New Surprises from Self-Reducibility

Eric AllenderRutgers University

New Surprises from Self-Reducibility

New Surprises from Self-Reducibility

CiE 2010, Ponta Delgada, Azores

New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores < 2 >

Why a “Fantastic Voyage”?Why a “Fantastic Voyage”?

It’s apt. It’s a bad pun on “self-reduction”. It is contemporary with the birth of self-

reducibility.


40 Years of Self-Reducibility40 Years of Self-Reducibility

Boris A. Trakhtenbrot, On Autoreducibility, Dokl. Akad. Nauk. SSSR 11,

1970.


Self-ReducibilitySelf-Reducibility

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



A set B is said to be “self-reducible” if B ≤r 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).



A set B is said to be “self-reducible” if B ≤r B via a reduction that, on input x, does not ask about whether x is in B.

Very well-studied notion. In fact, this is such a simple notion, the really

surprising thing is that, for four decades, slight variations on this theme have yielded surprising and powerful insights.

We will not survey all 40 years of work on this topic! (See [Selke].)




Plan for TodayPlan for Today

Give a brief review of some (historical) settings where self-reducibility has been useful in complexity theory.

Present a few recent examples of work at the intersection of complexity theory and computability theory, where self-reducibility plays a central role.

But first, let’s recall some of the grand challenges in complexity theory that motivate these investigations.


What Crypto Needs from ComplexityWhat Crypto Needs from Complexity

Factoring (or some other suitable trap-door function) is hard for some fixed input size (corresponding to the size of a public key).

That is: we need to talk about hardness of finite functions.

Complexity theory can do this: 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)


Circuits vs Turing MachinesCircuits vs Turing Machines

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.





– 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).


Two Fundamental Questions:Two Fundamental Questions:

SAT є P SAT є P/poly

coNPNP = NPNP

[Karp-Lipton, 1980]


Two Fundamental Questions:Two Fundamental Questions:

SAT є P SAT є P/poly

coNPNP = NPNP

[Karp-Lipton, 1980]

Guess a circuit, and use the NP oracle to see if it

computes SAT.


Autoreducibility of Complete SetsAutoreducibility of Complete Sets

Here are a few longstanding open questions in complexity theory:

– EXP = NP

– EXP = PH (= NP U NPNP U NPNPNP …)

– PSPACE = NP

– PSPACE = PH (= NP U NPNP U NPNPNP …)

[Buhrman, Fortnow, van Melkebeek, Torenvliet] showed that resolving some innocent-sounding questions about auto-reducibility would solve these questions!


Autoreducibility of Complete Sets Autoreducibility of Complete Sets

[BFvMT]: All ≤P-Complete sets for EXP are autoreducible.

There is an oracle A, relative to which not all ≤P-Complete sets for EXP are autoreducible.

– Thus the proof of the preceding theorem does not “relativize”. (That’s a good thing!)

Not all ≤P-Complete sets for EEXPSPACE (doubly-exponential space) are autoreducible.

How about classes between EXP and EEXPSPACE? (E.g., EXPSPACE & EEXP.)



Are all ≤P-Complete sets for EEXP autoreducible?

– If YES, then PH ≠ EXP.

– If NO, then P ≠ PSPACE. Are all ≤P-Complete sets for EXPSPACE

autoreducible? Usually questions about “big” classes like

EXPSPACE and EEXP are not too hard to answer. Diagonalization techniques work there, that don’t work for “smaller” classes.





– If NO, then P ≠ PSPACE. Are all ≤P-Complete sets for EXPSPACE

autoreducible?

– If YES then PH ≠ PSPACE.





– If NO, then P ≠ PSPACE & NL ≠ NP. Are all ≤P-Complete sets for EXPSPACE

autoreducible?

– If YES then PH ≠ PSPACE.

– If NO, then NL ≠ NP.


Big Complexity ClassesBig Complexity Classes

NP P . . NC NL (Nondeterministic Logspace) 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

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°)

Amazingly, even with restricted reductions, the classes of complete sets for “big” complexity classes (EXP, NP, …) are essentially unchanged.

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.


Other Longstanding Open ProblemsOther Longstanding 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]




[Goldwasser et al]: Many of the important problems in (or near) NC1 have a special self-reducibility property:



[Goldwasser et al]: Many of the important problems in (or near) NC1 have a special self-reducibility property: Instances of length n are AC0-Turing reducible to instances of length n½ via reductions of linear size.

