Perspective on Lower Bounds: Diagonalization

Lance Fortnow

NEC Research Institute

A TheoremPermanent is not in uniform TC0.

Papers: Allender ’96. Caussinus-McKenzie-Thérien-Vollmer ’96. Allender-Gore ’94.

Counting HierarchyPP – Class of languages accepted by

probabilistic machines with unbounded error.

Counting Hierarchy

PPPPPPP

Counting Hierarchy in TC0

If Permanent is in uniform TC0 then Permanent is in P and PP in P.

Counting Hierarchy collapses to P.Permanent is AC0-hard for P.All of P and thus the entire counting

hierarchy collapses to uniform TC0.

Threshold MachinesAlternating machines that ask “Do a

majority of my paths accept?”Polynomial-time unbounded thresholds

is equivalent to PSPACE.Polynomial-time constant thresholds is

the counting hierarchy.Log-time constant thresholds is uniform-

TC0.

Almost doneFor any k, there exists a language L

accepted by a polynomial-time k-threshold machine that is not accepted by any log-time k-threshold machine.

Not yet done: Could be that L is accepted by a log time

r-threshold machine for some r > k.

Finishing UpSAT is accepted by log-time k-threshold

machine.All of NP is accepted by some log-time

k-threshold machine.All of the counting hierarchy is accepted

by some log-time k-threshold machine.Contradiction!

DiagonalizationWant to prove separation.Assume collapse.Get other collapses.Keep collapsing until we have collapsed

two classes that can be separated by diagonalization.

Diagonalization - PositivesDiagonalization works!Diagonalization is not “natural” or at

least it avoids the Razborov-Rudich natural proof issues.

Proofs are simple—sometimes require clever ideas but rarely hard combinatorics.

Diagonalization - NegativesOnly weak separations so far.Relativization

Probably will not settle P = NP. Can only get nonrelativizing separations by

using nonrelativizing collapses.Hard to diagonalize against nonuniform

models of computation.

DiagonalizationCantor (1874) – There is no one-to-one

function from the power set of the integers to the integers.

Proof: Suppose there was. Then we could enumerate the power set of the integers: S1, S2, S3, …

Proof of Cantor’s Theorem

1 2 3 4 5 6

S1 In Out In Out In In

S2 Out In Out Out In Out

S3 Out Out Out Out Out Out

S4 In Out In Out In Out

S5 In In In In In In

S6 Out In Out Out Out In

Proof of Cantor’s Theorem

1 2 3 4 5 6

S1 In Out In Out In In

S2 Out In Out Out In Out

S3 Out Out Out Out Out Out

S4 In Out In Out In Out

S5 In In In In In In

S6 Out In Out Out Out In

Proof of Cantor’s Theorem

1 2 3 4 5 6

S1 Out Out In Out In In

S2 Out Out Out Out In Out

S3 Out Out In Out Out Out

S4 In Out In In In Out

S5 In In In In Out In

S6 Out In Out Out Out Out

A Brief History600 BC - Epimenides Paradox.

All cretans are liars…One of their own poets has said so.

400 BC - Eubulides Paradox. This statement is false.

1200 AD – Medieval Study of Insolubia. I am a liar.

A Brief History1874 – Cantor.

The set of reals is not countable.1901 - Russell’s Paradox.

The set of all sets that does not contain itself as a member.

1931 - Gödel’s Incompleteness. This statement does not have a proof.

A Brief History1936 – Turing.

The halting problem is undecidable.1956 – Friedberg-Muchnik.

There exist incomplete Turing degrees.1965 – Hartmanis-Stearns.

More time gives more languages.

Time and Space HierarchiesNondeterministic Space Hierarchy.

Ibarra (1972), IS (1988). First to use multiple collapses.

Nondeterministic Time Hierarchy. Cook (1973), SFM (1978), Žàk (1983). Unbounded collapses necessary.

Almost-everywhere Hierarchies.Open: Probabilistic, Quantum.

Delayed DiagonalizationLadner ’75

If P NP then there exists a set in NP that is not in P and not complete.

To keep the language in NP we wait until we have fulfilled the previous diagonalization step.

Diagonalization is it!Kozen (1980)

Any proof that P is different from NP is a diagonalization proof.

Says more about the difficulty of formalizing the notion of diagonalization than of the possibility of other types of proofs.

Nonrelativizing SeparationsBuhrman-Fortnow-Thierauf (1998).

Exponential version of MA does not have polynomial size circuits.

Relativized world where it does have polynomial size circuits.

Proof uses EXP in P/poly implies EXP in MA (Babai-Fortnow-Lund).

The Next Great ResultLogspace is strictly contained in NP.

No good reason to think this is hard. Several possible approaches.

