Exponential Lower Bounds for Monotone Span Programs Stephen A. Cook - - PowerPoint PPT Presentation

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Exponential Lower Bounds for Monotone Span Programs Stephen A. Cook - - PowerPoint PPT Presentation

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Exponential Lower Bounds for Monotone Span Programs Stephen A. Cook Toniann Pitassi FOCS 2016 Robert Robere Benjamin Rossman University of Toronto Familiar Picture 2 Familiar Picture Formulas 3 Familiar Picture Switching Networks

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Exponential Lower Bounds for Monotone Span Programs

Stephen A. Cook Toniann Pitassi Robert Robere Benjamin Rossman University of Toronto FOCS 2016

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Familiar Picture

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Familiar Picture

Formulas

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Familiar Picture

Formulas Switching Networks (Branching Programs)

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Familiar Picture

Formulas Switching Networks (Branching Programs) Directed Switching Networks (Non-det. Branching Programs)

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Familiar Picture

Formulas Switching Networks (Branching Programs) Directed Switching Networks (Non-det. Branching Programs) Polylog-depth Circuits

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Familiar Picture

Formulas Switching Networks (Branching Programs) Directed Switching Networks (Non-det. Branching Programs) Polylog-depth Circuits Circuits

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Familiar Picture

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(Less) Familiar Picture

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(Less) Familiar Picture

Span Programs over fjeld F [KW '90]

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Span Programs [KW '90]

What is a Span Program over a fjeld F?

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A Matrix over F What is a Span Program over a fjeld F?

Span Programs [KW '90]

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What is a Span Program over a fjeld F? 1 1 1 1 1 1

Span Programs [KW '90]

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What is a Span Program over a fjeld F? 1 1 1 1 1 1 Rows labelled with input literals.

Span Programs [KW '90]

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What is a Span Program over a fjeld F? 1 1 1 1 1 1

Span Programs [KW '90]

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What is a Span Program over a fjeld F? 1 1 1 1 1 1 Accept assignment if the consistent rows span all-1s vector

Span Programs [KW '90]

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What is a Span Program over a fjeld F? 1 1 1 1 1 1

Span Programs [KW '90]

Accept assignment if the consistent rows span all-1s vector

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What is a Span Program over a fjeld F? 1 1 1 1 1 1

Span Programs [KW '90]

Accept assignment if the consistent rows span all-1s vector

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What is a Span Program over a fjeld F? 1 1 1 1 1 1 ACCEPT!

Span Programs [KW '90]

Accept assignment if the consistent rows span all-1s vector

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What is a Span Program over a fjeld F? 1 1 1 1 1 1

Span Programs [KW '90]

Accept assignment if the consistent rows span all-1s vector

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What is a Span Program over a fjeld F? 1 1 1 1 1 1 ACCEPT!

Span Programs [KW '90]

Accept assignment if the consistent rows span all-1s vector

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What is a Span Program over a fjeld F? 1 1 1 1 1 1

Span Programs [KW '90]

Accept assignment if the consistent rows span all-1s vector

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What is a Span Program over a fjeld F? 1 1 1 1 1 1 REJECT!

Span Programs [KW '90]

Accept assignment if the consistent rows span all-1s vector

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(Less) Familiar Picture

Span Programs over fjeld F [KW '90] Capture logspace counting classes.

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(Less) Familiar Picture

Span Programs over fjeld F [KW '90] Capture logspace counting classes. Comparator Circuits

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(Less) Familiar Picture

Span Programs over fjeld F [KW '90] Capture logspace counting classes. Comparator Circuits ~ Sorting networks.

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(Less) Familiar Picture

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Familiar Picture

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Familiar Picture

How many separations do we have?

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Familiar Picture

How many separations do we have?

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Familiar Picture

How many separations do we have? Fortunately, this is easy to fjx.

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Familiar Picture

How many separations do we have? Fortunately, this is easy to fjx. Monotone = No Negations in Circuit Models

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Familiar Picture

How many separations do we have?

