Optimal%20Proof%20Systems%20and%20Sparse%20Sets

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Optimal Proof Systems andSparse Sets

• Harry Buhrman, CWI
• Steve Fenner, South Carolina
• Lance Fortnow, NEC/Chicago
• Dieter van Melkebeek, DIMACS/Chicago

2
Convergence of Theory

• This talk talks about relating some very
  different looking concepts in complexity theory
• Optimal proof systems.
• Complete sets for the class of sparse NP
  languages.
• Reductions of sparse sets to tally sets.

3
The Great Book

• Does Erdös great book really exist?

4
Tautologies

• A tautology is a formula that is true no matter
  what assignment is used.
• Formulas that are not tautologies have easy
  proofs of this fact.

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Tautologies

• If we set x1 to TRUE, x2 to TRUE and x3 to FALSE
  then formula is false.
• Focus on tautologies in Disjunctive Normal
  FormOR of ANDs.
• How about proofs that a formula is a tautology?

6
Proof Systems

• Cook and Reckhow (1979) defined proof systems for
  tautologies. A proof system is a way of
  describing easily verifiable proofs that a
  formula is a tautology.
• For example, a truth-table of all the possible
  inputs will prove that a formula is a tautology.
• These proofs are quite large though.

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Resolution Proofs

• Consider the following two formula

• The first formula is a tautology if and only if
  the second one is a tautology.
• This process is called resolution.

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Resolution Proofs

• Every tautology can be resolved to a DNF with an
  empty clause.
• The list of resolutions forms a proof system.
• Haken (1985) showed that resolution requires
  large proofs.

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Proof Systems

• A proof system is an efficiently computable
  function mapping onto the tautologies.
• For a given proof system f and tautology f, the
  size of a proof for f is the length of the
  shortest x such that f(x)f.

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Proof Systems and Complexity

• Cook and Reckhow Tautologies have
  polynomial-size proof systems if and only if NP
  co-NP.
• Idea Guess polynomial-size proof.
• Can separate NP and co-NP and thus P from NP by
  showing that tautologies do not have small proof
  systems.

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Comparing Proof Systems

• We say a proof system f is as good as a proof
  system g if for every proof of a tautology in g
  there is a proof in f that is not much longer.
• Formally There is a polynomial p such that for
  all strings x there is a y, y lt p(x), and
  f(y) g(x).
• Resolution is as good as truth-table.

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f is as good as g
g-proofs
Formula
f-proofs
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Optimal Proof Systems

• A proof system is optimal if it is as good as any
  other proof system.
• Similar to the notion of NP-completeness, because
  it measures the largest member of a class.
• If you have an optimal proof system f, then NP
  co-NP if and only if f has polynomial-size proofs
  for all tautologies.

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Do optimal proof systems exist?

• If NP co-NP then tautology has polynomial-size
  proof which are trivially optimal.
• Even if tautology has no short proof systems,
  there still might be an optimal one.
• Let us first look at a variation of optimal proof
  systems.

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P-optimal Proof Systems

• A proof system f is P-optimal if for any proof
  system g, tautology f, and proof p for f, (g(p)
  f), we can efficiently compute from p a f-proof q
  of f.
• Every P-optimal proof system is optimal though
  the other direction is not clear.
• Do there exist P-optimal proof systems?

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f is an optimal proof system
g-proofs
Formula
f-proofs
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f is a p-optimal proof system
g-proofs
Formula
f-proofs
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UP-Complete Sets

• UP consists of the languages accepted by
  nondeterministic Turing machines having at most
  one accepting path.
• Examples include primality, factoring.
• One-way functions exist if and only if P ? UP.
• L is UP-complete if L is in UP and for every A in
  UP there is a function f such that

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Do UP-complete sets exist?

• The typical complete set
• L lti,x,1jgt Mi(x) accepts in j steps
• If M1, M2, enumerate the NP machines then L may
  not be in UP.
• We need to enumerate UP machines, i.e., machines
  that have at most one accepting path for all
  inputs.

