On the nature and future of ToC

On the nature and future of ToC
Avi Wigderson Institute for Advanced Study Picking t-shirts מיום שחרב בית-המקדש ניתנה הנבואה לשוטים Since the destruction of the Temple prophecy was given to fools Det er vanskeligt at spaa, især naar det gælder Fremtiden It is difficult to make predictions, especially about the future

About my 2 cents 40 years in the field: my views evolved
Not so much… past plenaries: Depth through Breadth STOC ’04 The art of Reduction FCRC ‘07 Position paper with Oded Goldreich: ToC: a scientific perspective ‘96 - The arguments got far stronger - The world is changing much faster

Independent major Theory of Computation academic discipline
Mathematics Computer Science Statistics Social Science Independent major academic discipline Theory of Computation Optimization core intersects other spheres question marks in areas – we don’t understand Biology Economics Information& Coding Physics Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Mathematics Computer Science Statistics Social Science Optimization
I can put in the center anything I like. Indeed, this is an ancient tradition Biology Economics Information& Coding Physics Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Ptolemy's geocentric model
Could be dogmatic – hold for 1000 years with the strength of religion Ptolemy's geocentric model

ML Transportation Emotions Medicine Food Decisions Technology Law
Or economic forces Technology Law Science Recreation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Computer Science & Technology
Transportation Medicine Emotions Food Computer Science & Technology ML Decisions Or economic forces Technology Law Science Recreation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Computer Science & Technology
transportation Medicine Emotions Computer Science & Technology Food Decisions Or economic forces Technology Law Science recreation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Computer Science & Technology Science
Transportation Medicine Computer Science & Technology Emotions Food Science Decisions Or economic forces Technology Law Science Recreation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Theory of Computation Mathematics Computer Science Statistics Social
Optimization core intersects other spheres question marks in areas – we don’t understand Biology Economics Information& Coding Physics Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Must grow! ToC Mathematics Computer Science Statistics Social Science
Optimization We are a factor smaller than each of the fields we interact with. As Christos says “ToC replaces Math as the queen of sciences” Biology Economics Information& Coding Physics Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Proof by exhaustion: Laundry lists
Tons of papers

Beginnings: The dual life of ToC
CS&E MATH Young Old Impatient Patient Practice Aesthetics Utility Abstraction Study computer Rigorously! systems! Freely! Modeling Complexity challenges zoo ToC

ToC & CS: Great theories
Turing machine (toc  cs) Cellular automata (life, replication, tolerance) Automata (Non-deterministic,Probabilistic) Formal languages (programming, compilation) Verification (model def & checking) Data bases (relational, hierarchical,…) Complexity (NP, completeness, reductions) Efficient algs & data structures Cryptography (Internet, e-commerce) Distributed sytems (asynchrony, atomicity) Randomness (algs, PRGs, weak sources) Learning (Perceptrons, PAC,…)

ToC & CS: great theories
- Turing machine (toc  cs) Automata (Non-deterministic,Probabilistic) Cellular automata (life, replication, tolerance) Formal languages (programming, compilation) Verificaion (model def & checking) Data bases (relational, …) Complexity (NP, completeness, reductions) Efficient algs & data structures Cryptography (Internet, e-commerce) Distributed sytems (asynchrony, atomicity) Randomness (algs, PRGs, weak sources) Learning (Perceptrons, PAC,…) Parallel algs (NC, networks) Space comp. (NL=coNL, SL=L,…) Networks (Sorting, comm, expanders,…) Quantum computing (crypto,info,physics) Online algs (competitive analysis) Approx algs (PCP, hardness, hierarchies,…) Sublinear algs (Property testing, local decoding) Differential Privacy (Adaptive science,…) Learn Why & How these theories developed

Turing Award for ToC work
Turing machine Cellular Automata Automata Formal languages Verificaion Data bases Complexity Algorithms Cryptography Randomness Distributed systems Learning Minsky’69, Dijkstra’72, Knuth’74, Rabin-Scott’76, Floyd’78, Hoare’80, Codd’81, Cook’82, Karp’85, Hopcroft-Tarjan’86, Milner’91, Hartmanis-Stearns’93, Blum’95, Pnueli’96 Yao’00 Rivest-Shamir-Adleman’02 Liskov’08 Valiant’10 Pearl’11 Micali-Goldwasser’12 Lamport’13 Diffie-Hellman’15 40% of the TA’s are for pure ToC work.