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Familiar Picture

[Karchmer-Wigderson '88] (Undirected st-connectivity) [Raz-Mckenzie '97] (GEN) [Potechin '10] (Directed st-connectivity) [Babai, Gal, Wigderson '99] (Odd Factor)

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Familiar Picture

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Familiar Picture

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[Babai et al '96] Quasipolynomial lower bounds against mNP .

Familiar Picture

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[Babai et al '96] Quasipolynomial lower bounds against mNP . [Gal '98] Improved lower bounds using rank measure (still quasipolynomial).

Familiar Picture

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[Babai et al '96] Quasipolynomial lower bounds against mNP . [Gal '98] Improved lower bounds using rank measure (still quasipolynomial). [BW '05] Quasipolynomial against nonmonotone NC

Familiar Picture

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[Babai et al '96] Quasipolynomial lower bounds against mNP . [Gal '98] Improved lower bounds using rank measure (still quasipolynomial). Extra Motivation: [BW '05] Quasipolynomial against nonmonotone NC

Familiar Picture

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[Babai et al '96] Quasipolynomial lower bounds against mNP . [Gal '98] Improved lower bounds using rank measure (still quasipolynomial). Extra Motivation: Equivalent to Linear Secret Sharing Schemes (!) [KW '90] [BW '05] Quasipolynomial against nonmonotone NC

Familiar Picture

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Familiar Picture

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Familiar Picture

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Familiar Picture

Essentially nothing known! Exponential bounds for Clique Cannot even prove it contains mNL

  • r mL
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Familiar Picture

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Familiar Picture

Natural Questions:

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Familiar Picture

Can we separate mSPAN from mP? mNL? Natural Questions:

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Familiar Picture

Can we separate mSPAN from mP? mNL? Can we separate mCC from mP? mNL? Natural Questions:

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Familiar Picture

Can we separate mSPAN from mP? mNL? Can we separate mCC from mP? mNL? Natural Questions: Yes --- also unify nearly all lower bounds in mP .

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Rank Measure

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Rank Measure

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Rank Measure

monotone

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Rank Measure

monotone

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Rank Measure

monotone Matrix Not the 0-1 Communication Matrix

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Rank Measure

monotone Matrix Not the 0-1 Communication Matrix For any input index i, take submatrix of

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Rank Measure

monotone Matrix Not the 0-1 Communication Matrix For any input index i, take submatrix of

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Rank Measure

monotone For any input index i, take submatrix of

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Rank Measure

monotone Ranging over inputs...

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Rank Measure

monotone Ranging over inputs...

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Rank Measure

monotone All rectangles cover A! Ranging over inputs...

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Rank Measure

monotone Rank Measure [Razborov '90]: Ranging over inputs... All rectangles cover A!

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Rank Measure

Rank Measure [Razborov '90]:

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Rank Measure

Rank Measure [Razborov '90]: Theorem [R '90, KW '90, G '98, CPRR '16]: For any fjeld F, any boolean function f, and any matrix A over F,

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Rank Measure

Rank Measure [Razborov '90]: Theorem [R '90, KW '90, G '98, CPRR '16]: For any fjeld F, any boolean function f, and any matrix A over F, in NP! Best prior lower bounds:

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Main Theorem

Theorem: There is a function f (GEN) in mP and a real matrix A such that There is a function g (STCONN) in mNL and a real matrix B such that

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Main Theorem

Theorem: There is a function f (GEN) in mP and a real matrix A such that There is a function g (STCONN) in mNL and a real matrix B such that Prior Work: Unifjed proof of many previous monotone separations between classes within P . Simplifjcation of [Potechin '10]

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Span Programs: First exponential lower bounds for monotone span programs and linear secret sharing schemes.

