff Edmonds, Venkatesh Medabalimi, Toniann Pitassi Motivation Branching Programs Lower Bounds against Function Composition Open Problems

Thank You !