Theory of Computational Complexity: Computational Complexity(A Modern Approach)

24 Pages·2013·0.22 MB·English

by  Christian Stegemann

TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
Theory of Computational Complexity:
Computational Complexity(A Modern Approach)
by Arora and Barak, Chapter 6.5 - 6.7
Christian Stegemann
January 7, 2013
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
Outline
1 Definitions From Previous Presentations
2 Chapter 6.5: Circuit Lower Bounds
3 Chapter 6.6: Nonuniform Hierarchy Theorem
4 Chapter 6.7: Finer Gradations Among Circuit Classes
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
DefinitionsFromPreviousPresentations
n-Input, single-output Boolean Circuit C
Directed, acyclic graph with n sources, 1 sink. Nonsource vertices
are called gates and labeled with one of ∨ (OR), ∧ (AND) and ¬
(NOT). |C| is number of vertices. For Input x ∈ {0,1}n,C(x) is
the natural output.
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
DefinitionsFromPreviousPresentations
Boolean circuit
A circuit computing the XOR function:
∨
∧ ∧
¬ ¬
x x
1 2
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
DefinitionsFromPreviousPresentations
Circuit families
For T : N → N, a T(n)-size circuit family is a sequence {Cn}n∈N
of boolean circuits, where C is n input, single-output and for
n
every n ∈ N holds: |C | ≤ T(n). A language L is in SIZE(T(n)), if
n
there exists a T(n)-size circuit family {Cn}n∈N such that for every
x ∈ {0,1}n holds x ∈ L ⇐⇒ C (x) = 1
n
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
DefinitionsFromPreviousPresentations
The class P
/poly
P is the class of languages that are decidable by
/poly
polynomial-sized circuit families. This means:
P = ∪ SIZE(nc)
/poly c
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
Chapter6.5:CircuitLowerBounds
Chapter 6.5: Circuit Lower Bounds: Existence of hard
functions
Since P ⊆ P , proving NP (cid:42) P would prove P 6= NP.
/poly /poly
Theorem
For every n ≥ 1, there exists a function f : {0,1}n → {0,1} that
cannot be computed by a circuit C of size 2n/(10n).
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
Chapter6.5:CircuitLowerBounds
Proof: Count number of possible configurations:
1 Number of functions from {0,1}n → {0,1} is 22n
2 Every circuit of size at most S can be represented as a string
of 9×Slog(S) bits, so the number of such circuits is at most
29×Slog(S).
Setting S = 2n/(10n), the number of circuits of size S is at most
29×Slog(S) ≤ 22n9n/10n < 22n
Then there must be a hard function.
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
Chapter6.5:CircuitLowerBounds
An alternative proof uses the probabilistic method. For a given
n ∈ N, we can prove the probability of the existence of a circuit C
computing f to be at most 2−0.1×2n. This also shows that for big n
most of the functions are hard.
TheoryofComputationalComplexity:ComputationalComplexity(AModernApproach)byAroraandBarak,Chapter6.5-6.7
Chapter6.6:NonuniformHierarchyTheorem
Chapter 6.6: Nonuniform Hierarchy Theorem
Theorem
For every functions T,T0 : N → N with
2n/n > T0(n) > 10T(n) > n,
SIZE(T(n)) (cid:40) SIZE(T0(n))
This means that larger circuits can compute strictly more functions.