Main Theorem

First separations between monotone span programs and monotone P, monotone NL Example of a function computable by non-monotone span programs over GF(2), not computable by monotone span programs over reals Theorem: There is a function f (GEN) in mP and a real matrix A such that There is a function g (STCONN) in mNL and a real matrix B such that

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Comparator Circuits: First exponential lower bounds for comparator circuits computing a function in monotone P .

Main Theorem

First separations between monotone comparator circuits and monotone P, monotone NL Example of a function computable by non-monotone comparator circuits, not effjciently computable by monotone comparator circuits Theorem: There is a function f (GEN) in mP and a real matrix A such that There is a function g (STCONN) in mNL and a real matrix B such that

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Pause! Breathe!

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The Proof

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The Proof

Previous Proofs:

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The Proof

Previous Proofs: Direct combinatorial constructions

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The Proof

Previous Proofs: Direct combinatorial constructions Resulting matrices have {0,1} entries, for which we have quasipolynomial upper bounds [Razborov '90].

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The Proof

Previous Proofs: Direct combinatorial constructions Resulting matrices have {0,1} entries, for which we have quasipolynomial upper bounds [Razborov '90]. Our Proof:

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The Proof

Previous Proofs: Direct combinatorial constructions Resulting matrices have {0,1} entries, for which we have quasipolynomial upper bounds [Razborov '90]. Our Proof: Prove a new lifting theorem to reduce the lower bound to bounding a new algebraic query measure on search problems.

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The Proof

Previous Proofs: Direct combinatorial constructions Resulting matrices have {0,1} entries, for which we have quasipolynomial upper bounds [Razborov '90]. Our Proof: Prove a new lifting theorem to reduce the lower bound to bounding a new algebraic query measure on search problems. Our matrices have entries in , and so we can avoid the above obstacle.

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The Proof

Overview Rank Measure [Razborov '90]:

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

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(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

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(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f)

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

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(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f)

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

2

(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f) Follows from [Raz-Mckenzie '97] [Goos-Pitassi '15]

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

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(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f)

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The Proof

Lifting Theorem

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The Proof

Lifting Theorem (Communication Setting)

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The Proof

Lifting Theorem (Communication Setting) Search Problem S = Search(f)

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The Proof

Hard for Weak Complexity Measure

Decision Tree

Lifting Theorem (Communication Setting) Search Problem S = Search(f)

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The Proof

Hard for Weak Complexity Measure

Decision Tree

Lifting Theorem (Communication Setting) Search Problem S = Search(f)

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The Proof

Hard for Weak Complexity Measure

Decision Tree

Compose S with some two input function g Lifting Theorem (Communication Setting) Search Problem S = Search(f) Alice gets x inputs Bob gets y inputs

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The Proof

Hard for Weak Complexity Measure

Decision Tree

Compose S with some two input function g Hard for Strong Complexity Measure Lifting Theorem (Communication Setting) Search Problem S = Search(f) Alice gets x inputs Bob gets y inputs

Communication Matrix

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The Proof

Lifting Theorem (Our Setting)

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The Proof

Lifting Theorem (Our Setting) Search Problem S = Search(f)

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The Proof

Lifting Theorem (Our Setting) Hard for Strong Complexity Measure Search Problem S = Search(f)

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The Proof

Lifting Theorem (Our Setting) Hard for Weak Complexity Measure Hard for Strong Complexity Measure Search Problem S = Search(f)

?

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The Proof

Lifting Theorem (Our Setting) Hard for Weak Complexity Measure Hard for Strong Complexity Measure Search Problem S = Search(f) ?

?

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The Proof

Lifting Theorem (Our Setting) Hard for Weak Complexity Measure Hard for Strong Complexity Measure Search Problem S = Search(f) Polynomial certifying a large algebraic gap for S ?

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The Proof

Lifting Theorem (Our Setting) Hard for Weak Complexity Measure Hard for Strong Complexity Measure Search Problem S = Search(f) Polynomial certifying a large algebraic gap for S Compose p with two-input function g instead!

