eric allender rutgers university 27 and still counting: iterated product, inductive counting, and...

Eric AllenderRutgers University

27 and Still Counting: Iterated Product, Inductive

Counting, and the Structure of P

27 and Still Counting: Iterated Product, Inductive

Counting, and the Structure of P

ImmermanFest, Vienna, July 13, 2014

Eric Allender: 27 and Still Counting < 2 >

Why 27??Why 27??

33 years ago this month, the

earth shifted:


Why 27??Why 27??

Let us recall the landscape prior to

July, 1987.


In the Beginning (1956) …In the Beginning (1956) …

…there was the Chomsky Hierarchy.

c.e.

CSL

CFL

Regular

co-c.e.

co-CFL

co-CSL??


In the Beginning (1956) …In the Beginning (1956) …

…there was the Chomsky Hierarchy.

Σ01

CSL

CFL

Regular

Π01

co-CFL

co-CSL??


In the very Beginning (1943) …In the very Beginning (1943) …

…there was the Arithmetic Hierarchy.

Σ03

Σ02

Σ01

Π03

Π01

Π02


…and it was good!…and it was good!

Alternative characterizations in terms of

– Logic (Alternating quantifiers and recursive predicates)

– Alternating Turing machines.

– Oracle Turing machines.

– FO(Halting Problem) [not really]

– AC0-Turing reductions to the Halting Problem [not really]


AC0 ReductionsAC0 Reductions

B

B B

A ≤AC° B means that there is a constant-depth circuit computing A that has the usual AND, OR, and NOT gates, and also has ‘oracle gates’ for B.


The Arithmetic Hierarchy begatThe Arithmetic Hierarchy begat

…the Polynomial Hierarchy.

Σp3

Σp2

Σp1=NP

Πp3

coNP=Πp1

Πp2


…which was also pretty good!…which was also pretty good!


– Logic (Alternating quantifiers and recursive predicates)



– FO(SAT) [not really]

– AC0-Turing reductions to SAT [not really]

– Some fairly natural complete problems at levels 2 and 3.


The Polynomial Hierarchy begatThe Polynomial Hierarchy begat

…the NL Alternation Hierarchy.

Σlog3

Σlog2

Σlog1=NL

Πlog3

coNL=Πlog1

Πlog2


…and it was not so great.…and it was not so great.


– Logic [if you played with the definitions]



– FO(GAP)

– AC0-Turing reductions to GAP

– Some fairly natural complete problems at levels 2 and 3. [Rosier]


So what was the problem?So what was the problem?

You’d like NLNL to be a subclass of P. Unfortunately, it’s NP! So Ruzzo, Simon, and Tompa introduced

“RST” relativization. (The oracle machine must work deterministically while writing a query.)

May seem artificial – but it corresponds to AC0- and FO-Turing reducibility.

So of course, this gives us another hierarchy:


The NL Oracle HierarchyThe NL Oracle Hierarchy

Where is the Alternation Hierarchy?

NLNLNL

NLNL

NL

coNLNLNL

coNL

coNLNL

ALH



A lovely structure?

NLNLNL

NLNL

NL

coNLNLNL

coNL

coNLNL

ALH



A lovely structure? Or a fine mess?

NLNLNL

NLNL

NL

coNLNLNL

coNL

coNLNL

ALH


And the walls came a tumblin’ down

And the walls came a tumblin’ down

And here’s where you expect me to mention NL=coNL…

…but this collapse happened in 1986! In two phases:

– The NL Alternation Hierarchy = LNL [Lange, Jenner, Kirsig]

– The NL Oracle Hierarchy = LNL [Schöning, Wagner][Buss, Cook, Dymond, Hay]


The NL “Hierarchy”The NL “Hierarchy”

But true enlightenment had not yet arrived. Within the year, the world would know that

NL=coNL.

NLcoNL

LNL


Impact of Inductive CountingImpact of Inductive Counting

The discovery that NL=coNL provided the single most significant insight into the nature of space-bounded computation since the 60’s.

The list of complexity classes that have been impacted by these new insights into nondeterminism includes

– LogCFL, VP, VP(2), DET, PL, #L, UL, ModkL, SAC1(log), RUL, CNL, …

– Some of these are not so important…but some assuredly are!


The NC HierarchyThe NC Hierarchy

AC0

TC0

NC1

AC1

TC1

NC2

L

NL


The NC HierarchyThe NC Hierarchy

AC0

TC0

NC1

AC1

TC1

NC2

L

NL These have

natural complete

sets.

These …not so much.

But there are other important problems

in the vicinity.