Crypto revolutions Mind games! Public-key crypto Predated RSA
& enabled the Internet & e-commerce Public-key crypto RSA Formal foundations Zero Knowledge proof for every theorem - Oblivious computation for every function - Completeness theorems for protocols - Universal composability (Internet setting) - Constant-depth crypto - Non-Black-Box crypto - PIR/Delegation (interactive proofs for clouds) - Homomorphic encryption - Indistinguishability Obfuscation (?) Circuit comp, Learning, Pseudorandomnes,PCP,…… Zooming in on only one of the many theories, crypto, we see it is made of many important separate breakthroughs (as many others) Most invented and developed without any direct practical motivation! Enabled huge practical impact – e.g. without promised security of the late 70s early 80s work, the would exist no e-commerce applications, and hence no economic motivations to explain the huge industry investments in the creation of the Internet. True for all other theory developments.

Other ZK appearances Physical ZK proofs: - Child education
Nuclear disarmament Anonymous DNA testing Practical ZK proofs: Anonymous cash: Bitcoin, Z-cash,… Public ledgers … Science- Quantum SZK proofs: Black-hole firewall paradox,… We can gain zoom in on any breakthrough, such as the definition and then power of Zero-Knowledge proofs. Here we list only much later developments, very differently motivated and applied than the original idea. [Naor, Reinglod,…] [Barak,…] [Philippe,… ] [Ben Sasson, Walfish,…] [Harlow, Hayden]

ToC & Math Users Collaborators Suppliers Theory of Computation Logic
Combinatorics Algebra Users Collaborators Suppliers Theory of Computation Number theory ToC evolution required results from more and more diverse areas of Math, and gradually many became collaborations and interactions. Geometry Lattice theory Analysis Algebraic Geometry Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

IM ?? ToC  Math Isoperimetric inequalities
Stability (inverse) theorems Metric embeddings Explicit constructions Structure vs. randomness thms Exponential sum estimates (Un)Definability results Statistical sample bounds Lots of new questions & collaborations with practically all math areas! ToC was naturally lead to investigate and prove math results in diverse areas. An integral part of understanding computation!

Two recent breakthroughs
Kadison-Singer (open 65 years) Linear solvers, spectral sparsifiers, Insistence on prob O(nlogn)  det O(n) Hyperbolic polynomials, approx permanent Free probability, Electrical networks,… Explicit Ramsey graphs (open 70 years) Voting, influence, pseudorandomness, circuit complexity, extractors, finding entropy, sum-product theorem,… [Marcus-Spielman-Srivastava,…] [Chattopadhyay-Zuckerman, Cohen]

Two more Efficient characterization of primes (200 y)
Taylor-made prob alg, derandomization, polynomials, PIT, number theory estimates Reverse Minkowaski inequality (100 years) Lattices, Integer programming, Additive combinatorics, Convex geometry, Crypto, Brownian motion,… [Agrawal-Kayal-Saxena] [Dadush-Regev]

Two more First-order variable hierarchy strict (20yrs)
Constant depth circuits, random restrictions, sensitivity, cliques in G(n,p) Operator scaling algorithm (20 years) PIT, matrix scaling, perfect matching, non commutative fields, commutative invariant theory, quantum information theory, Brascamp-Lieb ineuqalities,… [Rossman] Past results in the same spirit [ Fagin, Immerman, Ajtai, Gurevich,..] [Garg-Gurvits-Olivera-Wigderson]

Million $$ questions (7 Clay millennium problems)
The Riemann hypothesis (150 years) About pseudorandom walks Navier-Stokes equations (250 years) About Turing universality of water (?) P=NP? (50 years) Also about computational complexity [Mertens] Characterization of RH via Mobius function cancellations [Tao] See Arxiv & Quanta magazine article by Erica Klarreich

