Church, Kolmogorov and von Neumann:
Their Legacy Lives in Complexity
Lance Fortnow
NEC Laboratories America
1903 – A Year to Remember
1903 – A Year to Remember
1903 – A Year to Remember
Kolmogorov
Church
von Neumann
Andrey Nikolaevich Kolmogorov


Born:
April 25, 1903
Tambov, Russia
Died:
Oct. 20, 1987
Alonzo Church


Born:
June 14, 1903
Washington, DC
Died:
August 11, 1995
John von Neumann


Born:
Dec. 28, 1903
Budapest, Hungary
Died:
Feb. 8, 1957
Frank Plumpton Ramsey



Born:
Feb. 22, 1903
Cambridge, England
Died:
January 19, 1930
Founder of Ramsey
Theory
Ramsey Theory
Ramsey Theory
Applications of Ramsey Theory






Logic
Concrete Complexity
Complexity Classes
Parallelism
Algorithms
Computational Geometry
John von Neumann







Quantum
Logic
Game Theory
Ergodic Theory
Hydrodynamics
Cellular Automata
Computers
The Minimax Theorem (1928)


Every finite zero-sum two-person game
has optimal mixed strategies.
Let A be the payoff matrix for a player.
max min X AY  min max X AY
T
X
Y
T
Y
X
The Yao Principle (Yao, 1977)


Worst case expected runtime of
randomized algorithm for any input equals
best case running time of a deterministic
algorithm for worst distribution of inputs.
Invaluable for proving limitations of
probabilistic algorithms.
Making a Biased Coin Unbiased

Given a coin with an unknown bias p, how
do we get an unbiased coin flip?
Making a Biased Coin Unbiased

Given a coin with an unknown bias p, how
do we get an unbiased coin flip?
HEADS
TAILS
or
Flip Again
Making a Biased Coin Unbiased

Given a coin with an unknown bias p, how
do we get an unbiased coin flip?
HEADS
p(1-p)
TAILS
(1-p)p
or
Flip Again
Weak Random Sources



Von Neumann’s coin flipping trick (1951)
was the first to get true randomness from
a weak random source.
Much research in TCS in 1980’s and 90’s
to handle weaker dependent sources.
Led to development of extractors and
connections to pseudorandom generators.
Alonzo Church


Lambda Calculus
Church’s Theorem


No decision procedure
for arithmetic.
Church-Turing Thesis

Everything that is
computable is
computable by the
lambda calculus.
The Lambda Calculus




Alonzo Church 1930’s
A simple way to define and manipulate
functions.
Has full computational power.
Basis of functional programming
languages like Lisp, Haskell, ML.
Lambda Terms



x
xy
lx.xx


lxy.yx


Function Mapping x to xx
Really lx(ly(yx))
lxyz.yzx(luv.vu)
Basic Rules

a-conversion


b-reduction


lx.xx equivalent to ly.yy
lx.xx(z) equivalent to zz
Some rules for appropriate restrictions on
name clashes

(lx.(ly.yx))y should not be same as ly.yy
Normal Forms


A l-expression is in normal form if one
cannot apply any b-reductions.
Church-Rosser Theorem (1936)


If a l-expression M reduces to both A and B
then there must be a C such that A reduces to
C and B reduces to C.
If M reduces to A and B with A and B in
normal form, then A = B.
Power of l-Calculus


Church (1936) showed that it is impossible
in the l-calculus to decide whether a term
M has a normal form.
Church’s Thesis


Expressed as a Definition
An effectively calculable function of the
positive integers is a l-definable function of
the positive integers.
Computational Power

Kleene-Church (1936)


Computing Normal Forms has equivalent
power to the recursive functions of Turing
machines.
Church-Turing Thesis

Everything computable is computable by a
Turing machine.
Andrei Nikolaevich Kolmogorov



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
Measure Theory
Probability
Analysis
Intuitionistic Logic
Cohomology
Dynamical
Systems
Hydrodynamics
Kolmogorov Complexity

A way to measure the amount of
information in a string by the size of the
smallest program generating that string.
K  x   min  p : U  p   x
p
Incompressibility Method



For all n there is an x, |x| = n, K(x)  n.
Such x are called random.
Use to prove lower bounds on various
combinatorical and computational objects.