Determinant

CFLs


Linear Algebra and LogspaceLinear Algebra and Logspace

The connection between linear algebra and logspace-bounded computation was discovered rather late, and via excessively difficult arguments – primarily because inductive counting was discovered so late.

The relevant logspace classes were initially studied without any motivation from natural problems.

What are these classes?

– PL, #L, GapL, C=L


Probabilistic LogspaceProbabilistic Logspace

PL was introduced by [Gill, 1977], by analogy with PP (defined in the same paper).

The history of upper bounds on the complexity of PL:

– PSPACE [Gill, 1977]

– SPACE(log6n) [Simon, 1981]

– NC2 [Borodin Cook Pippenger, 1982]

– L#L [Jung, 1985] What was the problem??


The problem with PLThe problem with PL

In a nutshell, the problem is that PL machines can continue to do useful work after exponential time.

For example: NL = RL!


The problem with PLThe problem with PL

In a nutshell, the problem is that PL machines can continue to do useful work after exponential time.

For example: NL = RL! (If “RL” is defined without a polynomial time bound.)

But with Inductive Counting as a tool, it’s easy to see that PL is the same class, with or without a polynomial-time restriction. (Jung did this the hard way, in 1985.)

Thus PL is characterized by NL machines with more accepting than rejecting paths.


Linear Algebra and #LLinear Algebra and #L

The connection between #P and the Permanent was made in 1979.

#L was explicitly defined and studied in 1990. The fact that Determinant is complete for

GapL (= #L - #L) was not discovered until 1991-1992.

An immediate consequence was: {M : Det(M) > 0} is complete for PL.

…but we much more often ask: Is Det(M)=0?


Singular matricesSingular matrices

The set of singular matrices is complete for C=L (characterized by NL machines where the number of accepting and rejecting computations are equal.)

Hierarchies:

AC0(C=L) = C=L U C=LC=L U …

AC0(PL) AC0(#L)




Hierarchies:

AC0(C=L) = C=L U C=LC=L U …

AC0(PL) = PL Collapse! [Beigel, Fu, 1997] AC0(#L)




Hierarchies:

AC0(C=L) = LC=L Collapse! [A. Beals, Ogihara]

AC0(PL) = PL Collapse! [Beigel, Fu, 1997] AC0(#L)




Hierarchies:

AC0(C=L) = LC=L Collapse! [A. Beals, Ogihara]

AC0(PL) = PL Collapse! [Beigel, Fu, 1997] AC0(#L) = ???? (No collapse known.)


The C=L HierarchyThe C=L Hierarchy

C=LcoC=L

LC=L


The C=L HierarchyThe C=L Hierarchy

Singular MatricesNonsingular Matrices

Rank

So this is strong evidence that

these three classes are distinct.


Another view of #LAnother view of #L A complete problem for #L is: Counting the

number of paths from s to t in a directed graph.

Equivalently: it’s the problem of computing the (1,1) entry of a product of several nxn matrices with entries in the Natural Numbers.

Similarly, iterated product of integer matrices is complete for GapL (i.e., the determinant class).

Immerman and Landau highlighted the connection between complexity classes and iterated product over several algebras.


Immerman & LandauImmerman & Landau

Curiously, missing from this list

is the most “algebraic” class: VP

Complexity Class Algebra for which Iterated Product is complete

TC0 Integers

NC1 5x5 Boolean matrices

NC1 Permutations on 5 elements (S5)

#NC1 6x6 Integer matrices

L Permutations on n elements (Sn)

NL nxn Boolean matrices

GapL nxn integer matrices


VP and Polynomial DegreeVP and Polynomial Degree

Valiant introduced the study of the arithmetic complexity of n-variate polynomials with algebraic degree bounded by poly(n). Later, this came to be called VP (or VPR for algebra R).


VP and Polynomial DegreeVP and Polynomial Degree

VP has a nice circuit characterization: log depth poly-size circuits with unbounded fan-in + gates, fan-in 2 * gates. (Semiunbounded fan-in circuits). [VSBR ‘83, Vinay ‘91]

Over the Boolean ring, this was discovered earlier [Ruzzo ‘79, Venkateswaran ‘87].

VP({0,1},V,Λ) is also known as LogCFL and SAC1.

– LogCFL = problems logspace-reducible to CFLs.

– SAC1 = Semi-unbounded fan-in circuits of depth log1 n.


