perspective on lower bounds: diagonalization

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Perspective on Lower Bounds: Diagonalization. Lance Fortnow NEC Research Institute. A Theorem. Permanent is not in uniform TC 0 . Papers: Allender ’96. Caussinus-McKenzie-Th érien-Vollmer ’96. Allender-Gore ’94. Counting Hierarchy. - PowerPoint PPT Presentation

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  • 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?