Assume no lower bound.
Choose random x.
Get contradiction by giving
a short program for x.
Incompressibility Method

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Ramsey Theory/Combinatorics
Oracles
Turing Machine Complexity
Number Theory
Circuit Complexity
Communication Complexity
Average-Case Lower Bounds
Complexity Uses of K-Complexity



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Li-Vitanyi ’92: For Universal Distributions
Average Case = Worst Case
Instance Complexity
Universal Search
Time-Bounded Universal Distributions
Kolmogorov characterizations of
computational complexity classes.
Rest of This Talk


Measuring sizes of sets using Kolmogorov
Complexity
Computational Depth to measure the
amount of useful information in a string.
Measuring Sizes of Sets

How can we use
Kolmogorov
complexity to
measure the size
of a set?
Measuring Sizes of Sets

Strings of length n
How can we use
Kolmogorov
complexity to
measure the size
of a set?
Measuring Sizes of Sets

n
A
Strings of length n
How can we use
Kolmogorov
complexity to
measure the size
of a set?
Measuring Sizes of Sets

n
A
Strings of length n

How can we use
Kolmogorov
complexity to
measure the size
of a set?
The string in An of
highest Kolmogorov
complexity tells us
about |An|.
Measuring Sizes of Sets

n
A
Strings of length n

There must be a
string x in An such
that K(x) ≥ log |An|.
Simple counting
argument, otherwise
not enough programs
for all elements of An.
Measuring Sizes of Sets

n
A
Strings of length n

If A is computable, or
even computably
enumerable then
every string in An has
K(x) ≤ log |An|.
Describe x by A and
index of x in
enumeration of
strings of An.
Measuring Sizes of Sets

If A is computable
enumerable then
max K  x   log A
n
n
A
Strings of length n
xA
Measuring Sizes of Sets in P


n
A
Strings of length n
What if A is efficiently
computable?
Do we have a clean
way to characterize
the size of A using
time-bounded
Kolmogorov
complexity?
Time-Bounded Complexity

Idea: A short description is only useful if
we can reconstruct the string in a
reasonable amount of time.
K
t
p : U  p   x in t  x 
 x   min

p

steps
Measuring Sizes of Sets in P

n
A
Strings of length n

It is still the case that
some element x in An
has Kpoly(x) ≥ log |A|.
Very possible that
there are small A with
x in A with Kpoly(x)
quite large.
Measuring Sizes of Sets in P

n
A
Strings of length n
Might be easier to
recognize elements in
A than generate
them.
Distinguishing Complexity

Instead of generating the string, we just
need to distinguish it from other strings.
U ( p, x )  1




t
KD  x   min  p :
U ( p, y )  0 for y  x

p


U
(
p
,
y
)
uses
t
(|
x
|

|
y
|)
time


Measuring Sizes of Sets in P

Ideally would like
max KD
n
A
xA


Strings of length n
poly
 x   log
True if P = NP.
Problem: Need to
distinguish all pairs of
elements in An
n
A
Measuring Sizes of Sets in P

Intuitively we need
max KD
xA

poly
 x   log
n
A
2
 2 log A
n
Buhrman-Laplante-Miltersen (2000) prove
this lower bound in black-box model.
Measuring Sizes of Sets in P

Buhrman-Fortnow-Laplante (2002) show
max KD
poly
xA

 x   2 log
n
A
We have a rough approximation of size
log A  max KD
n
xA
poly
 x   2 log
n
A
Measuring Sizes of Sets in P

Sipser 1983: Allowing randomness gives a
cleaner connection.
max KD
xA

poly
 x | r   log
n
A
Sipser used this and similar results to
show how to simulate randomness by
alternation.
Useful Information

Simple strings convey small amount of
information.


Random string have lots of information


00000000000000000000000000000000
00100011100010001010101011100010
Random strings are not that useful
because we can generate random strings
easily.
Logical Depth


Chaitin ’87/Bennett ’97
Roughly the amount of time needed to
produce a string x from a program p
whose length is close to the length of the
shortest program for x.
Computational Depth




Antunes, Fortnow, Variyam and van
Melkebeek 2001
Use the difference of two Kolmogorov
measures.
Deptht(x) = Kt(x) – K(x)
Closely related to “randomness deficiency”
notion of Levin (1984).
Applications

Shallow Sets



Generalizes random and sparse sets with
similar computational power.
L is “easy on average” iff time required is
exponential in depth.
Can easily find satisfying assignment if
many such assignments have low depth.
1903 – A Year of Geniuses

Several great men that helped create the
fundamentals of computer science and set the
stage for computational complexity.
2012 - The Next Celebration



Alan Turing
Born:
June 23, 1912
London, England
Died:
June 7, 1954
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Church, Kolmogorov and von Neumann: Their Legacy …