Abstract Models of Computation

1
Abstract Models of Computation

• Capturing the Essence of Computation

gregory.taylor_at_trincoll.edu
2
What is Computation?

• Paradigm instances of computation
• Operations on numeric data
• Operations on character data
• Function computation

3
Arithmetic Operations

• 34239 23478378
• 2347 - 293874
• ?43.3421
• 3.24
• 72 of the population of Hartford

4
Function Computation

• Preparatory Integers 14 and 25 are relatively
  prime since gcd(14, 25) ____
• Example Eulers ? function defined as
• ?(n) def. the number of integers k lt n such
  that k and n are relatively prime (k ? 1)
• Thus ?(20) ______.

5
String Manipulation

• Find the length of string ababababbbaba
• What is abababbabbabR?
• Is abababbbbbababa a palindrome?
• What is the longest substring common to GCATCCTAA
  and ATACTAATCCTTA?

6
Three Computational Paradigms

• Function computation (natural numbers only)
• Language acceptance (recognition)
• Transduction (transformation of an input into an
  appropriate output)
• Turing machines implement each of these paradigms

7
The Turing Machine Concept (Informal Description)
8
Example 1.2.1

• State diagram (transition diagram) full
  description of machine Ms behavior
• Machine states and instructions
• Initial state q0
• Machine instructions

9
Example 1.2.1 (cont.)

• Read/write tape (tape squares)
• Read/write head scans one tape square
• Initial tape configuration

10
M after One Computation Step

• Machine configuration In state q1 scanning an a
  (all other squares blank).
• What happens next?

11
Summary of Ms Behavior

• When started scanning square on a completely
  blank tape, M writes ab and halts scanning a.
• We makes no claims for M if tape not initially
  blank.

12
Important Concepts

• Tape configuration
• Machine configuration tape configuration
  current state

13
Turing Machines Can Compute Forever

• Machine on left moves off to the right forever
• Machine on the right writes 1s off to the right
  forever

14
The Copying Machine (Icon)

• Load tape8.tt
• 11111111 on tape initially
• 11111111B11111111 on tape when M halts
• Scanning leftmost 1

15
Exercise 1

• Design a Turing machine M that, starting on a
  blank tape, writes a short, unbroken string of as
  and bs of your choice on the tape and then halts
  scanning the leftmost of these symbols.
• Example Design M so that it writes aba on the
  tape and then halts.
• M can work from left to right or from right to
  left (slightly more efficient).

16
Language Acceptance

• Produces accepting 1 for word w just in case
  na(w) nb(w)

17
A Turing Machine May Accept Word w

• word finite string of symbols
• Word abab accepted
• Word baaa not accepted
• Same Number of as and bs
• Starts scanning leftmost symbol of input input
  word w
• Halts scanning a 1 on otherwise blank tape

18
Another Example

• Accepted a, ab, bb, aab, abb, bbb, ...

• Not accepted a, aa, ba, aaa, aba, bba, ...

19
Language Acceptance

• Both machines on previous slide accept all and
  only words in language L anbmn³0,m³1
• We say that these machines accept language L
  anbmn³0,m³1
• We dont care what M does for input words not in
  L so long as it does not write accepting 1.

0 or more as followed by 1 or more bs
20
Exercise 2

• Alphabet just a and b
• Output alphabet just 1
• Design a Turing machine that accepts the language
  consisting of all words of even length.
• Should write a 1 in response to aa, ab, ba, bb,
  etc.
• Should not write a 1 for a, aba, etc.

21
Language Recognition

• M recognizes language L
• If word w ? L, then M writes accepting 1 and
  halts. (M accepts w.)
• If w ? L, then M writes rejecting 0 and halts.
  (M rejects w.)
• Note that 0, 1 ? ?.
• Special case ?
• Example under icon Recognizer.

22
Exercise 3

• Input alphabet just a and b
• Output alphabet 1 and 0
• Design a Turing machine that recognizes the
  language consisting of all words of even length.
• Should write a 1 in response to aa, ab, ba, bb,
  etc.
• Should write a 0 for a, aba, etc.

23
Terminology

• Language L is Turing-acceptable if some Turing
  machine accepts L.
• Language L is Turing-recognizable if some Turing
  machine recognizes L.
• If M recognizes L, then M accepts L, but not
  necessarily vice versa.

24
Representing Natural Numbers

• 111 represents natural number 2
• What represents 4? __________
• What represents 0? __________

25
Computing the Successor Function

• If M started scanning input 1111, then halts
  scanning 11111.

26
Exercise 4

• Design a Turing machine that computes the
  function defined by f(n) n 3
• Assumed to start scanning leftmost 1 in an
  unbroken string of 1s
• Should halt scanning leftmost 1 left on tape

27
Computing Binary Functions

• binary functions of two arguments
• What would it be for a Turing machine to compute
  addition function f(n, m) n m?
• Answer If started scanning 1111B111
  representing arguments 3 and 2, then halts
  scanning 111111 representing argument 5.

28
Exercise 5

• Design a Turing machine M that computes the
  binary addition function in the sense of the
  previous slide.
• M should halt scanning the leftmost 1 on tape.
• Hint fill in intermediate blank and delete two
  1s on the left

29
Terminology

• Natural numbers
• Members of N 0, 1, 2,
• Functions from natural numbers to natural numbers
  only
• A function f is Turing-computable if there
  exists a Turing machine M that computes f.

30
Turing-Computable Functions

• Addition is Turing-computable (Exercise 5).
• Multiplication is Turing-computable (icon).
• Exponentiation is Turing-computable (icon).
• Every function youve ever heard of is
  Turing-computable.
• Turing computability is a powerful concept.
• Those functions that arent Turing-computable are
  complicated to describe.

31
Turing Machines as Transducers

• Icon Reverse Word
• Input word abbb is transformed into bbba.
• Input word ? is transformed into ? (limiting
  case).

32
Exercise 6

• Input alphabet just a and b
• Design a Turing machine that transposes its
  input string, changing every a to b, and vice
  versa
• So abba is turned into baab
• Halt at far left

33
Summary

• We have now seen that Turing machines implement
  each of our three paradigms.
• Language acceptance (recognition)
• Number-theoretic function computation
• Transduction