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Church, Kolmogorov and von Neumann: Their Legacy Lives in Complexity

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Church, Kolmogorov and von Neumann:Their Legacy Lives in Complexity

Lance Fortnow

NEC Laboratories America

Kolmogorov

Church

von Neumann

- Born:April 25, 1903Tambov, Russia
- Died:Oct. 20, 1987

- Born:June 14, 1903Washington, DC
- Died:August 11, 1995

- Born:Dec. 28, 1903Budapest, Hungary
- Died:Feb. 8, 1957

- Born:Feb. 22, 1903Cambridge, England
- Died:January 19, 1930
- Founder of Ramsey Theory

- Logic
- Concrete Complexity
- Complexity Classes
- Parallelism
- Algorithms
- Computational Geometry

- Quantum
- Logic
- Game Theory
- Ergodic Theory
- Hydrodynamics
- Cellular Automata
- Computers

- Every finite zero-sum two-person game has optimal mixed strategies.
- Let A be the payoff matrix for a player.

- 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.

- Given a coin with an unknown bias p, how do we get an unbiased coin flip?

- Given a coin with an unknown bias p, how do we get an unbiased coin flip?

HEADS

TAILS

Flip Again

or

- Given a coin with an unknown bias p, how do we get an unbiased coin flip?

HEADS

p(1-p)

TAILS

(1-p)p

Flip Again

or

- 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.

- Lambda Calculus
- Church’s Theorem
- No decision procedure for arithmetic.

- Church-Turing Thesis
- Everything that is computable is computable by 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.

- x
- xy
- lx.xx
- Function Mapping x to xx

- lxy.yx
- Really lx(ly(yx))

- lxyz.yzx(luv.vu)

- a-conversion
- lx.xx equivalent to ly.yy

- b-reduction
- 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

- 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.

- 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.

- 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.

- Measure Theory
- Probability
- Analysis
- Intuitionistic Logic
- Cohomology
- Dynamical Systems
- Hydrodynamics

- A way to measure the amount of information in a string by the size of the smallest program generating that string.

- 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 givinga short program for x.

- Ramsey Theory/Combinatorics
- Oracles
- Turing Machine Complexity
- Number Theory
- Circuit Complexity
- Communication Complexity
- Average-Case Lower Bounds

- Li-Vitanyi ’92: For Universal Distributions Average Case = Worst Case
- Instance Complexity
- Universal Search
- Time-Bounded Universal Distributions
- Kolmogorov characterizations of computational complexity classes.

- Measuring sizes of sets using Kolmogorov Complexity
- Computational Depth to measure the amount of useful information in a string.

- How can we use Kolmogorov complexity to measure the sizeof a set?

- How can we use Kolmogorov complexity to measure the sizeof a set?

Strings of length n

- How can we use Kolmogorov complexity to measure the sizeof a set?

An

Strings of length n

- How can we use Kolmogorov complexity to measure the sizeof a set?
- The string in An of highest Kolmogorov complexity tells us about |An|.

An

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.

An

Strings of length n

- If A is computable, or even computably enumerable then every string in An hasK(x) ≤ log |An|.
- Describe x by A and index of x in enumeration of strings of An.

An

Strings of length n

- If A is computable enumerable then

An

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?

An

Strings of length n

- Idea: A short description is only useful if we can reconstruct the string in a reasonable amount of time.

- 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.

An

Strings of length n

- Might be easier to recognize elements in A than generate them.

An

Strings of length n

- Instead of generating the string, we just need to distinguish it from other strings.

- Ideally would like
- True if P = NP.
- Problem: Need to distinguish all pairs of elements in An

An

Strings of length n

- Intuitively we need
- Buhrman-Laplante-Miltersen (2000) prove this lower bound in black-box model.

- Buhrman-Fortnow-Laplante (2002) show
- We have a rough approximation of size

- Sipser 1983: Allowing randomness gives a cleaner connection.
- Sipser used this and similar results to show how to simulate randomness by alternation.

- Simple strings convey small amount of information.
- 00000000000000000000000000000000

- Random string have lots of information
- 00100011100010001010101011100010

- Random strings are not that useful because we can generate random strings easily.

- 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.

- 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).

- 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.

- Several great men that helped create the fundamentals of computer science and set the stage for computational complexity.

- Alan Turing
- Born:June 23, 1912London, England
- Died:June 7, 1954