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Lifting Theorem (ST-CONN)

Theorem: (Lifting Theorem for Rank Measure) Consider layered ST-CONN on the grid, and let k be the algebraic gap complexity of the ST-CONN search problem. There is a real matrix A such that

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Lifting Theorem (ST-CONN)

Theorem: (Lifting Theorem for Rank Measure) Consider layered ST-CONN on the grid, and let k be the algebraic gap complexity of the ST-CONN search problem. There is a real matrix A such that Proof: Intuition on previous slide, extension of the Pattern Matrix Method [Sherstov '08].

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

2

(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f)

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

2

(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f)

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The Proof

Lifting Theorem Algebraic Gaps

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The Proof

Lifting Theorem Algebraic Gaps Def: Let be an unsatisfjable CNF . Then Search(F) is the following problem: Given an assignment x to the variables of F,

  • utput the name of a clause falsifjed by x.
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The Proof

Lifting Theorem Algebraic Gaps Def: Let be an unsatisfjable CNF . Then Search(F) is the following problem: Given an assignment x to the variables of F,

  • utput the name of a clause falsifjed by x.

Def: Let be a total search

  • problem. The algebraic gap complexity
  • f Search(F) is the maximum k for which there

is a polynomial such that for each certifjcate C of Search(F).

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The Proof

Lifting Theorem Algebraic Gaps Def: Let be a total search

  • problem. The algebraic gap complexity
  • f Search(F) is the maximum k for which there

is a polynomial such that for each certifjcate C of Search(F).

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The Proof

Lifting Theorem Algebraic Gaps Def: Let be a total search

  • problem. The algebraic gap complexity
  • f Search(F) is the maximum k for which there

is a polynomial such that for each certifjcate C of Search(F). We give lower bounds on the algebraic gap complexity for the search problems corresponding to GEN and ST-CONN by reducing to reversible pebbling.

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

2

(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f)

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The Proof

Overview Rank Measure [Razborov '90]:

1

Associate with certain special functions f (like GEN and ST-CONN) a search problem Search(f)

2

(Lift) Reduce constructing a good matrix A for f to lower bounding a complexity measure on Search(f)

3

Actually prove the query lower bounds against Search(f)

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Conclusion

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Conclusion

Unifjed lower bounds against monotone models by “lifting”.

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Conclusion

Unifjed lower bounds against monotone models by “lifting”. Algebraic gaps → other applications?

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Conclusion

Unifjed lower bounds against monotone models by “lifting”. Algebraic gaps → other applications? Average case lower bounds?

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Conclusion

Unifjed lower bounds against monotone models by “lifting”. Algebraic gaps → other applications? Average case lower bounds? Sharpen lifting theorems further?

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Conclusion

Unifjed lower bounds against monotone models by “lifting”. Algebraic gaps → other applications? Average case lower bounds? Sharpen lifting theorems further? Other algebraic query complexity measures for search problems?

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Conclusion

Unifjed lower bounds against monotone models by “lifting”. Algebraic gaps → other applications? Average case lower bounds? Sharpen lifting theorems further? Other algebraic query complexity measures for search problems?

Thanks for listening!

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References

Babai, Gal, Kollar, Ronyai, Szabo, Wigderson. Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs. STOC '96.

  • Gal. A characterization of span program size and improved lower bounds for

monotone span programs. STOC '98.

  • Potechin. Bounds on monotone switching networks for directed connectivity.

FOCS '10. Chan, Potechin. Tight bounds for monotone switching networks via Fourier

  • analysis. STOC '12.

Karchmer, Wigderson. Monotone circuits for connectivity require super- logarithmic depth. STOC '88. Karchmer, Wigderson. On span programs. Structure in Complexity Theory '93. Raz, Mckenzie. Separation of the monotone NC hierarchy. FOCS '97.

  • Razborov. Applications of matrix methods to the theory of lower bounds in

computational complexity. Combinatorica '90.

  • Sherstov. The pattern matrix method for lower bounds on quantum
  • communication. STOC '08.