SemiUnbounded Fan-InSemiUnbounded Fan-In

Immediately after NL=coNL was established, SAC1 was shown to be closed under complement, too (using inductive counting). [Borodin, Cook, Dymond, Ruzzo, Tompa]

Thus the following two models are equivalent:

– Unbounded fan-in V, bounded fan-in Λ

– Bounded fan-in V, unbounded fan-in Λ How about other algebras? A similar result over, say GF2 would be

remarkable: VP(2) would equal AC1!


VP and Iterated “Product”VP and Iterated “Product”

There aren’t that many “natural” complete problems for VP. (Using the connection to LogCFL, one class of complete problems is counting # of parse trees showing that x is in L, for certain CFLs L.)

Recently, a complete problem was added to this list, that looks a lot like an “iterated product”:

– Tensor Contraction [Capelli, Durand, Mengel]


VP and Iterated “Product”VP and Iterated “Product”

Matrices with b rows and c columns are 2-dimensional tensors of order [b,c].

Given 2 tensors A and B (of orders, say [a,b,c] and [c,d,e]), their contraction has order [a,b,d,e]

Given a list of 1-, 2-, and 3-dimensional tensors (or even poly-dimensional tensors), computing an entry of their iterated contraction is complete for VP.

But this is non-associative. (The nesting must be given, say, as a tree.)


Undirected ReachabilityUndirected Reachability

The same paper that showed LogCFL = coLogCFL also applied inductive counting to SL (the problems reducible to reachability in undirected graphs).

SL had its own hierarchy, inside the NL hierarchy.

SL was known to be in RL (with a poly run-time), and the new insight was a coRL algorithm. Many developments followed.

Reingold ended this saga, showing SL=L.


Where do these classes fit?Where do these classes fit?

NL

Mod2L

SAC1

Mod3L Mod5L

VP(2) VP(3) VP(5)

AC1

VP

#L


Gal and WigdersonGal and Wigderson

NL

Mod2L

SAC1

Mod3L Mod5L

VP(2) VP(3) VP(5)

AC1

VP

#L

Lots of nonuniform inclusions:


NL in #L is OpenNL in #L is Open

NL

Mod2L

SAC1

Mod3L Mod5L

VP(2) VP(3) VP(5)

AC1

VP

#L

But UL is in #L (by definition).

UL


[GW] + Inductive Counting[GW] + Inductive Counting

UL/poly = NL/poly

Mod2L

SAC1

Mod3L Mod5L

VP(2) VP(3) VP(5)

AC1

VP

#L

More nonuniform inculsions [Reihnardt, A]


[GW] + Inductive Counting[GW] + Inductive Counting

UL/poly = NL/poly

Mod2L

SAC1

Mod3L Mod5L

VP(2) VP(3) VP(5)

AC1

VP

#L

These hold uniformly if SAT needs size 2n/100.


A sampling of open questionsA sampling of open questions

Inductive counting (and related techniques) have thus far failed to show:

– UL = coUL

– C=L = coC=L

– NL = UL

– A collapse of the #L hierarchy

– Any relationship between AC1 and #L. (Immerman and Landau conjecture that TC1 reduces to #L.)


AC1 and VPAC1 and VP

TC1 = #AC1(mod pn) [Reif, Tate]

– Arithmetic degree nlog n.

AC1 is contained in #AC1(mod pn) where all multiplication gates have fan-in log n. [A, Jiao, Mahajan, Vinay]

– Arithmetic degree nloglog n. Thus if the degree could be lowered to poly(n),

working over Q instead of [Z mod pn], one would have AC1 contained in VP.

If also GapL = VP, we’d have AC1 in GapL.


A Continuing LegacyA Continuing Legacy

Inductive Counting continues to shape the research agenda.

Case in point: Catalytic Computing (STOC ‘14 [Buhrman, Cleve, Koucky, Loff, Speelman])

CL = problems solvable using logarithmic space augmented with a “full memory” that must be restored to its original state.

TC1 is contained in CL, which is contained in ZPP.

What about CNL?


CNLCNL

A new “nondeterministic” class. An unfamiliar model. How can one program in this model?

There is currently exactly one nondeterministic algorithm known in this model. It is used in order to show…

– CNL = coCNL.


LegacyLegacy

A lessening of confidence in the framework of complexity classes, and an increase in humility regarding popular conjectures.

An invaluable insight into the nature of nondeterminism, and space-bounded computation in general.

A fundamental shift in the way that we approach questions in complexity theory.


Thank you!Thank you!

…and Thank You, Neil! Congratulations on your 60th!

eric allender rutgers university 27 and still counting: iterated product, inductive counting, and...

Documents

nl log

nl oracle hierarchy

nl alternation hierarchy

p2p2 slide

inductive counting

rosier slide

oracle turing machines

turing reductions