Nearby fields Foundational discoveries Use interactivity of ToC
Ever since Yao, hybrid arguments use randomization to fix the suffix of the PRG

Optimization (ORSA/TIMS 3000 attn.)
PCP thm (prog testing, game th, coding th) Harness of approx (analysis Bool fns, long/ short codes, univ approx algs, dichotomies) LP/SDP hierarchies (rand rest, Fourier,…) Extended formulations (exp. lower bds! non-negative rank, quantum comm. Comp.) Continuous algs flows, spectral, descents interior pt, ellipsoid, scaling,… Algorithmic Game Theory Opt. aspects of auctions, mechanism design, equilibria,… [Arora-Safra-Lund-Motwani-Sudan-Szegedy,…] [Feige-Goldwasser-Lovasz-Safra-Szegedy,….] [Arora-Bollogs-Lovasz,……Lee-Raghavendra-Steurer] [Yannakakis, Fiorini et al, … Rothvoss,…] [Spileman, …Madry,…] [Nisan, Papadimitriou, Tardos,…]

Coding Theory (ISIT 700 attn)
Local decoding (PCP, hardness, prog/prop testing, geometry, Kakeya, PIT, alg comp) List decoding (Optimal bounds! ECCs, Crypto, extractors, PCPs,…) LDCP (linear time enc/dec! expanders, Tornado, Fountain, belief propagation,…) RS/RM/BCH (Wt. enumeration, capacity achieving, influence, threshold, add. comb ) [Blum,Luby,Rubinfeld,Kannan,Sudan,Dvir, Saraf,Kayal,…] [Goldreich-Levin,Guruswami-Rudra,Guruswami-Umans-Vadhan...] [Sipser,Spielman, Luby, Shokrollahi,…] [Sudan, Kaufman, Lovett, Abbe, Shpilka, W,…]

Information Theory Statistics Numerical analysis Control theory ………
Information & communication comp  Interactive information theory [Schulman, Braverman, Raz, Kol, Haeupler, Tauman-Kalai,…] Property testing of distributions, sample bounds, concentration inequalities [Rubinfeld, Valiant G+P, Daskalakis,…] Approximation theory [Viola, Servedia, De, Kane, Spileman, Madry, …] [Reif, Lichtenstein-Linial-Saks] Very old,….Incentives/prices in AGT, Control in social networks dynamics

Natural & Social Sciences

Computation Church-Turing Thesis Turing machine can emulate any computation! Computation: every evolution by a sequence of simple, local steps, of bits in computers computers in a network neurons in the brain atoms in matter cells in living tissue bacteria in Petri dish prices in a market individuals in populations deductions in a proof proteins in a cell qbits in entanglement friends on Facebook… ToC Methodology! info process: Considering practically all natural and artificial processes, as information processes, is the connection of all sciences to computation, and the source of numerous great challenges to ToC.

Understanding Computation great challenge & responsibility Computation: every evolution by a sequence of simple, local steps, of bits in computers computers in a network neurons in the brain atoms in matter cells in living tissue bacteria in Petri dish prices in a market individuals in populations deductions in a proof proteins in a cell qbits in entanglement friends on Facebook… Models Algorithms Resources Efficiency Upper bds Lower bds Reductions Completeness Abstraction Play ToC Methodology! Many of these principles are used in all sciences. In ToC, these principles are the core of the field, and are themselves objects of study. It is unlikely, but not out of the question, that to understand e.g. bacteria in Petri dishes, one may need to extend the model to study “quantum bactera” in “hyperbolic Petri dishes”, or “hyperbolic bacteria” in “quantum Petri dishes”. The freedom to do so has time and again Lead to leaps of understanding and unexpected connections in ToC. Quantum hyperbolic

ToC  Sciences Scientific revolution: fundamental, new
Questions Methodology Definitions - Principles Results Connections Insights Standards … Sciences are conservative! (decades-centuries) Heliocentric model, Evolution, Bacteria, General Relativity, Quantum mechanics, Quasi-periodic crystals… Early Successes: NP-completeness 1000s of papers, by scientists, in science journals! “P  NP” as a law of nature: Intractability as a modeling guide, like 2nd law of thermodynamics Common goals (understanding natural processes) but very different methodologies – promises great synergies and collaboration.