Examples:

– BFE

– the word problem over S5

– MAJORITY


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 other problems.


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.


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


Prospects for ProgressProspects 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.

Thus, it’s conceivable that a “natural” proof can be given of a modest lower bound of the form: BFE requires TC0 circuits of size n1.0000001. This would yield an “unnatural” proof separating NC1 from TC0.


Recall…Recall…

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., if SAT is in TC0, does it have TC0 circuits of size n1.0000001?)


Different Flavors of Self-ReducibilityDifferent Flavors of Self-Reducibility

If A is “word-decreasing self-reducible” (the self-reduction queries only lexicographically smaller strings) then A is in EXP. Some EXP-complete sets have this property.

If A is “downward self-reducible” (the self-reduction queries only shorter strings) then A is in PSPACE. Some PSPACE-complete sets have this property.

If A is “strongly downward self-reducible” (the self-reduction queries only very short strings) then A is in NC. Some NC1-complete sets have this property. (This is not tight!)


Different Flavors of Self-ReducibilityDifferent Flavors of Self-Reducibility

If A is “strongly downward self-reducible” (the self-reduction queries only very short strings) then A is in NC. Some NC1-complete sets have this property. (This is not tight!)

There are lots of classes between NC1 and NC (such as L and NL, among others).

Are there sets that are complete for L and NL that are strongly downward self-reducible? Would this imply something unlikely?


How Powerful is Randomness?How Powerful is Randomness?

Recall the basic definitions of Kolmogorov Complexity:

– C(x) = min {|d| : U(d) = x}.

– C(x) ≤ |x| + O(1).

– x is random if |x| ≤ C(x).

– RC is the set of Kolmogorov-random strings.

[ABKMR]: PSPACE is poly-time Turing reducible to RC.


How Useful is this Theorem?How Useful is this Theorem?

PSPACE is poly-time Turing reducible to RC.

Is it trivial? After all, RC isn’t even computable!

Note that RC is not hard for NP under poly-time many-one reductions, unless P=NP. (This follows, since RC has no infinite enumerable subset.)

No simple direct reduction from PSPACE to RC is known; the known proofs rely on techniques from derandomization, interactive proof systems, and …




Is it trivial? After all, RC isn’t even computable!

Note that RC is not hard for NP under poly-time many-one reductions, unless P=NP. (This follows, since RC has no infinite enumerable subset.)

No simple direct reduction from PSPACE to RC is known; the known proofs rely on techniques from derandomization, interactive proof systems, and … self-reducibility.




Is this inclusion optimal in some sense? Is there some larger complexity class that is reducible to RC?

An intriguing possibility: can PSPACE be characterized in some sense, in terms of efficient reductions to RC?

…or is the Halting Problem poly-time reducible to RC?


A Strange Characterization of PA Strange Characterization of P

Here is an illustration of what such a characterization might look like. Instead of poly-time truth-table reductions, consider poly-time dtt reductions. (I.e., in poly-time, output a list of queries, and accept if at least one of them is in RC.)

Fact: For every computable time bound t, there is a decidable set D that is not in Dtime(t) that is poly-time dtt-reducible to RC.

This would seem to kill any possibility of characterizing complexity classes.



Here is an illustration of what such a characterization might look like. Instead of poly-time truth-table reductions, consider poly-time dtt reductions. (I.e., in poly-time, output a list of queries, and accept if at least one of them is in RC.)

Fact: For every computable time bound t, there is a decidable set D that is not in Dtime(t) that is poly-time dtt-reducible to RC.

…but the set D crucially depends on the universal Turing machine that defines C(x)!



[A,Buhrman,Koucky]: P consists precisely of the decidable sets that are poly-time dtt-reducible to RC no matter which universal Turing machine is used in the definition of the Kolmogorov complexity function C(x).

It would be very interesting if a similar characterization of PSPACE could be obtained.

Conjecture: There is a decidable set that is not poly-time reducible to RC. (Self-reducibility may be necessary, to make use of RC.)


Closing RemarksClosing Remarks Self-Reducibility is a simple idea that has been

surprisingly useful over a span of four decades.

Self-Reducibility points to promising avenues to separate complexity classes.

– Autoreducibility of EEXP-complete sets.

– Non-natural proofs in circuit complexity. …and it may help us to forge a new

connection between complexity theory and computability, by clarifying the power of efficient reducibility to RC.


Closing RemarksClosing Remarks

Obrigado!

New Surprises from Self-Reducibility

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