Eric Allender Rutgers University Chipping Away at P vs NP: How Far Are We from Proving Circuit Size...
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Eric Allender Rutgers University Chipping Away at P vs NP: How Far Are We from Proving Circuit Size Lower Bounds? Joint work with Michal Kouckyʹ Czech Academy of Sciences
Eric Allender Rutgers University Chipping Away at P vs NP: How Far Are We from Proving Circuit Size Lower Bounds? Joint work with Michal Koucky ʹ Czech
Eric Allender Rutgers University Chipping Away at P vs NP: How
Far Are We from Proving Circuit Size Lower Bounds? Joint work with
Michal Koucky Czech Academy of Sciences
Slide 2
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 2 >< 2 > 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.
Slide 3
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 3 >< 3 > 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.
Slide 4
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 4 >< 4 > 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 c 2 n d
steps, for some constants c and d. n -by- n Checkers is complete
for EXP.
Slide 5
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 5 >< 5 > 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 c 2 n 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.
Slide 6
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 6 >< 6 > 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 c 2 n 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).
Slide 7
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 7 >< 7 > 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 c 2 n 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).
Slide 8
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 8 >< 8 > 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 10 123 gates. (Stockmeyer, 1974) (Proof sketch): There is
a problem requiring exponential circuit size that is efficiently
reducible to WS1S.
Slide 9
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? < 9 >< 9 > 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 10 123 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.
Slide 10
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Big Complexity Classes NP PP .. .. NC L (Deterministic
Logspace)
Slide 11
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? TC 0 O(1)-Depth Circuits of MAJ gates AC 0 [6] NC 1
Log-Depth Circuits AC 0 cant compute Mod 2 [FSS,A] AC 0 O(1)-Depth
Circuits of AND/OR gates The Main Objects of Interest: Small
Complexity Classes
Slide 12
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? TC 0 O(1)-Depth Circuits of MAJ gates AC 0 [6] NC 1
Log-Depth Circuits AC 0 cant compute Mod 2 [FSS,A] AC 0 O(1)-Depth
Circuits of AND/OR gates The Main Objects of Interest: Small
Complexity Classes
Slide 13
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? TC 0 O(1)-Depth Circuits of MAJ gates NC 1 Log-Depth
Circuits AC 0 [2] cant compute Mod 3 [R,S] AC 0 [2] AC 0 O(1)-Depth
Circuits of AND/OR gates The Main Objects of Interest: Small
Complexity Classes
Slide 14
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? NC 1 Log-Depth Circuits TC 0 O(1)-Depth Circuits of MAJ
gates AC 0 [6] AC 0 [2] AC 0 O(1)-Depth Circuits of AND/OR gates
The Main Objects of Interest: Small Complexity Classes
Slide 15
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? NC 1 poly-size formulae TC 0 O(1)-Depth Circuits of MAJ
gates AC 0 [6] AC 0 [2] AC 0 O(1)-Depth Circuits of AND/OR gates
The Main Objects of Interest: Small Complexity Classes
Slide 16
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? NP has complete sets (under polynomial time reducibility P
) These small classes have complete sets, too (under AC ) Complete
Problems
Slide 17
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? 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
Slide 18
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? NC 1 TC 0 AC 0 [6] AC 0 [2] AC 0 Complete Problems sorting,
multiplication, division [Naor,Reingold] Pseudorandom
Generator
Slide 19
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? NC 1 TC 0 AC 0 [6] AC 0 [2] AC 0 Complete Problems BFE:
Balanced Boolean Formula Evaluation (AND,OR,XOR) Word problem over
S 5
Slide 20
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? The Word Problem Over S 5 A regular set complete for NC 1
=
Slide 21
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Complexity Classes are not Invented Theyre Discovered NP
(SAT, Clique, TSP,) P (Linear Programming, CVP, ) NL (Connectivity,
Shortest Paths, 2SAT, ) L (Undirected Connectivity, Acyclicity, )
NC 1 (BFE, Regular Sets) TC 0 (Sorting, Multiplication, Division)
Were interested in NC 1 (for instance) not because we want to build
formulae for these functions
Slide 22
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Complexity Classes are not Invented Theyre Discovered NP
(SAT, Clique, TSP,) P (Linear Programming, CVP, ) NL (Connectivity,
Shortest Paths, 2SAT, ) L (Undirected Connectivity, Acyclicity, )
NC 1 (BFE, Regular Sets) TC 0 (Sorting, Multiplication, Division)
but because we want to know if the blocks of this partition are
distinct.
Slide 23
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Complexity Classes are not Invented Theyre Discovered NP
(SAT, Clique, TSP,) P (Linear Programming, CVP, ) NL (Connectivity,
Shortest Paths, 2SAT, ) L (Undirected Connectivity, Acyclicity, )
NC 1 (BFE, Regular Sets) TC 0 (Sorting, Multiplication, Division)
These classes are real. Theyre important.
Slide 24
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? 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?
Slide 25
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Longstanding Open Problems Is P = NP? Is AC 0 [6] = NP? Is
depth 3 AC 0 [6] = NP? Well focus on questions such as : Is BFE in
TC 0 ? Is BFE in AC 0 [6]?
Slide 26
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? 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 TC 0, this is bad
news.
Slide 27
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? More Modest Goals Problems requiring formulae of size n 3
[Hstad] Problems requiring branching programs of size nearly n
loglog n [Beame, Saks, Sun, Vee] Problems requiring depth d TC 0
circuits of size n 1+ c [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.
