eric allender rutgers university cracks in the defenses: scouting out approaches on circuit lower...

Eric AllenderRutgers University

Cracks in the Defenses: Scouting Out Approaches on

Circuit Lower Bounds

Cracks in the Defenses: Scouting Out Approaches on

Circuit Lower Bounds

CSR 2008

Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 2 >

IntroductionIntroduction

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, we’ll make some observations that may cause some of you to be less pessimistic.


But First…Why Circuits?But First…Why 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 can’t.


An example: the Game of CheckersAn 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.





– Thus any program solving this problem must run very slowly on large inputs. This is the essence of asymptotic analysis.





– This is a much stronger statement about complexity than we are able to prove for most problems (such as NP-complete problems).





– but…Conceivably, there is a hand-held device that computes optimal moves, even for Checker boards of size 1000-by-1000!

– …because we don’t know if EXP is in P/poly (the class of problems with small circuits).


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.




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.


Big Complexity ClassesBig Complexity Classes

NP P . . NC L (Deterministic Logspace)


TC0 O(1)-Depth Circuits of MAJ gates

AC0 [6] NC1 Log-Depth Circuits

AC0 can’t compute Mod 2 [FSS,A]

AC0 O(1)-Depth Circuits of AND/OR gates

The Main Objects of Interest:Small Complexity Classes




AC0 [6] NC1 Log-Depth Circuits

AC0 can’t compute Mod 2 [FSS,A]

AC0 O(1)-Depth Circuits of AND/OR gates





NC1 Log-Depth Circuits

AC0 [2] can’t compute Mod 3 [R,S]

AC0 [2] AC0 O(1)-Depth Circuits of AND/OR gates




NC1 Log-Depth Circuits


AC0 [6] AC0 [2] AC0 O(1)-Depth Circuits of AND/OR gates




NC1 poly-size formulae


AC0 [6] AC0 [2] AC0 O(1)-Depth Circuits of AND/OR gates




NP has complete sets (under polynomial time reducibility ≤P)

These small classes have complete sets, too (under ≤AC°)

Complete ProblemsComplete Problems


ReductionsReductions

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


NC1

TC0

AC0 [6] AC0 [2] AC0


sorting, multiplication, division

[Naor,Reingold] Pseudorandom Generator


NC1

TC0

AC0 [6] AC0 [2] AC0


BFE: Balanced Boolean Formula Evaluation (AND,OR,XOR)

Word problem over S5


The Word Problem Over S5The Word Problem Over S5

A regular set complete for NC1

=


Complexity Classes are not Invented – They’re 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)

We’re interested in NC1 (for instance) not because we want to build formulae for

these functions…






… but because we want to know if the blocks of this partition are distinct.






These classes are real.

They’re important.


How far are we in this talk?How far are we in this talk?

We’ve explained why circuit lower bounds are important.

…even for restricted classes of circuits. What is currently known about these classes?


Longstanding Open ProblemsLongstanding Open Problems

Is P = NP? Is AC0[6] = NP? Is depth 3 AC0[6] = NP?

We’ll focus on questions such as:

Is BFE in TC0?

Is BFE in AC0[6]?


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 can’t compute a pseudorandom function generator.

Since the [Naor, Reingold] generator is computable in TC0, this is bad news.


More Modest GoalsMore 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 n1+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.


How close are the following two statements?

TC0 Circuits for BFE must be of size n1+Ω(1)

For some c>0, TC0 Circuits for BFE must be of size n1+c.




How close are the following two statements?

TC0 Circuits for BFE must be of size n1+Ω(1)

For some c>0, TC0 Circuits for BFE must be of size n1+c



This is known [IPS’97]

This implies TC0 ≠ NC1 [A, Koucky]


Self-ReducibilitySelf-Reducibility

A set B is said to be “self-reducible” if B ≤P B



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, φ is in SAT if and only if

(φ0 is in SAT) or (φ1 is in SAT)



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



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


Self ReducibilitySelf Reducibility

BFE

A subformula near the root

Subformulae near inputs



S5



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(k+1)/2).

Similar arguments hold for other classes (such as AC0[6] and NC1).

More complicated self-reductions can be presented for MAJORITY and Iterated Product of 3-by-3 matrices.


A CorollaryA Corollary

If BFE has TC0 or AC0[6] circuits, then it has such circuits of nearly linear size.

If S5 has TC0 or AC0[6] circuits, then it has such circuits of nearly linear size.

If MAJ has AC0[6] 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.


A CorollaryA Corollary

If BFE has TC0 or AC0[6] circuits, then it has such circuits of nearly linear size.

If S5 has TC0 or AC0[6] circuits, then it has such circuits of nearly linear size.

If MAJ has AC0[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 ≠ TC0 by proving that SAT requires TC0 circuits of size n1.0000001?)


Limitations of Self-ReducibilityLimitations 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.




d levels of oracle gates




d2 levels of oracle gates




d3 levels of oracle gates

After log log rounds,

the depth is logO(1)n


Prospects for ProgressProspects 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”.

Don’t the results of [Razborov & Rudich] indicate that further progress will require very different approaches?

Not necessarily!



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 n1+ε lower bound to a superpolynomial lower bound. This self-reducibility property holds for only a vanishingly small fraction of all functions.



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.


Other Avenues for ProgressOther Avenues for Progress

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


DiagonalizationDiagonalization

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.


Diagonalization + Algebraic Techniques

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 can’t

compute Mod 2” are not known to be provable using these techniques.


Other Avenues for ProgressOther Avenues for Progress

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


Lower Bounds via DerandomizationLower 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 2Ω(n).

He then outlines a program, of how one might build on this result, to separate P from NP.





How hard might it be to prove this first step?





Neither the Natural Proofs framework, nor the notions of Relativization and Algebrization explain why this should be difficult.


ConclusionsConclusions

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.

eric allender rutgers university cracks in the defenses: scouting out approaches on circuit lower...

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