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

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.

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?

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

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.

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.

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.

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.

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.

Europe in 1970

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

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.

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.

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

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.

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.

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

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.

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

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”

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.

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.

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.

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

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

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.

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.

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.

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

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

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

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

The Future P = NP?

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.

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

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!