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## History of Complexity

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**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. • 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 ofComputational Complexity**General Electric Research LaboratoryNiskayuna, New York November 11, 1962**Birth ofComputational 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.**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 ofMAX-SNP. • Proof complexity studies limitations of various logical systems to prove tautologies.**Probability**• Probabilistic Models • 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!