1. Lance Fortnow NEC Research Institute
History of Complexity
Lance Fortnow
NEC Research Institute

2. History of Logic
Edited by Dirk van Dalen, John Dawson and Akihiro Kanamori.
Published by Elsevier.
Chapter: History of Complexity
Authors: Lance Fortnow and Steve Homer
This talk
Lessons learned from writing this chapter.

3. Lesson One Impossible to please everyone. Resolutions
Often disagreements on who is responsible for what and which results are important.
Everyone wants a mention.
Resolutions
Can’t mention everything in 75 minutes.
Opinions in this talk are due to me alone.
How do I mention everyone?

4. Birth of Computational Complexity
General Electric Research Laboratory Niskayuna, New York
November 11, 1962

5. Birth of Computational Complexity
Juris Hartmanis and Richard Stearns 1965
On the Computational Complexity of Algorithms, Transactions of the AMS
Measure resources, time and memory, as a function of the size of the input problem.
Basic diagonalization results: More time can compute more languages.

6. No “Immaculate Conception”
Idea of algorithm goes back to ancient Greece and China and beyond.
Cantor developed diagonalization in 1874.
Kleene, Turing and Church formalized computation and recursion theory in 30’s.
Earlier work by Yamada (1962), Myhill (1960) and Smullyan (1961) that looked at specific time and space bounded machines.

7. Complexity in the ’60s Better simulations and hierarchies
Relationships between time and space, deterministic and nondeterministic.
Savitch’s Theorem
Blum’s abstract complexity measure
Union, speed-up and gap theorems.

8. Polynomial Time
Cobham (1964) – Independence of polynomial-time in deterministic machine models.
Edmonds (1965)
Argues that polynomial time represents efficient computation.
Gives informal description of nondeterministic polynomial time.

9. P versus NP Gödel to von Neumann letter in 1956.
Cook showed Boolean formula satisfiability NP-complete in 1971.
Karp in 1972 showed several important combinatorial problems were NP-complete.
Industry in the 1970’s of showing that problems were NP-complete.

10. Europe in 1970

11. Complexity in the Soviet Union
Perebor – Brute Force Search
1959 – Yablonski – On the impossibility of eliminating Perebor in solving some problems of circuit theory.
1973 – Levin – Universal Sequential Search Problems

12. Importance of P versus NP Today
Thousands of natural problems known to be NP-complete in computer science, biology, economics, physics, etc.
A resolution of the P versus NP question is the first of seven $1,000,000 prizes offered by Clay Mathematical Institute.
We are further away than ever from settling this problem.

13. Structure of NP
Ladner – 1975 – If P different than NP then there are incomplete sets in NP.
Berman-Hartmanis – 1977 – Are all NP-complete sets isomorphic?
Mahaney – 1982 – Sparse complete sets for NP imply P = NP.

14. Alternation
Development of the polynomial-time hierarchy by Meyer and Stockmeyer in 1972.
Chandra-Kozen-Stockmeyer – 1981
Alternating Time = Space
Alternating Space = Exponential Time

15. Relativization Baker-Gill-Solovay – 1975
All known techniques relativize.
There exists oracles A and B such that
PA = NPA
PB  NPB
Many other relativization results followed.

16. Oracles and Circuits
Is there an oracle where the polynomial-time hierarchy is infinite or at least different than PSPACE?
Sipser relates to question about circuits:
Can parity be computed by a constant-depth circuit with quasipolynomial number of gates?
In 1983, Sipser solves an infinite version of this question.

17. Oracles and Circuits
Furst, Saxe Sipser/Ajtai - Parity does not have constant depth poly-size circuits.
Yao – 1985 – Separating the polynomial-time hierarchy by oracles
Håstad – 1986 – Switching lemma and nearly tight bounds for parity

18. Circuits and Polytime Machines
1975 – Ladner – Every language in P has polynomial-size circuits.
1980 – Karp-Lipton – If NP has poly-size circuits then polytime hierarchy collapses.
To show P  NP, need only show that some problem in NP does not have poly-size circuits.

