Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds

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Cracks in the Defenses Scouting Out Approaches
on Circuit Lower Bounds

• CSR 2008

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Introduction

• How far are we from proving circuit lower bounds?
• I have no idea!
• There is a lot of pessimism, based on
• The lack of any good circuit lower bounds
• The Razborov,Rudich natural proofs obstacle
• Today, well make some observations that may
  cause some of you to be less pessimistic.

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But FirstWhy Circuits?

• 2 Basic models of computation
• Programs (one program works for every input
  length)
• Circuits (different circuit for each input
  length)
• One crucial difference circuit lower bounds can
  be used to prove intractability results for fixed
  input sizes.
• Program run-time lower bounds cant.

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An example the Game of Checkers

• Computing strategies for Checkers requires
  exponential time.
• More precisely, given an n-by-n Checkers board
  with checkers on it, no program can compute an
  optimal next move in fewer than c2n d steps,
  for some constants c and d.
• n-by-n Checkers is complete for EXP.

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An example the Game of Checkers

• Computing strategies for Checkers requires
  exponential time.
• More precisely, given an n-by-n Checkers board
  with checkers on it, no program can compute an
  optimal next move in fewer than c2n d steps,
  for some constants c and d.
• Thus any program solving this problem must run
  very slowly on large inputs. This is the essence
  of asymptotic analysis.

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An example the Game of Checkers

• Computing strategies for Checkers requires
  exponential time.
• More precisely, given an n-by-n Checkers board
  with checkers on it, no program can compute an
  optimal next move in fewer than c2n d steps,
  for some constants c and d.
• This is a much stronger statement about
  complexity than we are able to prove for most
  problems (such as NP-complete problems).

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An example the Game of Checkers

• Computing strategies for Checkers requires
  exponential time.
• More precisely, given an n-by-n Checkers board
  with checkers on it, no program can compute an
  optimal next move in fewer than c2n d steps,
  for some constants c and d.
• butConceivably, there is a hand-held device
  that computes optimal moves, even for Checker
  boards of size 1000-by-1000!
• because we dont know if EXP is in P/poly (the
  class of problems with small circuits).

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An Example of what can be done, given a circuit
size lower bound

• Theorem Any circuit that takes as input a
  logical formula (in WS1S) of length 616 and
  produces as output a correct answer, saying if
  the formula is valid or not, has at least 10123
  gates. (Stockmeyer, 1974)
• (Proof sketch) There is a problem requiring
  exponential circuit size that is efficiently
  reducible to WS1S.

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An Example of what can be done, given a circuit
size lower bound

• Theorem Any circuit that takes as input a
  logical formula (in WS1S) of length 616 and
  produces as output a correct answer, saying if
  the formula is valid or not, has at least 10123
  gates. (Stockmeyer, 1974)
• What we need Similar lower bounds, but for
  problems in NP such as SAT or FACTORING.
• We would even be glad to get lower bounds for
  restricted classes of circuits.

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Big Complexity Classes

• NP
• P
• .
• .
• NC
• L (Deterministic Logspace)

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The Main Objects of InterestSmall Complexity
Classes

• TC0 O(1)-Depth Circuits of MAJ gates
• AC0 6
• NC1 Log-Depth Circuits
• AC0 cant compute Mod 2 FSS,A
• AC0 O(1)-Depth Circuits of AND/OR gates

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The Main Objects of InterestSmall Complexity
Classes

• TC0 O(1)-Depth Circuits of MAJ gates
• AC0 6
• NC1 Log-Depth Circuits
• AC0 cant compute Mod 2 FSS,A
• AC0 O(1)-Depth Circuits of AND/OR gates

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The Main Objects of InterestSmall Complexity
Classes

• TC0 O(1)-Depth Circuits of MAJ gates
• NC1 Log-Depth Circuits
• AC0 2 cant compute Mod 3 R,S
• AC0 2
• AC0 O(1)-Depth Circuits of AND/OR gates

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The Main Objects of InterestSmall Complexity
Classes

• NC1 Log-Depth Circuits
• TC0 O(1)-Depth Circuits of MAJ gates
• AC0 6
• AC0 2
• AC0 O(1)-Depth Circuits of AND/OR gates

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The Main Objects of InterestSmall Complexity
Classes

• NC1 poly-size formulae
• TC0 O(1)-Depth Circuits of MAJ gates
• AC0 6
• AC0 2
• AC0 O(1)-Depth Circuits of AND/OR gates

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Complete Problems

• NP has complete sets (under polynomial time
  reducibility P)
• These small classes have complete sets, too
  (under AC)

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Reductions

• A AC B means that there is a constant-depth
  circuit computing A that has the usual AND and OR
  gates, and also has oracle gates for B.