Slide 28
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? How close are the following two statements? TC 0 Circuits
for BFE must be of size n 1+(1) TC 0 Circuits for BFE must be of
size n 1+ c for some c >0 How Close Are We to Proving Circuit
Lower Bounds?
Slide 29
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? How close are the following two statements? TC 0 Circuits
for BFE must be of size n 1+(1) TC 0 Circuits for BFE must be of
size n 1+ c for some c >0 How Close Are We to Proving Circuit
Lower Bounds? This is known [IPS97] This implies TC 0 NC 1
Slide 30
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Self-Reducibility A set B is said to be self-reducible if B
P B
Slide 31
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? 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, is in SAT if and
only if ( 0 is in SAT) or ( 1 is in SAT)
Slide 32
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Self-Reducibility Many of the important problems in (or
near) NC 1 have a special self-reducibility property:
Slide 33
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Self-Reducibility Many of the important problems in (or
near) NC 1 have a special self-reducibility property: Instances of
length n are AC 0 -Turing (or TC 0 - Turing) reducible to instances
of length n via reductions of linear size. Examples: BFE the word
problem over S 5 MAJORITY Iterated Product of 3-by-3 Integer
Matrices
Slide 34
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Self Reducibility BFE A subformula near the root
Subformulae near inputs
Slide 35
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Self Reducibility S5S5
Slide 36
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Self Reducibility The self-reduction of S 5, on inputs of
size n, uses ( n + 1) oracle gates of size n . Thus if S 5 has TC 0
circuits of size n k, it also has circuits of size ( n + 1) n k/ 2
= O(n (k+ 1)/2 ). Similar arguments hold for other classes (such as
AC 0 [6] and NC 1 ). More complicated self-reductions can be
presented for MAJORITY and Iterated Product of 3-by-3
matrices.
Slide 37
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? A Corollary If BFE has TC 0 or AC 0 [6] circuits, then it
has such circuits of nearly linear size. If S 5 has TC 0 or AC 0
[6] circuits, then it has such circuits of nearly linear size. If
MAJ has AC 0 [6] circuits, then it has such circuits of nearly
linear size. (Etc.) Thus, e.g., to separate NC 1 from TC 0, it
suffices to show that BFE requires TC 0 circuits of size n
1.0000001.
Slide 38
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? A Corollary If BFE has TC 0 or AC 0 [6] circuits, then it
has such circuits of nearly linear size. If S 5 has TC 0 or AC 0
[6] circuits, then it has such circuits of nearly linear size. If
MAJ has AC 0 [6] 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 TC 0 by proving that SAT requires TC
0 circuits of size n 1.0000001 ?)
Slide 39
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Limitations of Self-Reducibility Any problem for which
instances of length n are TC 0 -Turing reducible to instances of
length n 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 TC 0 then it has TC 0 circuits of nearly-linear
size.
Slide 40
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Limitations of Self-Reducibility Any problem for which
instances of length n are TC 0 -Turing reducible to instances of
length n via poly-size reductions lies in NC.
Slide 41
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Limitations of Self-Reducibility Any problem for which
instances of length n are TC 0 -Turing reducible to instances of
length n via poly-size reductions lies in NC. d levels of oracle
gates
Slide 42
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Limitations of Self-Reducibility Any problem for which
instances of length n are TC 0 -Turing reducible to instances of
length n via poly-size reductions lies in NC. d 2 levels of oracle
gates
Slide 43
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Limitations of Self-Reducibility Any problem for which
instances of length n are TC 0 -Turing reducible to instances of
length n via poly-size reductions lies in NC. d 3 levels of oracle
gates After log log rounds, the depth is log O(1) n
Slide 44
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Some Lower Bounds Recall that [IPS] showed: TC 0 Circuits
for BFE must be of size n 1+(1) Thus SAT also requires TC 0
circuits of this size. The [IPS] bound actually shows that PARITY
requires circuits of this size. We do NOT know similar bounds for
AC 0 [6].
Slide 45
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? A Lower Bound for AC 0 [6] For every d there is an >0
such that SAT requires depth d AC 0 [6] circuits of size n 1+ The
same proof shows that the same bound holds for TC 0.
Slide 46
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? A Lower Bound for AC 0 [6] For every d there is an >0
such that SAT requires depth d AC 0 [6] circuits of size n 1+ If no
such exists, then for all >0, SAT is in TimeSpace( n 1+ ,n 1- ).
This violates the time-space tradeoff results of [Van
Melkebeek].
Slide 47
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? A Lower Bound for AC 0 [6] For every d there is an >0
such that SAT requires depth d AC 0 [6] circuits of size n 1+ If no
such exists, then for all >0, SAT is in TimeSpace( n+n d,n 1- ).
Hint of proof: Do a depth-first evaluation of the circuit, using
the space bound to store the value of all gates having large fan-in
(and re- computing all other values as needed).
Slide 48
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? Prospects for Progress We have seen that existing
techniques prove bounds that are nearly good enough to separate NC
1 and TC 0. Some of these proofs are natural. Dont the results of
[Razborov & Rudich] indicate that further progress will require
very different approaches? Not necessarily!
Slide 49
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? 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 n 1+ lower bound to a superpolynomial
lower bound. This self-reducibility property holds for only a
vanishingly small fraction of all functions.
Slide 50
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? 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 NC 1 TC 0.) But
we do believe that this avenue deserves further exploration.
Slide 51
Eric Allender: How Close Are We to Proving Circuit Lower
Bounds? 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.