Four ways to separate NP from L. 1. Autoreducibility. 2. Intersections of Finite Automata. 3. Anti-Impagliazzo-Wigderson. 4. Trading Alternation, Time and Space.

1. AutoreducibilityAutoredubile sets are sets with a certain

amount of redundacy in them.Whether certain complete problems are

autoreducible can separate complexity classes.

Burhman, Fortnow, van Melkebeek and Torenvliet ’95

Reducibility

B ...

A set A (Turing) reduces to B if we can answer questions to A by asking arbitrary adaptive questions to B.

A ...

Autoreducibility

A ...

A set A is autoreducible if we can answer questions to A by asking arbitrary adaptive questions to A.

A ...

Autoreducibility

A set A is autoreducible if we can answer questions to A by asking arbitrary adaptive questions to A except for the original question.

A ...

A ...

Autoreducibility and NL NP If EXPSPACE-complete sets are all

autoreducible then NL NP. If PSPACE-complete sets are all

nonadaptively autoreducible then NL also differs from NP.

Diagonalization Helps!Assume NP = NL.We then create a set in A such that

A is in EXPSPACE. A is hard for EXPSPACE. A diagonalizes against all autoreductions.

NP = NL implies a EXPSPACE-complete sets that is not autoreducible.

2. Intersecting Finite AutomataFinite automata can capture pieces of a

computation. Intersecting them can capture the whole

computation.Karakostas-Lipton-Viglas 2000.

Intersecting Finite AutomataDoes a given automata ever accept?

Check in time linear in size.Do a given collection of k automata of

size s have a non-empty intersection? Can do in time sk. If one can do substantially better,

complexity separation occurs.

Simulating Computation

FiniteControl

Input Tape

Work Tape

Simulating Computation

FiniteControl

Input Tape

Work Tape

F1 F2 F3

Simulating Computation

FiniteControl

Input Tape

Work Tape

F1 F2 F3

G

ResultsGiven k finite automata with s states

and one finite automata with t states. If we can determine if there is a

common intersection in time

so(k)t

then P is different from L.

ResultsGiven k finite automata with s states

and one finite automata with t states. If we can determine if there is a

common intersection by a circuit of size

so(k)t

then NP is different from L.

Diagonalization HelpsQuick simulations of the intersections of

finite automata allow us to solve logarithmic space in time n1+ which is strictly contained in P.

3. Anti-Impagliazzo-Wigderson Impagliazzo-Wigderson ’97.

If deterministic 2O(n) time (E) does not have 2o(n) size circuits then P = BPP.

Assumption very strong: We are allowed to use huge amounts to nonuniformity to save a little time.

To prove assumption false would separate P from NP.

P = NP and Small Circuits for E

P = NP implies P = PH.P = PH implies E = EH.Kannan ’81: EH contains languages that

do not have 2o(n) size circuits.E does not have 2o(n) size circuits.

L = NP and Linear Space

If every language in linear space has 2o(n) size circuits then L is different than NP.

We don’t even know if SAT has 2o(n) size circuits.

If SAT does not have 2o(n) size circuits than L is different from NP.

How to show L NPAssuming SAT has very small, low-

depth circuits show that Linear Space has slightly small circuits.

4. Alternation, Time and SpaceUse relationships between alternation,

time and space to get the collapses needed for a contradiction. Kannan ’84. Fortnow ’97. Lipton-Viglas ’99. Fortnow-van Melkebeek ‘00. Tourlakis ‘00.

Lower Bounds on 2

2-Linear time cannot be simulated by a machine using n1.99 time and polylogarithmic space.

Suppose it could…logc n

n1.99

Suppose it could…logc n

n1.99

n0.995

n0.995

n0.995

n0.995

Suppose it could…logc n

n1.99

n0.995

n0.995

n0.995

n0.995

Separations

Generalize: j-linear time requires nearly nj

time on small space machines.

If one could show j-linear time requires nk time with small space for all k then NP is different from L.

Lower Bounds on SATSatisfiability cannot be solved by any

machine using no(1) space and na time

for any a less than the golden ratio, about 1.618.

Various time-space tradeoffs.

Razborov – “It’s not dead yet”Circuit Complexity – 5 yearsDiagonalization

Complexity Theory – 35 years Computability – 65 years Proof Technique – 125 years Concept – 2600 years … and “It’s not dead yet”

Steve Mahaney

“Diagonalization is a tool for showing separation results, but not a power tool.”

Steve Mahaney

“Diagonalization is a tool for showing separation results, but not a power tool.”

ConclusionsDiagonalization still produces new lower

bounds and possibilities for the future.The actual diagonalization step is easy.The trick is doing the collapses to get

the diagonalization.Hard combinatorics not required. Is NP L just around the corner?