19. Circuit Results
Razborov – 1985 – Clique does not have poly-size monotone circuits.
Razborov-Smolensky – 1987 – Lower bounds for constant depth circuits with modp-gates.

20. The Fall of Circuit Complexity
No major results in circuit complexity since 1987, particularly for non-monotone circuits.
Razborov – 1989 – Monotone techniques will not extend to non-monotone circuits.
Razborov-Rudich – 1997
“Natural Proofs”

21. Different Models
As technology changes so does the notion of what is “efficient computation”.
Randomized, Parallel, Non-uniform, Average-Case, Quantum computation
Complexity theorists tackle these issues by defining models and proving relationships between these classes and more traditional models.

22. Randomized Computation
Solovay-Strassen – 1977 – Fast randomized algorithm for primality.
1977 – Gill
Probabilistic Classes: ZPP, R, BPP
Sipser – 1983 – A complexity theoretic approach to randomness
BPP in polynomial-time hierarchy.
Various oracle results like BPP = NEXP.

23. Derandomization
Cryptographic one-way functions give pseudorandom generators that can save on randomness.
Hard languages in nonuniform models give pseudorandom generators.
Derandomization results for space-bounded classes.

24. Randomness and Proofs Goldwasser-Micali-Rackoff – 1989
Cryptographic primitive for not releasing information.
Babai-Moran – 1988
Classifying certain group problems.
Interactive Proof Systems
Public = Private; One-sided error

25. Power of Interaction ’89-’91
IP = PSPACE
MIP = NEXP
FGLSS – Limits on approximation based on interactive proof results.
NP = PCP(log n,1)
Better bounds on PCPs and approximation

26. Audience Poll What was more surprising in early 90’s?
The power of interactive proofs and their applications to hardness of approximation.
The end of the cold war, the collapse of the Soviet Union and the Eastern Bloc, the fall of the Berlin wall and the reunification of Germany.

27. The Role of Mathematics
Computation Complexity has often drawn insights, definitions, problems and techniques from many different branches of mathematics.
As complexity theory has evolved, we have continued to use more sophisticated tools from our mathematician friends.

28. Logic Complexity has its foundations in logic.
Turing machines, Diagonalization, Reductions, and the polynomial-time hierarchy.
Logical characterizations of classes have led to NL = coNL and formalization of MAX-SNP.
Proof complexity studies limitations of various logical systems to prove tautologies.

29. Probability Probabilistic Models Basic Techniques Probabilistic Method
BPP, Interactive Proofs, PCPs
Resource-Bounded Measure
Basic Techniques
Chernoff Bounds
Probability of OR bounded by Sum of Prob
Dependent Variables
Probabilistic Method

30. Algebra NC1 = Bounded-Width Branching Programs
Polytime Hierarchy reduces to Permanent
Mod3 requires large constant-depth parity circuits.
Interactive Proofs/PCPs
Coding Theory

31. Discrete Math/Combinatorics
Lower Bounds
Circuit Complexity
Branching Programs
Proof Systems
Ramsey Theory/Probabilistic Method
Expanders/Extractors

32. Information Theory Entropy Kolmogorov Complexity Cryptography
VLSI/Communication Complexity
Parallel Repetition
Quantum

33. The Future
P = NP?

34. Showing P  NP Other areas of mathematics
Algebraic Geometry
“Higher Cohomology”
New techniques for circuits, branching programs or proof systems.
Completely new model for P and NP.
Diagonalization.

35. Besides P = NP? Same Old, Same Old Handling new models
Complex Systems: The Other “Complexity”
Financial Markets, Biological Systems, Weather, The Internet
The Big Surprise

36. Conclusions Juris Hartmanis Notebook Entry 12/31/62:
“This was a good year.”
This was a good forty years.
Who knows what the future will bring?
Fasten your seatbelts!