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Do UP-complete sets exist?

• We need to enumerate UP machines, i.e., machines
  that have at most one accepting path for all
  inputs.

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Do UP-complete sets exist?

• We need to enumerate UP machines, i.e., machines
  that have at most one accepting path for all
  inputs.
• Determining whether a given nondeterministic
  machine M is a UP machine is undecidable.

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Do UP-complete sets exist?

• We need to enumerate UP machines, i.e., machines
  that have at most one accepting path for all
  inputs.
• Determining whether a given nondeterministic
  machine M is a UP machine is undecidable.
• For a better understanding we turn to oracles and
  relativization.

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Turing Machine
INPUT TAPE
M
WORK TAPE
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Oracle Turing Machines
INPUT TAPE
M
WORK TAPE
q? qy qn
ORACLE TAPE
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Oracle Turing Machines
INPUT TAPE
M
WORK TAPE
q? qy qn
QUERY
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Oracle Turing Machines
INPUT TAPE
M
WORK TAPE
q? qy qn
QUERY
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Oracle Turing Machines
INPUT TAPE
M
WORK TAPE
q? qy qn
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Oracle Turing Machines
INPUT TAPE
M
WORK TAPE
q? qy qn
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Oracle Turing Machines
INPUT TAPE
M
WORK TAPE
q? qy qn
ORACLE TAPE
The Oracle is the set of Yes answers.
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Relativization

• We appear quite far from separating any real
  complexity classes such as P and NP.
• Baker, Gill and Solovay (1975) noticed that
  proofs in complexity theory relativize, that is
  the proofs go through if all the machines
  involved have access to same oracles.

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Relativization and P vs NP

• Baker, Gill and Solovay (1975) show there are
  oracles A and B such that
• PA NPA
• PB? NPB
• Techniques currently used would not settle the P
  versus NP question.

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Interpreting Relativization

• Be careful in interpreting these results
• A very few number of results do not relativize,
  most notably in the area of interactive proofs.
• Space and large time classes do not have clean
  enough oracle models for these results.
• Relativization results are not impossibility
  results, nor do they give an indication whether a
  particular statement is true or false.

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UP and Relativization

• Hartmanis and Hemachandra (1984) show that UP
  does not have complete sets relative to an
  oracle.
• Note that if P NP then P UP NP and UP does
  have complete sets.
• How does this relate to P-optimal proof systems?

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P-Optimal and UP

• Messner and Torán (1998) show that ifP-optimal
  proof systems exist then UP has complete sets.
• Combining with Hartmanis-Hemachandra gives
  relativized world where there do not exist
  P-optimal proof systems.

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Sparse Sets and NP

• A set of strings over 0,1 can have 2n strings
  of length n.
• A sparse set is a small set with at most nc
  strings at length n for some fixed c.
• Are there complete sets for the sparse NP sets?

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NP?SPARSE-complete Sets

• Mahaney (1978) shows that if there is a sparse
  set that is NP-complete then P NP.
• Is there a set that is NP, sparse and hard for
  only the other sparse sets in NP?
• Similar to the UP case, it is impossible to
  decide whether a given NP machine accepts a
  sparse set.

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Optimal Proof Systems

• Messner and Torán also show that if optimal proof
  systems exist then NP?SPARSE has complete sets.
• They could not conclude that there exists
  relativized worlds where no optimal proof systems
  exist because the oracle question for NP?SPARSE
  remained open.

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Our Result

• There exists a relativized world where NP?SPARSE
  does not have complete sets.
• Corollary
• There exists a relativized world where there are
  no optimal proof systems.

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Other Types of Reductions

• Results described so far are for many-one
  reductions, where we say that A reduces to B if
  there exists a polynomial-time function f such
  that
• We can also consider other reductions.

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Turing Reducibility

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

A
...
...
B
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Truth-Table Reducibility

• A set A Truth-Table reduces to B if we can answer
  questions to A by asking arbitrary nonadaptive
  questions to B.

