Simple PCPs

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Probabilistically Checkable Proofs
Madhu Sudan MIT CSAIL
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Can Proofs Be Checked Efficiently?
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Proofs and Theorems

• Conventional belief Proofs need to be read
  carefully to be verified.
• Modern constraint Dont have the time (to do
  anything, leave alone) read proofs.
• This talk
• New format for writing proofs.
• Efficiently verifiable probabilistically, with
  small error probability.
• Not much longer than conventional proofs.

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Outline of talk

• Quick primer on the Computational perspective on
  theorems and proofs (proofs can look very
  different than youd think).
• Definition of Probabilistically Checkable Proofs
  (PCPs).
• Overview of a new construction of PCPs due to
  Irit Dinur.

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Theorems Deep and Shallow

• A Deep Theorem
• Proof (too long to fit in this section).
• A Shallow Theorem
• The number 3190966795047991905432 has a divisor
  between 25800000000 and 25900000000.
• Proof 25846840632.

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Computational Perspective

• Theory of NP-completeness
• Every (deep) theorem reduces to shallow one.
• Shallow theorem easy to compute from deep one.
• Shallow proofs are not much longer.

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More Broadly New formats for proofs

• New format for proof of T Divisor D (A,B,C dont
  have to be specified since they are known to
  (computable by) verifier.)
• Theory of Computation replete with examples of
  such alternate lifestyles for mathematicians
  (formats for proofs).
• Equivalence (1) new theorem can be computed from
  old one efficiently, and (2) new proof is not
  much longer than old one.
• Question Why seek new formats? What
• benefits can they offer?

Can they help ?
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Probabilistically Checkable Proofs

• How do we formalize formats?
• Answer Formalize the Verifier instead. Format
  now corresponds to whatever the verifier accepts.
• Will define PCP verifier (probabilistic, errs
  with small probability, reads few bits of proof)
  next.

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T
010010100101010101010

• PCP Verifier
• Reads Theorem
• 2. Tosses coins
• 3. Reads few bits of proof
• 4. Accepts/Rejects.

V
HTHTTH
0
1
0
P
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Features of interest

• Number of bits of proof queried must be small
  (constant?).
• Length of PCP proof must be small (linear?,
  quadratic?) compared to conventional proofs.
• Optionally Classical proof can be converted to
  PCP proof efficiently. (Rarely required in
  Logic.)
• Do such verifiers exist?
• PCP Theorem 1992 They do.
• Today New construction due to Dinur.

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Part II PCP Construction of Dinur
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Essential Ingredients of PCPs

• Locality of error
• If theorem is wrong (and so proof has an
  error), then error in proof can be pinpointed
  locally (since it is found by verifier that reads
  only few bits of proof).
• Abundance of error
• Errors in proof are abundant (i.e., easily seen
  in random probes of proof).
• How do we construct a proof system with these
  features?

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Locality From NP-completeness

• 3-Coloring

is NP-complete
T
P
Color vertices s.t. endpoints of edge
have different colors.
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3-Coloring Verifier
T

• To verify
• Verifier constructs
• Expects as
  proof.
• To verify Picks an edge and verifies endpoints
  distinctly colored.
• Error Monochromatic edge 2 pieces of proof.
• Local! But errors not frequent.

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Amplifying Error

• Dinur Transformation There exists a linear-time
  algorithm A

A
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Iterating the Dinur Transformation

• Logarithmically many iterations of the Dinur
  Transformation
• Leads to a polynomial time transformation.
• Preserve 3-colorability (valid theorems map to
  valid theorems).
• Convert invalid theorem into one where every
  proof has e0 fraction errors.

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Details of the Dinur Transformation

• Step 1
• Gap Amplification Increase number of
  available colors, but make coloring more
  restrictive.
• Goal Increase errors in this stage (at expense
  of longer questions).
• Step 2
• Color reduction Reduce number of colors
  back to 3.
• Hope Fraction of errors does not reduce by much
  (fraction will reduce though).
• Composition of Steps yields Transformation.

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Step 2 Reducing colors

• Form of classical Reductions similar to task
  of reducing k-coloring to 3-coloring.
• Unfortunately Classical reductions lose by
  factor k. Cant afford this.
• However Prior work on PCPs gave a simple
  reduction Lose only a universal constant,
  independent of k. This is good enough for Step 2.
• (So Dinur does use prior work on PCPs, but the
  simpler, more elementary, parts.)

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Step 1 Increasing Error

• Task (for example) Create new graph H (and
  coloring restriction) from G s.t. H is 3c-color
  if G is 3-colorable, but fraction of invalidly
  colored edges in H is twice the fraction in G.
• One idea Graph Products.

