Capabilities of computing systems

1
Capabilities of computing systems

• Numeric and symbolic Computations
• A look at Computability theory
• Turing Machines

2
Objectives

• What is numeric computation
• Popularity, applications
• Limitations
• What is symbolic computation
• Is is theoretically possible to solve all
  mathematical problems by computers?
• An intoduction to the study of this problem?

3
Numeric computation

• The earliest and most important application
  number cruching
• Virtually all applications that we come across
  are numeric.
• Eg virtual reality.- High resolution, very fast
  graphics with multimedia and specialized input
  output devices. Encryption, decryption/ Data
  bases

4
Need for more than numeric computing

• Many mathematical tasks are dealt with in a
  symbolic way by human beings.
• eg. Simplify a polynomial
• Solve equations
• factorize polynomials
• Similarly, computing pi to a large number
• of decimal places, factorials of large numbers
  would not fall within the range of numeric data
  handled by a computer.

5
Symbolic Computing

• Area dealing with a new type use of computers in
  mathematics, science and engineering.
• Can enhance significantly the types of problems
  humans can use computers for.

6
What is symbolic computing

• Solving problems expressed in terms of symbols
  or variables like x and y instead of numeric
  values is called symbolic computing
• Laborious tasks like expanding
• (1 x 3y) 4 or solving (x 10 1) can be
  given to computers

7
Mathematica

• It is a symbolic computing package.
• Wolfram Research Inc.
• http//www.wolfram.com/mathematica/
• Some commands
• Nexpr,I eg Npi,250
• Factorial example 200!
• Simplifyexpr, Factorexpr, Solveequation,unkno
  wn (systems)
• Expandexpr, Dexpr,x, Integrateexpr,x

8
Implementation

• Some of these tasks are simple, and have a simple
  interface to a numeric version of the problem and
  its solution
• Some tasks are more difficult and require
  techniques from artificial intelligence, using
• symbolic logic and complicated searches.

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An example
Curve fitting
Linear equation
Solvex2 3 0
x2 3 0
a1,b0,c3
Quadratic equation
User Interface
Equation Solver
Compute the two roots
User
1.732
Display Results on screen
1.732
Cubic equation
Plotting
10
A model of a computing agent

• Accepts input
• Store and retrieve information
• Can take actions according to algorithmic
  instructions actions depend on present state of
  the computing agent and the input
• 4. Produce output

11
Theory of computationEarly studies
on formalizing proofs, before modern
computers.Study of the nature of proofs.
Formalizing, validating and mechanizing
proofs.Many statements cannot be proved using a
system.Led to a study of the nature of
computation itself.Different models of
computation and computing agents.Turings model
ingenious - captures the essence of computing
12
A Turing machine

• A conceptual model
• An infinite tape , extending in both the
  directions left and right.
• Tape divided into cells, each can contain a
  symbol, from a finite set of symbols called the
  alphabet.
• Special symbol b blank, 1 , 0 and other
  symbols, placeholders or markers.

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The Turing Machine (contd)

• At any time only a finite number of cells contain
  nonblank symbols.
• Tape holds input to a TM/ input must be a finite
  string of nonblank symbols from the alphabet (a
  finite set of symbols).
• TM writes its output to tape using same set of
  symbols.
• Tape also serves as memory

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TM (contd)

• The TM has a state at any time, which is one of
  a finite set of states of a TM, labeled 1,2, .k.
• Also, at a time, its head is on a particular
  cell and it can read that cell.
• Based on state and input read, it does three
  things.
• Writes a symbol to the cell (replacing if needed)
• Go to new state
• Move one cell left or right

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Transitions or instructions

• A typical TM transition rule or instruction says
• If you are in state i and read symbol j
• then
• Write symbol k on tape, and
• go to state s, and
• move in direction d.
• It is of the form (i,j,k,s,d)
• (current state, input symbol, next symbol, next
  state, move)
• A set of such instructions defines a TM.

