Computability and Modeling Computation

1
Computability andModeling Computation

• What are some really impressive things that
  computers can do?
• Land the space shuttle (and other aircraft) from
  the ground
• Automatically track the location of a space or
  land vehicle
• Beat a grandmaster at chess
• Are there any things that computers can't do?
• Yes, and these are referred to as undecidable
  problems, unsolvable problems, or
  non-computable functions.

2
Some Difficult Problems

• For example, could a computer tell us what the
  meaning of life is?
• Suppose a computer outputs the string "You are
  born, live your life, and then you die.
• Has that computer given us the meaning of life?
• The statement may or may not be true
• Doesn't tell one how to live a happy life
• Doesn't tell one how to live a successful life
• Doesn't tell if there is an afterlife
• The program doesnt arrive at the solution by
  itself
• Point 1 Problems need to be precisely defined,
  before we can determine if they are solvable.

3
Some Difficult Problems, Cont.

• Similarly, could a computer tell us if there is
  life after death?
• Consider the following two programs the first
  simply outputs the string There is life after
  death, and the second simply outputs There is
  no life after death.
• Note that one of the two programs answers the
  question correctly, but which one? Also note that
  neither justifies or develops its own answer.
• Point 2 Before we can determine if something
  can be solved or computed, we have to define what
  we mean by compute.

4
A Formal Model of ComputationTuring Machines

• Before determining what computers are ultimately
  capable of, one must answer the following
  questions
• What is computation?
• What are reasonable steps that can be performed
  in a computation?
• Formally, computation is represented by a
  mathematical model called a Turing Machine.
• Proposed in 1926 by Alan Turing to model any
  possible computation
• Churchs hypothesis is that any general and
  reasonable way to compute, or any type of general
  purpose computer can be modeled/represented by a
  Turing machine.
• Detailed discussion of Turing machines is beyond
  the scope of this course.
• Other, less computer-like models of computation
  include recursive function theory and predicate
  calculus.

5
A Less Formal Model of Computation Algorithms

• Informally, an algorithm is a finite length
  sequence of operations, each of which is
  well-defined and finite. In addition, an
  algorithm must terminate after a finite number of
  operations.
• Well-defined means it should be clear at each
  step what is to be done.
• Compare x and y" doesn't say how x any are to be
  compared
• Finite means each step must involve a finite
  amount of information (i.e., data), which is
  manipulated to only a finite degree.
• Adding 1 to a number is finite
• Repeatedly dividing a number by 2 forever is not.

6
Algorithms, Cont.

• Intuitively, a well-defined and finite step could
  be performed by a person by pencil and paper in a
  fixed amount of time.
• Algorithms can be described by a computer
  program, flowchart, pseudo-code, or by some other
  similar method.
• Example
• Set x 0 reasonable
• Let x x 1 reasonable
• Set y x/0 unreasonable
• Place an arbitrary question in a memory
  location, wait 2 seconds for the answer,
  and then
  process the answer. unreasonable

7
Problems to be Solved

• What kinds of problems are there?
• Mathematical problems (simple and complex)
• Logical problems
• Data processing problems, etc.
• Consider just mathematical decision (I.e.,
  yes/no) problems
• Let x be an integer, is x even?
• Let x and y be integers, is xlty?
• Are there integers x, y and z such that x2y
  zy?

8
Unsolvable Problems

• Can all mathematical decision problems be solved
  with algorithms?
• Answer No! (by solved, we mean in all cases)
• Example of unsolvable problems
• Let P be a program. Does P contain an infinite
  loop, i.e., will P terminate eventually or loop
  forever?
• Let P be a program, and let x be a variable in P.
  Is x ever assigned a value when P is executed?
• Let P be a program, and let s be a statement in
  P. Is s ever executed when P is executed?

9
Problems vs. Computational Steps

• In general, problems fall into one of three
  categories
• Those which are provably solvable
• Those which are provable unsolvable
• Those whose status is unknown currently there is
  no algorithm to solve the problem, but it has not
  been proven that none exists
• Among this last category of problems, there is
  considerable debate as to their ultimate status,
  e.g., is there a program that can understand a
  natural language such as English?
• Note however, that this is an entirely different
  question from what constitutes a reasonable
  step in a computation, and on that issue there is
  no debate.
• A reasonable step in a computation is
  well-defined and finite, and can be performed by
  pencil and paper in a fixed amount of time.

10
Efficiency vs. Inefficiency

• In addition, solvable problems fall into one of
  three categories
• Solvable problems which have been proven to be
  solvable efficiently, i.e., solvable in
  (deterministic) polynomial time.
• Solvable problems which have been proven not to
  be solvable efficiently, i.e., NP-hard or
  NP-complete (NP stands for Nondeterministic
  Polynomial).
• Solvable problems for which there is no efficient
  solution, but which have not been proven not be
  be solvable efficiently.