Initial, Formal, Predictive !
Biology Morphogenesis, Self-Replication, Tolerance Genome, Comput. biology, Phylogenetics Natural algs / Influence systems Neurobiology: Circuits of the Mind,… Evolution as (a learning) algorithm Sex is an algorithm High level, foundational questions. Initial models incorporating computational complexity, which are formal, and predictive, and in some cases deviate from classical models. [Turing, von Neumann] [Karp, Meyers, Pevzner, Shamir]: no genome w/out the sequencing algorithms [Warnow,Mossell,…] Sharp bounds on number of samples needed for reconstruction under noise (mutations) [Livnat, Papadimitriou, Pippenger…] what is the “fittest” in survival of the fittest? Algs at the genes level [Chazelle] E.g. Flocking, Facebook,…Time evolving markov chains, depending of evolving spheres of influence. Algs vs. dynamical systems! [TuringValiant] A concrete model, consistent with physiology, implementing basic cognitive functions. [Darwin-Kelvin, Valiant] Age of the universe debate, Initial, Formal, Predictive !

Physics Ever since Yao, hybrid arguments use randomization to fix the suffix of the PRG

ToC & Stat Phys 1-dim Cellular Automata with eternal memory under constant noise rate MCMC rigorous analysis, conductance, expanders, approximating the permanent,… Spatial & temporal equivalences of Statistical Physics systems Threshold phenomena (spin, K-sat) rigorous! (PCP, Approximation, Game Theory, Stability in Comb., Message passing, Belief prop, Coding Th) Fundamental, general questions. [Gacs] Local system, constant noise rate, 2 Gibbs states [Broder, Jerrum-Sinclair, Dyer-Frieze-Kannan…] Markov chains – rigorous convergence analysis [Weitz, Sinclair,…]Long range correlations vs fast convergence of Glauber dynamics [Achlioptas, Peres, Sly, Sun

Quantum mechanics Factoring algorithm  Billions of $$
Fundamental questions about QM BQP = Adiabatic Hamiltonian dynamics, completeness Area Law & entanglement Black hole firewall paradox AdS/CFT correspondence Recent stuff I heard at IAS Physics talks: Circuits Quantum circuits P, NP, BQP, SZK Quantum state complexity? Do these belong in Quantum Mechanics? [Shor] Amazing technologies, challenging questions on quantum mechanics, decoherence,… [Aaronson] Other examples – Boson sampling etc. [Aaronov, Vazirani,…] Adiabatic- - gradually changing the ground state of a local Hamiltonian [Kitaev,…] [Hastings, Vazirani,…Verlinde] Rigorous, sharp quantitative results…micro explanations of entanglement & qubits to Einstein’s equations [Hayden,Harlow] Computational explanation, via quantum SZK. Assumptions  prediction [Suskingd, Kitaev, Maldacena, Aaronson} AdS (gravity) = CFT (quantum field theory). Use quantum circuit complexity as a measure of time. Many growing connections in which computational complexity seems to make its way into quantum mechanics and more generally theoretical physics.

Economics Complexity of equilibria (hardness)
Algorithmic Game Theory (& optimization) Auctions, Mechanism design Price of anarchy 2008 crash (information asymmetry arising from computational intractability) [Papadimitriou,Ding-Chen],… Recent communication complecity and crypto hardness [Nisan, Tardos,…]Lower bounds and algorithms. [Daskalakis, Weinberg,…] Very general reductions between truthful auctions and opt problems, illuminating both. [Papadimitriou, Tardos, Roughgarten,..] [Arora-Barak-Ge]Efficient, transparent markets: Info th vs Computational (Financial Derivatives, CDOs…), Market crash 2008

Social Sciences Six degrees of separation Markets, crowds and networks
Internet, WWW, Social, Financial nets Processes in nets (epidemics, gossip, influence,…) Dynamics of nets (growth, individual, aggregate…) Differential privacy (policy, law, health…) [Milgram, Kleinberg] The algorithmic puzzle is far more interesting than the graph theoretic one (diameter!!) [Easley,Kleinberg] Making some aspects of social science a quantitative study: processes & games on(information) networks … [Dwork,….] Surveys on the science, and it social impacts