B
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Complete Problems

• NC1
• TC0
• AC0 6
• AC0 2
• AC0

• sorting, multiplication, division
• Naor,Reingold Pseudorandom Generator

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Complete Problems

• NC1
• TC0
• AC0 6
• AC0 2
• AC0

• BFE Balanced Boolean Formula Evaluation
  (AND,OR,XOR)
• Word problem over S5

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The Word Problem Over S5

• A regular set complete for NC1

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Complexity Classes are not Invented Theyre
Discovered

• NP (SAT, Clique, TSP,)
• P (Linear Programming, CVP, )
• NL (Connectivity, Shortest Paths, 2SAT, )
• L (Undirected Connectivity, Acyclicity, )
• NC1 (BFE, Regular Sets)
• TC0 (Sorting, Multiplication, Division)

Were interested in NC1 (for instance) not
because we want to build formulae for these
functions
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Complexity Classes are not Invented Theyre
Discovered

• NP (SAT, Clique, TSP,)
• P (Linear Programming, CVP, )
• NL (Connectivity, Shortest Paths, 2SAT, )
• L (Undirected Connectivity, Acyclicity, )
• NC1 (BFE, Regular Sets)
• TC0 (Sorting, Multiplication, Division)

but because we want to know if the blocks of
this partition are distinct.
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Complexity Classes are not Invented Theyre
Discovered

• NP (SAT, Clique, TSP,)
• P (Linear Programming, CVP, )
• NL (Connectivity, Shortest Paths, 2SAT, )
• L (Undirected Connectivity, Acyclicity, )
• NC1 (BFE, Regular Sets)
• TC0 (Sorting, Multiplication, Division)

These classes are real. Theyre important.
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How far are we in this talk?

• Weve explained why circuit lower bounds are
  important.
• even for restricted classes of circuits.
• What is currently known about these classes?

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Longstanding Open Problems

• Is P NP?
• Is AC06 NP?
• Is depth 3 AC06 NP?

Well focus on questions such as Is BFE in
TC0? Is BFE in AC06?
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How Close Are We to Proving Circuit Lower Bounds?

• Conventional Wisdom Not Close At All!
• No new superpolynomial size lower bounds in over
  two decades.
• Razborov and Rudich Any natural argument
  proving a lower bound against a circuit class C
  yields a proof that C cant compute a
  pseudorandom function generator.
• Since the Naor, Reingold generator is
  computable in TC0, this is bad news.

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More Modest Goals

• Problems requiring formulae of size n3 Håstad
• Problems requiring branching programs of size
  nearly n loglog n Beame, Saks, Sun, Vee
• Problems requiring depth d TC0 circuits of size
  n1c Impagliazzo, Paturi, Saks
• Time-Space Tradeoffs Fortnow, Lipton, Van
  Melkebeek, Viglas
• There is little feeling that these results bring
  us any closer to separating complexity classes.

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How Close Are We to Proving Circuit Lower Bounds?

• How close are the following two statements?
• TC0 Circuits for BFE must be of size n1O(1)
• For some cgt0, TC0 Circuits for BFE must be of
  size n1c.

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How Close Are We to Proving Circuit Lower Bounds?

• How close are the following two statements?
• TC0 Circuits for BFE must be of size n1O(1)
• For some cgt0, TC0 Circuits for BFE must be of
  size n1c

This is known IPS97
This implies TC0 ? NC1 A, Koucky
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Self-Reducibility

• A set B is said to be self-reducible if B P B

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Self-Reducibility

• A set B is said to be self-reducible if B P B
  via a reduction that, on input x, does not ask
  about whether x is in B.
• Very well-studied notion.
• For example, f is in SAT if and only if
  (f0 is in SAT) or (f1 is in SAT)

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Self-Reducibility

• Many of the important problems in (or near) NC1
  have a special self-reducibility property

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Self-Reducibility

• Many of the important problems in (or near) NC1
  have a special self-reducibility property
  Instances of length n are AC0-Turing (or
  TC0-Turing) reducible to instances of length n½
  via reductions of linear size.
• Examples
• BFE
• the word problem over S5
• MAJORITY
• Iterated Product of 3-by-3 Integer Matrices

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Self Reducibility

• BFE

A subformula near the root
Subformulae near inputs
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Self Reducibility

• S5

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Self Reducibility

• The self-reduction of S5, on inputs of size n,
  uses (n½ 1) oracle gates of size n½.
• Thus if S5 has TC0 circuits of size nk, it also
  has circuits of size (n½ 1)nk/2 O(n(k1)/2).
• Similar arguments hold for other classes (such as
  AC06 and NC1).
• More complicated self-reductions can be presented
  for MAJORITY and Iterated Product of 3-by-3
  matrices.

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A Corollary

• If BFE has TC0 or AC06 circuits, then it has
  such circuits of nearly linear size.
• If S5 has TC0 or AC06 circuits, then it has
  such circuits of nearly linear size.
• If MAJ has AC06 circuits, then it has such
  circuits of nearly linear size. (Etc.)
• Thus, e.g., to separate NC1 from TC0, it suffices
  to show that BFE requires TC0 circuits of size
  n1.0000001.

