Lecture on Mar 2, 2004 Introduction to Theory of Computation

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Lecture on Mar 2, 2004Introduction to Theory of
Computation

• Topics
• Introduce Computability Complexity
• Decision Problems Languages
• Decidability Undecidability
• Reading Sipser, Chapter 0, Sections 3.3 4.2

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What is TOC?

• Studies mathematical laws governing computation.
• Fundamental capabilities and limitations of
  computers.
• Computability
• Started in1930s by logicians Gödel, Turing,
  Church,
• Is a given problem computable (or solvable) by an
  algorithm?
• Complexity
• Hartmanis Stearns 66, Blum 66, Cook 70, Karp
  71, Levin 72,
• Is a given computable problem efficiently
  computable? How much resource is required?
• Resources Time, Space, Randomness, Communication
  Bandwidth,
• Both focus on negative results, i.e. on
  limitations.

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Why Study Negative Results

• Knowing that a problem is incomputable prevents
  us from attempting an algorithm for it (which
  does not exist).
• Knowing that a problem is not efficiently
  computable tells us not to attempt an algorithm
  that solves it exactly, but go for one that gives
  an approximate solution.
• Leads to Approximation Algorithms. Later in the
  course.
• CS 6550, CS 7520
• In some cases, negative results turn into
  positive results.
• Example Cryptography
• AD CS 8803 Foundations of Cryptography, Fall
  2004
• TOC has had tremendous influence on advances of
  technology.

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Decision Problems

• Problems whose answers are YES or NO.
• Examples
• Is a given array A sorted?
• Is a given graph G connected?
• Does a given weighted undirected graph G has a
  spanning tree of weight at most w?
• Does a given graph G have a path of length at
  most l?
• Is a given integer n a prime?

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Decidability Informal Definition

• Def A decision problem is decidable if there is
  an algorithm that solves (or decides) the problem
  correctly.
• To show that a decision problem is decidable,
  just give a deciding algorithm.
• Examples
• Is a given array A ? (A1, , An) sorted? is
  decidable. Deciding
  Algorithm Just check if Ai ? Ai ?1 for each 1 ?
  i ? n?1.
• Is a given graph G connected? is decidable.
  Deciding
  Algorithm DFS.

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Decidable Problems

• Examples (Continued)
• Does a given weighted undirected graph G has a
  spanning tree of weight at most w? is decidable.
  
  Deciding Algorithm Compute a MST of G
  (using Kruskal or Prim), check whether weight of
  MST is ? w.
• Does a given graph G have a path of length at
  most l? is decidable.
  
  Deciding Algorithm Compute all-pair
  shortest paths in G (using Floyd-Warshall), check
  if there are two vertices u, v s.t. d(u,v) ? l.
• Is a given integer n a prime? is decidable.
  (Naïve)
  Deciding Algorithm For each 2 ? k ? n ?1, check
  whether k divides n.

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Languages Formalizing Decision Problems

• Alphabet ?. Example ? ? 0,1.
• Everything (e.g. integers, sets, graphs,
  polynomials, programs ) is represented as a
  string of symbols in ?.
• Notation For an object O, ?O? denotes the
  encoding of O as a string over ?.
• Def ? ? set of all strings over ?, including
  the empty string ?. Example 0,1
  ? ?, 0, 1, 00, 01, 10, 11, 000, 001, .
• Def A language over ? is a subset of ? .

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Languages

• Decision Problem ? Language
• Examples
• SORTED ? ?A? A is a sorted array
• CONNECTED ? ?G? A is a connected graph
• SHORTPATH ? ?G, l? G is a graph that has a
  path of length ? l
• PRIMES ? ?n? n is a prime integer

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Decidable Languages

• Def A language L is decidable if there is an
  algorithm A that decides L, i.e.
• ? x ? L, A(x) ? 1
• ? x ? L, A(x) ? 0
• Each of SORTED, CONNECTED, SHORTPATH, PRIMES is
  decidable.
• A language is undecidable if it is not decidable.

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Is There an Undecidable Language?

• YES! Some very natural problems are undecidable!
• History Hilberts 10th Problem, 1900
• Devise an algorithm whether any given
  multivariate polynomial with integer coefficient
  has an integer root.
• Undecidable!!! Matijasevic, 1970.

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Halting Problem Another Undecidable Language

• Decide whether a given program M accepts a given
  input w.
• ATM ? ?M, w? M is a program and M accepts w.
• Theorem ATM is undecidable.
• Proof Next lecture.

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Remark on Reading Sipser

• In Sipser, decidability defined using Turing
  Machines.
• Turing Machine ? Algorithm (or Program).
• Replace each occurrence of Turing Machine by
  Algorithm (or Program).
• Skip details (e.g. tape, states, state
  transitions, head movements) of Turing Machines.