Key%20Challenges%20for%20Theoretical%20Computer%20Science

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Key Challenges for Theoretical Computer Science

• Richard M. Karp

NSF, August 31, 2005
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What is Theoretical Computer Science ?

• TCS applies abstract models and mathematical
  reasoning to problems related to computation.

• Provides a set of tools, and ways of thinking
  applicable to a wide variety of applied problems.

• Contributes to national security through
  cryptographic protocols and to computational
  science through fundamental algorithms.

Its core fundamental and interrelated questions
about the nature of computation.
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Topics at the Core of TCS

• Algorithms and complexity of computation
• Computational limits of proof methods
• Logic and program verification
• The power of randomization
• Cryptography
• Quantum computation
• Distributed computation and communication
• Computational learning theory

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The Revolutionary Impact of Algorithms

• Optimization
• Scientific computing
• Cryptography
• Genome sequencing
• Compiler construction
• Algebraic computation
• Data structures

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Fundamental Questions Complexity and Algorithms

• Two possible worlds

• (P NP) Combinatorial search easy but
  cryptography impossible.

• (P ? NP) Combinatorial search hard but
  unbreakable cryptography possible

• Find best algorithms for multiplying numbers,
  Discrete Fourier transform and matrix
  multiplication.

• Which tautologies in propositional logic have
  short proofs?

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Fundamental Questions Nature Limits of Proofs

• Find limits of computationally sound interactive
  proofs, which prove a statement by performing a
  computation that would be infeasible if the
  statement were false.

• Can we prove that statement is true without
  revealing any additional information?

(Prove you earned lt100K without revealing
salary.)

• Can we design proofs that can be verified by
  spot checking rather than checking every step?

(Traditional proofs are only strong as weakest
link.)
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Fundamental Questions Randomness, Quantum, Crypto

• Does access to random numbers give more computing
  power?

• Are hypothetical computers based on principles of
  quantum mechanics more powerful?

• Is there cryptosystem where everyone can send
  encrypted message to Alice, but only she can
  read it?

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Fundamental Questions
Mechanism Design, Distributed Comp., Learning

• Can we cause self-interested agents to co-operate
  over the Internet?

• Can we reach agreement in the face of
  asynchronicity and faulty parties?

• What are the inherent limits on the ability of
  computers to infer patterns from examples?

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Evolution of TCS

• 60s

70s
80s
90s
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Evolution of TCS

• 60s

70s
80s
90s
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TCSs Greatest Strength Unexpected Pay-offs
Zero Knowledge
Machine learningHardness amp.
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Emerging Challenges

• Theory of networked computation
• Security and privacy
• Incentives, pricing and sharing
• Reliable communication
• Massive distributed data sets

• Formal methods for reliable systems.

• Ties to physical and biological sciences
• Statistical physics.
• Quantum computing
• Computational biology

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Theory of Networked Computation

• Emergence of large networks (e.g. the Web) is
  profound shift in focus of CS.

• Networks built, operated and used by parties w/
  diverse interests and varying degrees of
  cooperation and competition.

• Challenges build and manage large systems
  consisting of autonomous parties.

• Ensure rights of individuals and full and fair
  exploitation of shared resources.

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Internet Algorithmics

• Emerged with the spread of the Web.

• Produced significant results on

• Search and information retrieval
• Network protocols
• Error correction
• Peer-to-peer networks
• E-commerce
• Internet-based auctions
• Mechanism design
• Massive distributed data sets.

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Theory of Networked Computation Agenda

• Theoretical complement to GENI Initiative and
  Cyber-infrastructure program.

• Close in spirit to Pattersons SPUR
  manifestoSecurity, Privacy, Usability,
  Reliability.

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Security and Privacy

• Users today invoke complex financial interactions
  with as single click.

• Current design of the Internet based on trust.
  Inadequate protection against worms, viruses,
  spam and identity theft.

• Must ensure appropriate use of information by
  dynamic and potentially large set of authorized
  users.

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Formal Models of Security

• Security can not be tested by experimentation or
  simulation.

• We need quantitative measures of security with
  respect to realistic models of user behavior.

• Past TCS work cryptographic primitives (RSA,
  Diffie-Hellman, DES), protocols (signatures,
  e-commerce, secure interactions), study of
  protocol composition.

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Security Ongoing TCS Work

• Expand protocol design to address scale,
  complexity and interactivity of modern
  environment.

• Use economic theory to obtain security through
  positioning incentives

• New techniques for sanitizing public data,
  traceback, intrusion detection, etc..

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Incentives, Pricing and Sharing

• Networks are built, operated and used by multiple
  parties with diverse goals and interests.

• Algorithmic distributed mechanism design studies
  economic mechanisms that induce globally
  efficient behavior in self-interested agents.

• Builds on algorithms, economic theory and game
  theory.

• Areas of study auctions, routing, congestion
  control, caching, border gateway protocol,
  pricing of multicast, network design, price of
  anarchy.

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Massive Distributed Data Sets

• Robust trends in IT ever-decreasing cost of data
  storage, ever-increasing ubiquity of computers
  and networks, accelerating deployment of sensor
  networks and surveillance systems.

