1 IDT Open Seminar ALAN TURING AND HIS LEGACY 100 Years Turing celebration Gordana Dodig Crnkovic, Computer Science and Network Department Mälardalen University March 8 th

2 Finite Automata Push-down Automata Turing Machines Chomsky Language Hyerarchy

TURING MACHINES “Turing’s "Machines". These machines are humans who calculate.” (Wittgenstein) “A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.” (Turing) 3

Tape Read-Write head Control Unit Turing Machine

Read-Write head No boundaries -- infinite length The head moves Left or Right The Tape

Read-Write head 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right The head at each time step:

7 Head starts at the leftmost position of the input string Blank symbol head Input string The Input String

8 Determinism Allowed Not Allowed No lambda transitions allowed in TM! Turing Machines are deterministic

9 Determinism Note the difference between state indeterminism when not even possible future states are known in advance. and choice indeterminism when possible future states are known, but we do not know which state will be taken.

10 Halting The machine halts if there are no possible transitions to follow

11 Example No possible transition HALT!

12 Final States Allowed Not Allowed Final states have no outgoing transitions In a final state the machine halts

13 Acceptance Accept Input If machine halts in a final state Reject Input If machine halts in a non-final state or If machine enters an infinite loop

14 Formal Definitions for Turing Machines

15 Transition Function

16 Transition Function

17 Turing Machine Transition function Initial state blank Final states States Input alphabet Tape alphabet

18 For any Turing Machine Initial stateFinal state The Accepted Language

19 Standard Turing Machine Deterministic Infinite tape in both directions Tape is the input/output file The machine we described is the standard:

20 Computing Functions with Turing Machines

21 Initial Configuration Final Configuration Domain For all A function is computable if there is a Turing Machine such that

22 Example (Addition) The function is computable Turing Machine: Input string: unary Output string: unary are integers

23 Start Finish final state initial state

24 Turing machine for function

25 Execution Example: Time 0 Final Result (2)

26 Time 0

27 Time 1

28 Time 2

29 Time 3

30 Time 4

31 Time 5

32 Time 6

33 Time 7

34 Time 8

35 Time 9

36 Time 10

37 Time 11

38 HALT & accept Time 12

39 Universal Turing Machine

40 A limitation of Turing Machines: Turing Machines are “hardwired” they execute only one program

41 Solution:Universal Turing Machine Reprogrammable machine Simulates any other Turing Machine Characteristics:

42 Universal Turing Machine simulates any other Turing Machine Input to Universal Turing Machine: Description of transitions of Initial tape contents of

43 Universal Turing Machine Description of Three tapes Tape Contents of Tape 2 State of Tape 3 Tape 1

44 We describe Turing machine as a string of symbols: We encode as a string of symbols Description of Tape 1

45 Alphabet Encoding Symbols: Encoding:

46 State Encoding States: Encoding: Head Move Encoding Move: Encoding:

47 Transition Encoding Transition: Encoding: separator

48 Machine Encoding Transitions: Encoding: separator

49 Tape 1 contents of Universal Turing Machine: encoding of the simulated machine as a binary string of 0’s and 1’s

50 As Turing Machine is described with a binary string of 0’s and 1’s the set of Turing machines forms a language: Each string of the language is the binary encoding of a Turing Machine.

51 Language of Turing Machines L = { , , , …… } (Turing Machine 1) (Turing Machine 2) ……

52 Question: Do Turing machines have the same power with a digital computer? Intuitive answer: Yes There was no formal proof of Church-Turing thesis until 2008! CHURCH TURING THESIS

53 Dershowitz, N. and Gurevich, Y. A Natural Axiomatization of Computability and Proof of Church's Thesis, Bulletin of Symbolic Logic, v. 14, No. 3, pp (2008) This formal proof of Church-Turing thesis relies on an axiomatization of computation that excludes randomness, parallelism and quantum computing and thus corresponds to the idea of computing that Church and Turing had.

54 Turing’s thesis Any computation carried out by algorithmic means can be performed by a Turing Machine. (1930) The Origins of the Turing Thesis Myth Goldin & Wegner