A
...
...
B
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Truth-Table Reducibility

• A set A Truth-Table reduces to B if we can answer
  questions to A by asking arbitrary nonadaptive
  questions to B.

A
...
...
B
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Turing Reductions

• Hartmanis and Yesha (1984) show that there exists
  a tally set in NP that is Turing-hard for every
  sparse NP set.
• A tally set is a subset of 1.
• Every tally set is sparse.
• What is the relationship between sparse and tally
  sets?

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SPARSE to TALLY

• A sparse set has at most a polynomial number of
  strings at any length.
• A tally set can only have 1n at length n.
• In some sense both sets can encode same amount of
  information.
• However the strings in a sparse set could be
  hidden making more complex sets.

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SPARSE to TALLY

• Book and Ko (1988) show
• Every sparse sets truth-table reduces to some
  tally set.
• There is some sparse sets that does not
  truth-table reduce to a tally set if the number
  of queries is fixed.

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SPARSE to TALLY

• Ko (1989)
• There is some sparse sets that does not
  truth-table reduce to a tally set if the
  reduction is disjunctiveaccepts if any of the
  queries are in the tally set.
• Buhrman-Longpré-Spaan (1995)
• Every sparse set can be conjunctively reduced to
  a tally setaccepts if all queries are in the
  tally set.

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SPARSE to TALLY

• Schöning (1993) gives a probabilistic reduction
  from sparse to tally.
• If A is sparse and p a polynomial, there is a
  tally set B and a randomized efficiently
  computable function f such that
• If x is in A then f(x) is always in B.
• If x is not in A then the probability that f(x)
  is in B is at most 1/p(x).

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NP?SPARSE-complete Sets

• These proofs preserve NP-ness.
• If A is sparse and in NP and p a polynomial,
  there is a tally set B in NP and a randomized
  efficiently computable function f such that
• If x is in A then f(x) is always in B.
• If x is not in A then the probability that f(x)
  is in B is at most 1/p(x).

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NP?TALLY-complete Sets

• NP?TALLY has complete sets.
• T 1lti,n,kgt Mi(1n) accepts in k steps
• T is complete for NP?SPARSE via
• Turing-reductions
• Truth-table reductions
• Conjunctive truth-table reductions
• Randomized reductions

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NP?SPARSE-complete Sets

• Open Can NP?SPARSE have complete sets but no
  complete tally sets?
• Our relativization techniques force any
  NP?SPARSE-complete set to look like a tally set.
• We can then apply negative results for SPARSE to
  TALLY to the NP?SPARSE-complete set problem.

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Relativization Results

• There exists relativized worlds where
• There do not exist any NP?SPARSE-complete sets
  under disjunctive reductions.
• There do not exist any NP?SPARSE-complete sets
  under truth-table reductions asking only o(n/log
  n) queries.
• There exists a sparse set that does not reduce to
  any tally set by any truth-table reduction using
  o(n/log n) queries.

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Tight Result

• For any constant c gt 0, there exists a
  relativized world where NP?SPARSE has no complete
  sets under truth-table reductions using o(n/log
  n) queries and O(log n) bits of advice.

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Tight Result

• Under a reasonable assumption, for all values of
  k, NP?SPARSE has a complete set under conjunctive
  truth-table reductions using n/(k log n) queries
  and O(log n) bits of advice.
• Uses derandomization techniques of Klivans and
  van Melkebeek.
• Similar results for SPARSE to TALLY.

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Further Directions

• Tight bounds for Turing reductions?
• Eliminate reasonable assumption needed for
  derandomization.
• How does NP?SPARSE compare with other promise
  classes like UP, BPP and NP?co-NP.
• Differences in enumerations and time-hierarchy.

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Conclusions

• Often very different looking questions on
  complexity theory tie together.
• We also use many different techniques from
  Kolmogorov complexity to state-of-the-art
  derandomization results.
• Still no strong evidence for or against the
  existence of optimal proof systems.