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Presentation Transcript
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Bounded depth circuits cannot sample good codes Shachar Lovett (IAS)
Joint with Emanuele Viola (Northeastern)<br>
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Lower bounds Classic lower bounds: functions
Bounded families of circuits cannot compute (or approximate) some explicit function
This work: distributions
Bounded families of circuits cannot sample (or approximate) some explicit distribution<br>
In fact, AC0 circuits can (approximately) sample any symmetric function [Viola’10]<br>
04
Sampling lower bounds for AC0 [Viola’10]: challenge - explicit distribution that cannot be sampled (or approximated) by AC0
This work: uniform distribution over a code
Applications to data structure lower bounds and pseudorandomness
[Viola’11]: (x,b(x)) for a Boolean<br>
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Talk overview Describe result
Applications
Proof<br>
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Computational model: AC0 Constant depth circuits with AND,OR,NOT gates
Multiple outputs: maps m bits into n bits
Polynomial size (can handle sub-exponential size)<br>
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Codes (n,k,d) code:
Subset
Size
Minimal distance:
A code is good if
(our results extend also to )
Good codes exist<br>
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Main result: AC0 cannot sample codes Good code
AC0 circuit
Theorem: the uniform distribution over
and the output distribution of f
have statistical distance<br>
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Application 1: data structures (n,k,d) good code
Consider a data structure for messages which allows encoding in AC0
Claim: data structure has redundancy log(n)
(i.e. use at least bits)
Proof: encode a uniform string
Previous results : bounded query model (i.e. NC0)
[Gál-Miltersen’07]<br>
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Application 2: Nisan PRG against AC0 Nisan (1991) gave the first PRG against AC0
To fool AC0 circuits of depth d, a naïve implementation requires AC0 circuits of depth d+1
Is this necessary? Maybe the pseudorandom distribution can be otherwise sampled
k-wise independence, which also fools AC0, can be sampled by small depth 2 circuits
This work: depth d is necessary for Nisan’s PRG (at least for some setting of the underlying designs)<br>
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Main result: AC0 cannot sample codes Good code
AC0 circuit
Theorem: the uniform distribution over
and the output distribution of f
have statistical distance<br>
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Core idea: noise sensitivity AC0 circuits are noise insensitive:
flipping few input bits hardly changes output
Codes are sensitive:
Distinct codewords have large distance
Easily shows that AC0 cannot compute codes; sampling lower bound requires some more work<br>
Warm-up: AC0 cannot compute codes (n,k,d) code
Encoding map
Assume f can be computed by AC0 circuits
Coupled inputs:
Contradiction whenever<br>
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AC0 cannot (exactly) sample codes (n,k,d) code
uniform over
Assume f can be computed by AC0 circuits
Coupled inputs:
Can f be local: f(x)=f(x’) w.h.p ?<br>
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Local maps: isoperimetric inequality uniform over (n,k,d) code
f induces a partition of {0,1}m with 2k equal parts:
Coupled input (x,x’): edge in hypercube {0,1}m
Edge-isoperimetric inequality [Harper’64, Hart’76]:
Each part cannot contain too many edges
So, with good probability:
(hence )<br>
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Local maps: isoperimetric inequality uniform over (n,k,d) code
Edge-isoperimetric inequality:
Subset of {0,1}m of size 2m-k has edges<br>
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Putting it all together uniform over (n,k,d) code
AC0 sensitivity:
Isoperimetric inequality:
Contradiction when<br>
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AC0 cannot (approximately) sample codes The previous argument actually shows that the statistical distance between:
The uniform distribution over a code
The uniform distribution of AC0 circuit
is at least
To get statistical distance , use less coupled inputs (i.e. more noise)<br>