• New computational models data streaming,
  external memory and cache oblivious models,
  sampling, property testing, sublinear time
  algorithms.

• Randomization and approximation are essential.

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Massive Data Sets Challenges

• Data replication, placement, access and
  persistence.

• Security and privacy, strategic and adversarial
  behavior, complex data formats (images, video,
  audio)

• Personalized search, complex queries, full-text
  search, defenses against adversarial behavior by
  web page owners.

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Reliable Storage and Communication

• Maintaining integrity of data is a classical
  challenge to computing.
• Modern issues
• Explosion in amount of data
• Radical differences in nature of communication
  and storage media
• Communication medium Internet
• Storage medium Worldwide Web

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Reliable Storage Communications TCS
Achievements Challenges

• Achievements ability to correct more errors,
  faster error-correction algorithms, rateless
  codes, checkable codes, list decoding,
  computationally bounded channels.

• Challenges more powerful error-correction
  techniques, ultra-fast decoding, malicious
  errors, integration with network protocols such
  as multicast.

• Connections with probabilistically checkable
  proofs, cryptographic protocols, pseudorandom
  number generation.

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Complexity Theory of Networked Computation

• Needed A theory of the fundamental limits of
  networked computation.
• How is a networked computational problem
  involving multiple agents specified?
• What is meant by a correct solution?
• Require formal models capturing massive scale,
  user self-interest, subnetwork autonomy,
  distributed control, network failures.
• Must define computational resources and cost,
  reductions between problems, complexity classes,
  complete problems, intractable problems.

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Formal Models of Reliable Systems

• Classical approach to reliability is simulation
  and testing.

• Detects errors only in late stages of
  development coverage is only partial.

• More principled approach rigorous mathematical
  specification and formal verification of system
  behavior.

• Need certified software with precise and well
  understood specifications.

• Particularly critical in embedded systems and
  autonomous medical applications.

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Role of Logic

• Logic provides languages for formalizing
  requirements

• Floyd-Hoare logic for sequential programs
• Temporal and fixpoint logics for reactive
  programs
• Logics tailored for authentication and security
  properties of crypto protocols.
• Led to standardized industrial-strength formal
  specification languages.

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Role of Automata Theory

• Model checker SPIN uses linear temporal logic as
  requirement language and an automata-theoretic
  model checking algorithm.

• Research on timed and hybrid automata provides
  foundation for the emerging area of embedded
  systems.

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Role of Decision Procedures

• Modern solvers for propositional satisfiability
  used routinely on industrial-scale problems with
  hundreds of thousands of variables.

• Symbolic fixpoint evaluation research led to
  industrial interest in model checking.

• Decision procedures are continually being refined
  and improved for use in verification tools.

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TCS Connections with Biology and the Physical
Sciences

• Statistical physics
• Quantum computation
• Computational Biology

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Statistical Physics

• Studies macroscopic properties of large systems
  involving simple microscopic components
  undergoing local interactions. Examples
  freezing of water, ferromagnetism.

• CS analogy global properties of WWW emerge from
  local interactions structure of complex
  combinatorial problems derives from local
  constraints.

• Statistical physics studies random interactions
  TCS studies algorithms on random structures.

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Phase Transitions and Sharp Thresholds

• Infinitesimal change in the parameters governing
  local interactions causes a drastic change in
  macroscopic behavior
• Physics transformation from water to steam.
• CS random satisfiability instances switch from
  easy to hard when ratio of clauses to variables
  passes a critical value.

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Cross-Fertilization

• Spin glasses are fluid at high temperatures but
  at lower temperatures have many clusters of
  stable configurations. Similarly, constraint
  satisfaction problems with sparse constraints are
  fluid, but with dense constraints get stuck in
  suboptimal solutions.

• Algorithmic paradigm based on this analogy has
  been spectacularly successful.

• Markov Chain Monte Carlo used in physics as model
  of evolution of a physical system, and in CS as
  technique for approximation algorithms.

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Quantum Computation

• Quantum mechanics holds the promise of
  exponentially faster computers and perfectly
  secure communication channels.
• Large numbers can be factored rapidly, allowing
  RSA to be broken.
• To realize quantum computers, must guard against
  decoherence, the dissipation of quantum
  information into the environment.

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Challenges for Theory of Quantum Computation

• Defeat decoherence using quantum error-correcting
  codes.
• Understand structure of problems that can be
  solved exponentially faster on quantum computers
  than on classical ones.
• Use quantum computation as a test of the validity
  of quantum mechanics.

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Revolution in Biology

• Sequencing of human genome is a landmark event in
  history of science.
• Biology is becoming a quantitative,
  information-based science.
• Goals
• Detailed, predictive model of how cells work at
  molecular level.
• Understand mechanisms of cancer, global
  organization of physiological systems, processes
  of development from embryo to complex organism
• Tailor therapy to genetic makeup of individuals.

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Role of Algorithms in Computational Biology

• Understand the information hidden in the genome.
• Construct mathematical models of complex cellular
  processes.
• Extract patterns from large biological data sets.
• Determine associations between genetic variation
  and disease.