perspective on lower bounds: diagonalization

Perspective on Lower Bounds: DiagonalizationLance FortnowNEC Research Institute

A TheoremPermanent is not in uniform TC0.Papers:Allender 96.Caussinus-McKenzie-Thrien-Vollmer 96.Allender-Gore 94.

Counting HierarchyPP Class of languages accepted by probabilistic machines with unbounded error.Counting Hierarchy

Counting Hierarchy in TC0If 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 simplesometimes require clever ideas but rarely hard combinatorics.

Diagonalization - NegativesOnly weak separations so far.RelativizationProbably 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 Cantors Theorem

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Proof of Cantors Theorem

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Proof of Cantors Theorem

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A Brief History600 BC - EpimenidesParadox.All cretans are liarsOne of their own poets has said so.400 BC - EubulidesParadox.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 - Russells Paradox.The set of all sets that does not contain itself as a member.1931 - Gdels 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), k (1983).Unbounded collapses necessary.Almost-everywhere Hierarchies.Open: Probabilistic, Quantum.

Delayed DiagonalizationLadner 75If 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

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

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

AutoreducibilityA 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 NPIf 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 thatA 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 ComputationFiniteControlInput TapeWork Tape

Simulating ComputationFiniteControlInput TapeWork TapeF1F2F3

Simulating ComputationFiniteControlInput TapeWork TapeF1F2F3G

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 timeso(k)tthen 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 sizeso(k)tthen 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-WigdersonImpagliazzo-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 EP = 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 SpaceIf every language in linear space has 2o(n) size circuits then L is different than NP.We dont 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 22-Linear time cannot be simulated by a machine using n1.99 time and polylogarithmic space.

Suppose it couldlogc nn1.99

Suppose it couldlogc nn1.99n0.995n0.995n0.995n0.995

Suppose it couldlogc nn1.99n0.995n0.995n0.995n0.995

SeparationsGeneralize: 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 Its not dead yetCircuit Complexity 5 yearsDiagonalizationComplexity Theory 35 yearsComputability 65 yearsProof Technique 125 yearsConcept 2600 years and Its not dead yet

Steve MahaneyDiagonalization is a tool for showing separation results, but not a power tool.

Steve MahaneyDiagonalization 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?