Ajtai Open Problem Session: Descriptive Complexity

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Ajtai Open Problem Session Descriptive
Complexity

• Ron Fagin
• IBM Almaden

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Ajtais paper

• Ajtai wrote a famous 1983 paper
• - formulae on finite structures
• Many results from that paper have been
  rediscovered by others
• Example Furst, Saxe and Sipsers 1984 result
  that parity is not in AC0

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What is ?

• -sentence ?Q1 ? ?Qm f(P,Q1,?,Qm)
• Here f is first-order, and P is, say, a binary
  relation symbol, and so about graphs
• Can define for more general structures than
  graphs

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Example 3-colorability

• Let P represent the graph relation
• Represent the 3 colors by Q1, Q2, Q3
• Let f(P,Q1,Q2,Q3) say Each point has exactly one
  color, and no two points with the same color are
  connected by an edge
• Then ?Q1 ?Q2 ?Q3 f(P,Q1,Q2,Q3) says The graph is
  3-colorable
• So 3-colorability is

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Connecting logic and complexity

• Theorem (Fagin 1974) NP

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L-structures

• Let L be a signature, that is, a collection of
  relation symbols
• If L contains only a binary relation symbol, then
  L is the language of graphs, and an L-structure
  is simply a graph

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A descriptive complexity hierarchy

• Let Sk(L) be those NP properties of L-structures
  that can be defined using only k-ary
  existentially quantified relation symbols.
• Example If P is a binary relation symbol, then
  Sk(P) consists of graph properties definable by a
  sentence of the form
• ?Q1 ? ?Qm f(P,Q1,?,Qm)
• with Q1,?, Qm k-ary
• S1(L) ? S2(L) ? S3(L) ?
• ?k Sk(L) the NP properties of L-structures

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Monadic NP

• S1(L) is often called monadic NP
• Example 3-colorability
• Theorem (Fagin 1975) Graph connectivity is not
  in monadic NP (although nonconnectivity is)

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Is the hierarchy strict?

• Let P be a binary relation symbol
• Since graph connectivity is easily shown to be in
  S2(P), it follows that S1(P) ? S2(P)
• Open problem Does the hierarchy collapse to the
  second level, so that every NP graph property is
  in binary NP, that is, S2(P)?

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A partial result

• Theorem (Fagin 1975) If Sk(L) Sk1(L), then
  Sk (L) Sn(L), for all n with n ? k

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A theorem from Ajtais famous 1983 paper

• Theorem (Ajtai 1983) Let R be a (k1)-ary
  relation symbol. The parity of the number of
  tuples in R is even is not in Sk(R) (although it
  is in Sk1(R))
• In particular, NP goes beyond binary NP if we
  consider richer structures than graphs

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A sub-hierarchy of Sk(L)

• Let us fix k, the arity of the 2nd-order
  quantifiers, and consider the number of these
  quantifiers
• When k1
• Theorem (Otto 1995) More monadic NP graph
  properties can be expressed with r1 unary
  quantifiers than with r

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Refinements of open problem

• We asked Is there an NP graph property not
  expressible in the form
• ?Q1 ? ?Qm f(P,Q1,?,Qm),
• where the Qis are binary?
• Is there an NP graph property not expressible in
  the form ?Q f(P,Q), where Q is binary?
• Is there a co-NP graph property not expressible
  in the form ?Q f(P,Q), where Q is binary?
• Note that resolving these problems does not
  settle P ? NP or NP ? co-NP

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Open meta-problem

• What other wonderful jewels are hidden away in
  Ajtais 1983 paper?