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TM

• TM starts from state 1.
• Finds an instruction to apply, for present sate
  and input and applies the instruction.
• Repeats the above.
• If there is no instruction applicable, TM halts.
• Assumption For one state input pair, there is at
  most one instruction applicable.

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An exampleA TM for converting 011 to 100

• (1,0,1,2,R)
• (1,1,1,2,R)
• (2,0,1,2,R)
• (2,1,0,2,R)
• (2,b,b,3,L)
• s1.bb011bb. s2bb111bb.
• s2.bb101bb. s2bb100bb.
• s3.bb100bb.

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Tm as a computing agent

• Satisfies all criteria.
• Can be built for all computable problems
• - see later.
• Can solve more than real or practical computers,
  as infinite memory is there.

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TM as an algorithm

• An algorithm
• 1. Well ordered collection.
• 2. contains unambiguous and effectively
  computable operations.
• 3. Halt in finite amount of time. (Problem)
• For correct input halts.
• 4. Produce an output.

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TM

• TM starts from state 1.
• Finds an instruction to apply, for present sate
  and input and applies the instruction.
• Repeats the above.
• If there is no instruction applicable, TM halts.
• Assumption For one state input pair, there is at
  most one instruction applicable.

21
An exampleA TM for converting 011 to 100

• (1,0,1,2,R)
• (1,1,1,2,R)
• (2,0,1,2,R)
• (2,1,0,2,R)
• (2,b,b,3,L)
• s1.bb011bb. s2bb111bb.
• s2.bb101bb. s2bb100bb.
• s3.bb100bb.

22
Tm as a computing agent

• Satisfies all criteria.
• Can be built for all computable problems
• - see later.
• Can solve more than real or practical computers,
  as infinite memory is there.

23
TM as an algorithm

• An algorithm
• 1. Well ordered collection.
• 2. contains unambiguous and effectively
  computable operations.
• 3. Halt in finite amount of time. (Problem)
• For correct input halts.
• 4. Produce an output.

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Examples

• Bit inverter
• (1,0,1,1,R)
• (1,1,0,1,R)
• A parity bit machine (1,1,1,2,R)
• (1,0,0,1,R)
• (2,1,1,1,R)
• (2,0,0,2,R) (1,b,1,3,R) (2,b,0,3,R)
• An extra bit called odd parity bit is attached to
  each string such that the number of 1s in each
  string becomes odd.

25
Turing machines and algorithms

• Church Turing Thesis
• Church (Lambda calculus)
• Turing (Turing machines)
• Showed that algorithms are equivalent
• And unsolvability of problems.

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TMs

• Popularity
• For describing algorithms
• For showing unsolvabilities

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The Halting Problem of TMs

• Given a Turing machine M (encoded) and an input I
  can it be decided whether the Turing machine
  accepts or does not accept.
• --------------------------------------------------
  -
• (Special states accept)
• (Special states reject)
• Else loops

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Not decidable

• If it is , T is a TM to do this task.
• Construct H to read ltMgt and run T on ltM Mgt.
• Reject if T accepts , accept if T rejects.
• Run H with ltHgt as input
• Contradiction accepts if reject and reject if
  accepts
• Hence no T exists.

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Problems

• Find the output of the TM
• (1,1,1,2,R) (1,0,0,2,R) (1,b,1,2,R) (2,0,0,2,R)
• (2,1,0,1,R)
• When run on the tape. bbb1001bbbb
• Describe the behaviour of a TM (1,1,1,1,R)
  (1,0,0,2,L) (2,1,0,2,L) (2,b,1,3,L) (3,b,b,1,R)
  on the input bbb101bbb

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Solutions

• 10001 and halts
• Prints successively 1001, 10001, 100001, and so
  on without halting

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Problem

• Write a TM that begins on a tape containing a
  single 1 and never halts but successively
  displays the strings
• ..b1b
• 010
• 00100.
• And so on