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Graph Products and Gap Amplification

• Problem 1 Not clear that error amplifies.
  Non-trivial question. Many counter-examples to
  naïve conjectures. (But might work )
• Problem 2 Quadratic-blow up in size. Does not
  work in linear time!!!
• Dinurs solution Take a derandomized graph
  product

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Step 1 The final construction

• Definition of H (and legal coloring)

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Analysis of the construction.

• Does this always work?
• No! E.g., if G is a collection of disconnected
  graphs, some 3-colorable and others not.
• Fortunately, connectivity is the only bottleneck.
  If G is well-connected, then H has the right
  properties. (Intuition developed over several
  decades of work in expanders and
  derandomization.)
• Formal analysis Takes only couple of pages ?

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Conclusion

• A moderately simple proof of the PCP theorem.
  (Hopefully motivates you. Read original paper at
  ECCC. Search for Dinur, Gap Amplification).
• Matches many known parameters (but doesnt match
  others).
• E.g., Håstad shows can verify proof by reading
  3 bits, rejecting invalid proofs w.p. .4999
• Cant (yet) reproduce such constructions using
  Dinurs technique.

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Conclusions (contd.)

• PCPs illustrate the power of specifying a format
  for proofs.
• Can we use this for many computer generated
  proofs?
• More broadly Revisits the complexity of proving
  theorems vs. verifying proofs.
• Is PNP?

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Brief History

• Definition conceived in 1991 (based on extensive
  series of works in 1980s motivated principally by
  cryptography).
• Quick Series of Results (1990-1992) culminating
  in the PCP theorem
• There exists an efficient PCP verifier who checks
  proofs probabilistically by looking at constant
  number of bits of the proof accepts valid
  proofs rejects invalid theorems w.p. ½
• PCP proofs not much longer than conventional
  proofs.

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PCP constructions

• 1992 construction quite complex (wont discuss
  today).
• Since 1992-2004 Constructions got more
  efficient, and more complex!
• 2005 A new construction
• Irit Dinur, The PCP Theorem by Gap
  Amplification, ECCC, 2005.
• Much simpler exploits improved understanding of
  computational randomness (1970s-2000s).

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The Burden of Checking Proofs

• Editor sees articles that are hundreds of pages
  long. Sometimes, almost surely wrong. But how can
  we check.
• Probabilistically Checkable Proofs Can proofs be
  checkable efficiently, without reading the whole
  proof?
• First reaction NO! Proof may only have one
  error. How are we going to find it. (Recall the
  many proofs of 12 youve seen.)
• Goal of this talk A Better Answer YES!

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Logic, Computation, and Probability

• First clarification Proofs as we know it are not
  even checkable, leave alone probabilistically
  and efficiently. (I.e., an typo in proof in
  journal is not considered fatal.)
• Distinction between Informal Proofs and Formal
  Proofs Readability vs. Syntactic correctness.
• Reliance on Informal Proofs based on implicit
  belief that we can translate proofs to Formal,
  syntactically correct, ones.
• Study of such formal, syntactically correct,
  proofs and their verification Logic

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Logic and Theory of Computing

• Principal thesis of Logic There exists a format
  for writing theorems and proofs so that
• Proofs can be verified (easily).
• Completeness Every true theorem has a (short)
  proof.
• Soundness No false theorem has a proof.
• In fact, there are many formats.
• Theory of Computing If we interpret easily and
  short properly, then we can produce many
  unlikely formats!

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Easy vs. Hard Short vs. Long

• Underlying hypothesis Theorem (T), Proof (P) are
  just sentences/sequence of letters from some
  finite alphabet. Indeed w.l.o.g., alphabet
  0,1.
• Easy Verification Validity(T,P) computable in
  time polynomial in the length of T, P.
• Short Length of P is not much longer than
  length of P in any other reasonable format.

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Non-standard formats for Proofs.

• Standard Language T Riemann Hypothesis and
  say P is expected to be of length n.
• Non-standard language T (A,B,C) integers.
  Proof P D another integer.
• Valid?(T,P) D divides A, and B D C.
• Equivalence Given T and n, can efficiently
  compute T (A,B,C) with 0 A,B,C exp(nc)
  such that T valid in non-standard format iff T
  valid in standard format.

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Aside Theory of NP-completeness

• Theory provides a wide variety of formats for
  proof systems (proving they are all equivalent).
• Fundamental question P NP?
• Equivalent to Is proving a theorem as easy as
  verifying its proof?
• More provocative version Can we replace every
  mathematician by a computer?

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Logic, Computation, and Randomness

• What new formats emerge when we introduce
  randomness into the mix.
• Verifier allowed to
• Toss random coins.
• Make error with small probability.
• What benefits can this provide?
• Can we ensure verifier reads