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A Corollary

• If BFE has TC0 or AC06 circuits, then it has
  such circuits of nearly linear size.
• If S5 has TC0 or AC06 circuits, then it has
  such circuits of nearly linear size.
• If MAJ has AC06 circuits, then it has such
  circuits of nearly linear size. (Etc.)
• How widespread is this phenomenon? Is it true
  for SAT? (I.e., can we show NP ? TC0 by proving
  that SAT requires TC0 circuits of size
  n1.0000001?)

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Limitations of Self-Reducibility

• Any problem for which instances of length n are
  TC0-Turing reducible to instances of length n½
  via poly-size reductions lies in NC.
• Thus there is no obvious way to apply these
  techniques to SAT or to problems complete for P.
• but perhaps, rather than showing directly that
  SAT has this strong form of self-reducibility,
  one can argue that if SAT is in TC0 then it has
  TC0 circuits of nearly-linear size.

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Limitations of Self-Reducibility

• Any problem for which instances of length n are
  TC0-Turing reducible to instances of length n½
  via poly-size reductions lies in NC.

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Limitations of Self-Reducibility

• Any problem for which instances of length n are
  TC0-Turing reducible to instances of length n½
  via poly-size reductions lies in NC.

d levels of oracle gates
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Limitations of Self-Reducibility

• Any problem for which instances of length n are
  TC0-Turing reducible to instances of length n½
  via poly-size reductions lies in NC.

d2 levels of oracle gates
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Limitations of Self-Reducibility

• Any problem for which instances of length n are
  TC0-Turing reducible to instances of length n½
  via poly-size reductions lies in NC.

After log log rounds, the depth is logO(1)n
d3 levels of oracle gates
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Prospects for Progress

• We have seen that existing techniques prove
  bounds that are nearly good enough to separate
  NC1 and TC0. Some of these proofs are natural.
• Dont the results of Razborov Rudich indicate
  that further progress will require very different
  approaches?
• Not necessarily!

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Prospects for Progress

• The Razborov Rudich framework of natural
  proofs assumes that a natural proof of a lower
  bound will make use of a combinatorial property
  that (among other things) is shared by a large
  fraction of the functions on n bits.
• In contrast, we are making use of a
  self-reducibility property that allows us to
  boost a n1e lower bound to a superpolynomial
  lower bound. This self-reducibility property
  holds for only a vanishingly small fraction of
  all functions.

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Prospects for Progress

• These observations are simple, but
• they have forever changed the way that we look at
  quadratic (and smaller) lower bounds.
• We are not claiming to have found a way around
  the obstacles identified by Razborov Rudich.
  (Such a claim will have to wait until someone
  proves that NC1 ? TC0.) But we do believe that
  this avenue deserves further exploration.

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Other Avenues for Progress

• Diagonalization Algebraic Tools
• The Mulmuley-Sohoni Approach
• Lower Bounds via Derandomization

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Diagonalization

• The archtype of a relativizable proof technique
  unable to prove P ? NP, or even NEXP not
  contained in P/poly.
• Non-relativizing proof techniques have been
  developed, using algebraic techniques that were
  useful in analyzing interactive and
  probabilistically checkable proof systems.
• These proof techniques algebrize Aaronson,
  Wigderson, and hence also cannot prove that NEXP
  is not contained in P/poly.

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Diagonalization Algebraic Techniques

• There is no evidence that these techniques are
  unable to prove that NEXP is not contained in
  TC0.
• but there is also no evidence that they can.
• Even simple results such as AC0 cant compute
  Mod 2 are not known to be provable using these
  techniques.

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Other Avenues for Progress

• Diagonalization Algebraic Tools
• The Mulmuley-Sohoni Approach
• Lower Bounds via Derandomization

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Lower Bounds via Derandomization

• Nisan, Wigderson showed that probabilistic AC0
  can be simulated in quasipolynomial time.
• Agrawal observes that, if one could improve
  quasipolynomial to polynomial, then there is
  a problem in E ( DTIME(2O(n))) that requires AC0
  circuits of size 2O(n).
• He then outlines a program, of how one might
  build on this result, to separate P from NP.

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Lower Bounds via Derandomization

• Nisan, Wigderson showed that probabilistic AC0
  can be simulated in quasipolynomial time.
• Agrawal observes that, if one could improve
  quasipolynomial to polynomial, then there is
  a problem in E ( DTIME(2O(n))) that requires AC0
  circuits of size 2O(n).
• How hard might it be to prove this first step?

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Lower Bounds via Derandomization

• Nisan, Wigderson showed that probabilistic AC0
  can be simulated in quasipolynomial time.
• Agrawal observes that, if one could improve
  quasipolynomial to polynomial, then there is
  a problem in E ( DTIME(2O(n))) that requires AC0
  circuits of size 2O(n).
• Neither the Natural Proofs framework, nor the
  notions of Relativization and Algebrization
  explain why this should be difficult.

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Conclusions

• Circuit lower bounds are necessary.
• Program run-time lower bounds do not yield bounds
  for fixed input sizes.
• We even need circuit lower bounds for small
  circuit classes.
• Seemingly-modest improvements to existing lower
  bounds would yield exciting separations of
  complexity classes.
• There may be cause for renewed optimism.