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Time Hierarchieswith
One Bit of Advice
Konstantin Pervyshev
Steklov Mathematics Institute, St. Petersburg, Russia
joint work with
Dieter van Melkebeek
Time Hierarchy
An open question for probabilistic algorithms:
is there a time hierarchy ?
O(n2)
O(n3)
O(n)
(we still can’t disprove this)
Our result
• Previous results– [Barak 02] uses a modified notion of algorithm
• (algorithms with small advice)
– Under the modified notion of algorithm, a time hierarchy for the probabilistic algorithms exists
• Our result– Under the same modified notion of algorithm,
a time hierarchy exists for any class of algorithms• various classes of probabilistic algorithms• Arthur-Merlin and Merlin-Arthur games, other kinds of IP• NP ∩ co-NP
Outline
• Why standard techniques don’t work
• Where advice helps
• How to prove the generic time hierarchy
Diagonalization
• To separate deterministic time na+e from time na, consider a machine M such that– M(k) := not Nk(k), where na steps of Nk are
simulated• M runs in time na+
• M recognizes some languages L• L can’t be recognized in time na
• (!) deterministic algorithms are recursively enumerable
Probabilistic Algortithmsand Diagonalization
• Probabilistic Algorithms– Probabilistic Turing machines that satisfy some
condition on the error probability • Two-sided error (BPP)
– Pr[M(x) = 1] > 2/3 or Pr[M(x) = 0] > 2/3– “machine M is good on input x”– “machine M is good at length n”
• Need to enumerate only good machines• M(k) := not Nk(k) • Pr[Nk(k) = 1] = 1/2 => Pr[M(k) = 1] = 1/2
• It’s not possible
More Failures
• Various classes of probabilistic algorithms– bounded probability of error
• BPP – two-sided error• RR – one-sided error• ZPP – zero-sided error
• NP ∩ co-NP– two machines solve the same language
• Generally speaking, semantic classes• Diagonalization fails
– Nk(k) is bad => M(k) is bad
• To overcome this, M needs advice on whether Nk is good
Algorithms with Advice
• Turing machine M on input x of length n is provided with– some advice a(n) of length l(n)
• Advice is the same for every input of length n
• Depending on the advice provided,– M may recognize several languages– M may satisfy the promise or not
• Advice of length 1 bit• helps with time hierarchies
Time Hierarchies with Advice
• A time hierarchy exists for probabilistic algorithms with advice of length– O(log log n) bits – [Barak 02] – 1 bit – [Fortnow, Santhanam 04]
• Time hierarchy for any class of algorithms with advice of length– O(log n * log log n) bits – [Fortnow, Santhanam, Trevisan 05]– 1 bit – our result
Generic Time Hierarchy
– To separate na+e from na, it’s sufficient to prove that
• for any 1 ≤ a, there exists a language L
solvable in probabilistic polynomial time with 1 bit of advice
– machine M with advice a(n)
not solvable in probabilistic time na with 1 bit of advice
– any machine Nk with any advice b(n)
A Failed Approach
• Construct M with advice a(n) so that– for some inputs x(0) and x(1) of the same length n
M (x(0),a(n)) := not Nk(x(0),0)
M (x(1),a(n)) := not Nk(x(1),1)
• Both Nk(x,0) and Nk(x,1) may be bad
=> M needs 2 bits of advice in order to diagonalize safely
Another Failed Approach
• M can safely simulate Nk via deterministic simulation– needs exponentially more time
• To get exponentially more time,
we use delayed diagonalization
A Step of Delayed Diagonalization
x(0)
x(1)
z(1)
y(0)Advice on whetherN/0 is good on x’s
M(z(1)) = N(x(1),1)
M(y(0)) = “no”
Advice on whetherN/1 is good on x’s
N/0 is badN/1 is good
Tree-Like Delayed Diagonalization
x(00-11)
z(10,11)
y(00,01)
v(01)
w(11) M(z(01)) = N(x(01),1)M(z(11)) = N(x(11),1)
M(v(01)) = N(z(01),0)
M(y(00)) = “no”M(y(01)) = “no”
M(w(11)) = “no”
N/0 is badN/1 is good
N/0 is good N/1 is bad
Towards a Contradiction
– Assume• for some advice b(n), N is good and
solves the same language as M
– Then• N(v(s),b(|v|)) = M(v(s) ,a(|v|)) =• N(z(s),b(|z|)) = M(z(s) ,a(|z|)) =• N(x(s),b(|x|)) = M(x(s) ,a(|x|))
– Therefore,• N(v(s) ,b(|v|)) = M(x(s),a(|x|)) for some s
– So let• M(x(s),a(|x|)) := not N(v(s),b(|v|))• this can be done deterministically
– thus a contradiction
x(s)
z(s)v(s)
|x| ~ 2|v|a
Choice of the Input Lengths
• We need– parent’s length is polynomial in
children’s length• so that M runs in poly-time
– for any leaf v, roots length is greater than 2|v|a
• so that M can deterministically simulate N at leaves
• It’s possible to satisfy these conditions
• QED
x(s)
z(s)v(s)
Summary
• A time hierarchy exists for virtually any kind of algorithms with one bit of advice
• The probabilistic time hierarchy with advice is a property of algorithms with advice
Thank you!
Dieter van Melkebeek, Konstantin Pervyshev“A Generic Time Hierarchy for Semantic Models
with One Bit of Advice”(CCC’06)
Generic Time Hierarchy
– Theorem• for any 1 ≤ a < b, there exists a language L
solvable in probabilistic time nb with 1 bit of advicenot solvable in probabilistic time na with 1 bit of advice
– Only basic properties of algorithms are needed– Approach
• Construct probabilistic M with 1-bit advice a(n) that– works in time nb
– is good
• Prove that for any probabilistic N with any 1-bit advice b(n) that
– works in time na
– is good
• There exists x such that M(x,a(|x|)) ≠ N(x,b(|x|))
Non-Uniform World
• Previous results– [Barak02, FS04, FST05] a time hierarchy exists for 1 bit
non-uniform probabilistic algorithms with two- and one-sided error
• Our result– a time hierarchy exists for any class of 1 bit non-uniform
algorithms• various classes of probabilistic algorithms
• Arthur-Merlin and Merlin-Arthur games, other kinds of IP
• NP ∩ co-NP