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Data Complexity |
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Query Complexity and Combined Complexity |
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Data Complexity
the complexity of {A | A
² f} for fixed f |
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Query Complexity
the complexity of {f |
A ² f} for fixed A |
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Combined complexity
the complexity of {(A,
f) | A ² f} |
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For every f, the complexity of {A | A ² f} is in
PTIME |
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[Why ?] |
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However, it is much lower than PTIME (next) |
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Theorem The data complexity of FO is uniform AC0 |
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What is AC0 ? |
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What is uniform AC0 ? |
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We will review next, but most importantly: |
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uniform-AC0 µ LOGSPACE µ PTIME |
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Fix n ¸ 0. A boolean circuit with n inputs, C,
is a rooted DAG with nodes labeled with labels from:
{Æ,
Ç, :, x1, …, xn} |
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size(C)
= number of gates |
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depth(C)
= length of longest path |
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Definition A language L µ {0,1}* is
in non-uniform AC0 if there exists d > 0, a polynomial p(n),
and a family of circuits (Cn)n ¸ 0
s.t.:
size(Cn)
· p(n)
depth(Cn)
· d, and |
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Let All be a vocabulary consisting of all
relations on N
All = P(N) [
P(N2) [ P(N3) [ … |
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In All we have names for <, +, /, … |
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Definition FO(All) = FO over vocabulary All |
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Interpretation: consider only ordered domains,
assimilate with {0, 1, 2, …, n-1}. Each relation R on N is interpreted as
its restriction to {0, 1, …, n-1}. |
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Note: we can express EVEN in FO(All) [why ?] |
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Theorem FO(All) µ non-uniform AC0 |
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Proof sketch in class (hint: it’s simple…) |
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Let s = {U}
(a unary table) |
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The property PARITY is true on models A where |UA|
is even |
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PARITY and EVEN are very related, but… |
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Theorem [Furst-Saxe-Sipser, Ajtai] PARITY is not
expressible in non-uniform AC0 |
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Corollary PARITY is not expressible in FO(All) |
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Comment: i.e. there is no formula in FO over
vocabulary {U} [ All that checks if |U| is even. |
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Corollary Graph connectivity is not expressible
in FO(All) [why ?] |
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EVEN is not expressible in FO |
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But EVEN is expressible in FO(<, +) |
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PARITY is not expressible in
FO(<, +, exp, …, any-relation-on-N, ….) ! |
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Non-uniform AC0 can express
non-computable properties ! |
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Need to restrict the association n ! Cn
to something easily computable |
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Complex definitions in complexity theory
textbooks… |
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Better:
define uniform AC0 = FO(+, £) |
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Alternatively FO(+, £) = FO(<, BIT) |
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Theorem The combined complexity is in PSPACE |
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[proof: in classs] |
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Note: proof in book is wrong. |
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Theorem There exists a structure A s.t. the
query complexity {f | A ² f} is PSPACE complete |
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Proof.
Recall the Satisfiability of Quantified Boolean Formulas problem
(QBF): is a QB formula F true ? |
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Example:
F = 9 X1 8 X2 8 X3 (X1
Æ X2 Ç : X1 Æ X3)
is it true ? |
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Note: boolean satisfiability (the NP-hard
problem) is in QBF [why ?] |
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Theorem [Stockmeyer] QBF is PSPACE complete |
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Return to our proof: reduction from QBF. |
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Given QBF formula F, let s = {U}, where U =
unary |
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A = (A, UA), s.t. A = {0,1}, UA
= {1} |
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Translate F (a QB formula) to f (an FO formula)
s.t. F is true iff A ² f [how ?] |
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Notes