Philosophy Novel, formal meanings & uses -Add comp comp -Stir well
collusion learning coordination ontology conflict prediction creativity privacy entropy process equilibrium proof evolution randomness game secret induction simultaneity intelligence strategy interaction synchrony lnowledge truth language verification Novel, formal meanings & uses -Add comp comp -Stir well -Put in oven [Aaronson] Complexity Theory for Philosophers This list has to be read slowly. It is hard to think of a science which has embraced the challenge of understanding so many fundamental notions – for ToC they arose naturally.

What our small community has achieved
The reader is encouraged to study in depth the achievements of ToC, and the great scientific revolutions and their impacts. What our small community has achieved in a short period is comparable to the greatest scientific revolutions of all time!

What went right? What has changed? How should we act?
Our science & mission Our community Our education Our politics Ever since Yao, hybrid arguments use randomization to fix the suffix of the PRG

What kind of science is ToC?
Mathematical? Technological? Natural? Social? Experimantal? Philosophical? (all answers correct) Independent academic discipline! Ever since Yao, hybrid arguments use randomization to fix the suffix of the PRG

Mission: understand computation in all forms
Mathematics Computer Science Statistics Social Science Must grow! Must grow-up! ToC Optimization Are part in CS and Math are larger only for historical reasons – will grow in all others. We are a factor smaller than each of the fields we interact with. As Christos says “ToC replaces Math as the queen of sciences” Biology Economics Information& Coding Physics Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Communication Complexity
We have grown, organically Mathematics Computer Science Statistics Theoretical Computer Science Learning Randomness Verification Distributed Lower bounds Communication Complexity Quantum Computing Logic AGT Property testing Core ToC Comp. Comp. Structure, Insights, Interconnectivity,… Theoretical Computer Science Social Science Crypto We have grown beautifully, around conferences! What happens at the core happens at STOC & FOCS & ITCS & SODA and the many satellite/specific ToC conferences. Some we are conscious of, some not. I will discuss some of the interconnectivity later. Biology Economics Physics Must preserve common goals, language, confs,… Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation Core TCS Machine Learning Pseudorandomness Cryptography Coding Theory Sublinear Algorithms Quantum Computing Interactive Computation

Community: Culture & People key to past+future success
Open-minded Collegial Democratic Driven Diverse Cohesive Educational Agile Smart Self confidence ? Online dating profile examples Must preserve culture and quality Challenges we have a-plenty Chair Dean/Funding/ ask: Is your work practical? Does it explain reality? Answer: My science is!

Education, confidence & independence
College: ToC major! Teach our science and its history, as happened, in full glory, e.g. Algs Complexity/logic Data reps Modeling principles Modeling principles Graph algs Circuit comp Comp & math Approx algs Communication comp Comp & biology Parallel algs Crypto Comp & physics Dist algs Learning Comp & economics Prob algs Semantics Comp & Soc. Sci Sublinear algs Verification Comp & Philosophy Heuristics Formal languages …… Interactive, cross-fertilizing ideas across all I,II,III,… Great achievements by other sciences I,II,III,… Building self-confidence must start at an early age! Many students come with plenty of programming experience Imagine the value of such education in academia and industry

Education: Basic knowledge
K-12, teaching colleges: ToC principles Save CS ed. from programming-oriented ed Write books Design curricula Give popular lectures “Computers cannot solve every problem!” “Algs create industries & cure disease” “Turing = Einstein, but far more influential” “Mult harder than Add? Prove it!” “World economy rests on comp. assumption”

Politics (growing-up)
CATCS: ToC growth & health requires (besides great research!!), also funding, positions, education, advocacy… Education: PR, popular lectures, papers Advocacy: Congress, NSF, universities Service: Chairs, deans, NSF officers,… Do what you do best! Internal competition* on resources: Do what you like best and promote it! *Vast resources flow to CS

Thank you! Make ToC Great again!

On the nature and future of ToC

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