Perspective on Lower Bounds: Diagonalization

1
Perspective on Lower Bounds Diagonalization

• Lance Fortnow
• NEC Research Institute

2
A Theorem

• Permanent is not in uniform TC0.
• Papers
• Allender 96.
• Caussinus-McKenzie-Thérien-Vollmer 96.
• Allender-Gore 94.

3
Counting Hierarchy

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

4
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.

5
Threshold Machines

• Alternating 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.

6
Almost done

• For 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
  timer-threshold machine for some r gt k.

7
Finishing Up

• SAT 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!

8
Diagonalization

• Want to prove separation.
• Assume collapse.
• Get other collapses.
• Keep collapsing until we have collapsed two
  classes that can be separated by diagonalization.

9
Diagonalization - Positives

• Diagonalization 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.

10
Diagonalization - Negatives

• Only 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.

11
Diagonalization

• Cantor (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,

12
Proof of Cantors 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 ?
? ? ? ? ? ? ? ?

13
Proof of Cantors 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 ?
? ? ? ? ? ? ? ?

14
Proof of Cantors 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 ?
? ? ? ? ? ? ? ?

15
A Brief History

• 600 BC - Epimenides Paradox.
• All cretans are liarsOne 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.

16
A Brief History

• 1874 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 - Gödels Incompleteness.
• This statement does not have a proof.

17
A Brief History

• 1936 Turing.
• The halting problem is undecidable.
• 1956 Friedberg-Muchnik.
• There exist incomplete Turing degrees.
• 1965 Hartmanis-Stearns.
• More time gives more languages.

18
Time and Space Hierarchies

• Nondeterministic 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.

19
Delayed Diagonalization

• Ladner 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.

20
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.

21
Nonrelativizing Separations

• Buhrman-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).

22
The Next Great Result

• Logspace 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.

23
1. Autoreducibility

• Autoredubile 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

24
Reducibility

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

A
...
...
B
25
Autoreducibility

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

A
...
...
A
26
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
27
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.

28
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.

29
2. Intersecting Finite Automata

• Finite automata can capture pieces of a
  computation.
• Intersecting them can capture the whole
  computation.
• Karakostas-Lipton-Viglas 2000.

30
Intersecting Finite Automata

• Does 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.

31
Simulating Computation
Input Tape
Finite Control
Work Tape
32
Simulating Computation
Input Tape
Finite Control
F1
F2
F3
Work Tape
33
Simulating Computation
Input Tape
G
Finite Control
F1
F2
F3
Work Tape
34
Results

• Given 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.

35
Results

• Given 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.

36
Diagonalization Helps

• Quick simulations of the intersections of finite
  automata allow us to solve logarithmic space in
  time n1? which is strictly contained in P.

37
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.

38
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.

39
L NP and Linear Space

• If 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.

40
How to show L ? NP

• Assuming SAT has very small, low-depth circuits
  show that Linear Space has slightly small
  circuits.

41
4. Alternation, Time and Space

• Use 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.

42
Lower Bounds on ?2

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

43
Suppose it could
logc n
n1.99
44
Suppose it could
logc n
n0.995
n0.995
n1.99
n0.995
n0.995
45
Suppose it could
logc n
n0.995
n0.995
n1.99
n0.995
n0.995
46
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.

47
Lower Bounds on SAT

• Satisfiability 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.

48
Razborov Its not dead yet

• Circuit Complexity 5 years
• Diagonalization
• Complexity Theory 35 years
• Computability 65 years
• Proof Technique 125 years
• Concept 2600 years
• and Its not dead yet

49
Steve Mahaney

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

50
Steve Mahaney

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

51
Conclusions

• Diagonalization 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?