Turing Machines. COMP2600 Formal Methods for Software Engineering. Katya Lebedeva. Australian National University Semester 2, 2014

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1 Turing Machines COMP2600 Formal Methods for Software Engineering Katya Lebedeva Australian National University Semester 2, 2014 Slides created by Jeremy Dawson and Ranald Clouston COMP 2600 Turing Machines 1

2 Alan Turing ( ) Let s start by looking at the life and works of the extraordinary Alan Turing... COMP 2600 Turing Machines 2

3 Scientific contributions 1936: Introduced Turing machines and the study of computability (our final topics); 1950: The Turing test set up artificial intelligence as a concrete research problem; In June 2014, Eugene Goostman, a computer programme convinced 33% of judges that it was a 13-year-old boy. 1952: Pioneering work on computation in nature; Also: Key figure in the invention of the earliest modern computers. Remarkable as these advances were, there is another facet to his life, kept secret until well after his death... COMP 2600 Turing Machines 3

4 Turing at War In World War Two, Nazi Germany used the most sophisticated cryptography system ever devised to communicate secret orders - the Enigma machine. A British codebreaking effort was assembled at Bletchley Park, with Turing as their chief scientific genius. Their efforts to crack Enigma has been credited with shortening World War Two by as much as two years! Many of the ideas and technologies of Bletchley Park fed directly into the invention of modern computers. Short video: COMP 2600 Turing Machines 4

5 Death and Legacy In 1952, less than a decade after his heroic efforts for his country in WWII, Turing was prosecuted for the crime of homosexuality. He was sentenced to chemical castration, and subsequently committed suicide in He is now widely recognised as the father of computing: The Turing award is computer science s equivalent to the Nobel Prize; UK government apologised for his prosecution in 2009; COMP 2600 Turing Machines 5

6 Turing Machines Introduction In 1936, Alan Turing s paper On computable numbers, with an application to the Entscheidungsproblem claimed to solve a long-standing problem of David Hilbert, changed the world. It provides a remarkably simple mathematical machine that is claimed to be able to simulate any action of a computer. But wait a minute - wasn t 1936 before computers were even invented..? (ENIAC in 1946 is generally considered the first modern computer.) So what was Turing talking about? COMP 2600 Turing Machines 6

7 Computing as a profession Photo c/o Early Office Museum Archives COMP 2600 Turing Machines 7

8 A model for computers In 1936 a computer was not a machine, it was a profession. Turing set himself the task of giving a precise mathematical definition of everything that such computers were capable of. His 1936 paper justifies his definition by references to states of mind etc. (Section 9.I of his paper, justifying his definition, is quite readable; see cs.virginia.edu/~robins/turing_paper_1936.pdf ) But no computer (as we now understand the word) has ever been built that is more powerful than a Turing Machine; Turing seems to have discovered the limits of what is possible to compute by any means. COMP 2600 Turing Machines 8

9 Push Down Automata, reloaded a0 a1 a an read head input tape Finite State Control zk z2 z1 stack memory COMP 2600 Turing Machines 9

10 Turing Machines a0 a1 a an read head input tape Finite State zk Control read write head z2 tape memory z1 We generalise the PDA into a Turing Machine by using a tape memory for the store instead of a stack memory. We can access an arbitrary symbol by moving the tape head. COMP 2600 Turing Machines 10

11 Single tape Turing Machines input data scratch space a0 a1 an z0 z1 zk read / write head Finite State Control We can simplify the two tape TM by using a single tape for both input data and auxiliary storage. This is the standard form of Turing Machines. The tape is assumed to be infinite in both directions, so we will never run out of space. COMP 2600 Turing Machines 11

12 Turing Machines as language recognisers Turing Machines are deterministic - at each point there is at most one action that is legal. If there is no action that is legal, the TM halts. If a TM halts in a final state, it has accepted the original input. The set of strings accepted by a TM is its language. If it halts in a non-final state, this is an error, i.e. the input is rejected. A TM may also loop forever... The class of languages recognisable TMs are called the recursively enumerable languages. COMP 2600 Turing Machines 12

13 Output A TM is supposed to be able to perform any computable function. So sometimes we are interested in output more sophisticated than yes/no. If a TM halts in a final state then whatever is left on the tape will be its output. (The tape is infinite in both directions, but a halted computation can only have written on a finite part of it, so it is that non-blank bit in the middle that we regard as the output.) COMP 2600 Turing Machines 13

14 Turing Machine formal definition A Turing Machine has the form (Q,q 0,F, Γ,Σ,Λ, δ), where Q is the set of states, q 0 Q is the initial state and F Q are the final states; Γ is the set of tape symbols, Σ Γ is the set of input symbols, and Λ Γ/Σ is the blank symbol. δ is a (partial) transition function δ : Q Γ Q Γ {L,R,S} δ : (state, tape symbol) (new state, new tape symbol, direction) The direction tells the read/write head which way to go next: Left, Right, or Stay. COMP 2600 Turing Machines 14

15 Running a TM Initialisation: some input (a finite string over Σ) is written on the tape; every other tape cell is a blank Λ; the read/write head sits over the left-most cell of the input (or over any Λ is the input is ε); the state is the start state q 0. The TM then runs according to the deterministic transition function δ, halting if it reaches a state-symbol pair for which δ is undefined. COMP 2600 Turing Machines 15

16 Graphical Representation of the Transition Function S 0 1 1,L Λ 1,L S 1 Λ 1,S H Similar to FSA, annotate transition edges with commands for accessing tape. Numerator: symbol read from tape. Λ means the tape is blank at that position. Denominator: symbol written / direction of head movement. direction one of L, R, S for Left, Right, Stay. COMP 2600 Turing Machines 16

17 What does this one do? S 0 0 0,R Λ Λ,L 1 1,R S 1 0 1,L Λ Λ,R 1 0,L H Phase 1: initialisation. Phase 2: computation, in this case, complement a binary number. COMP 2600 Turing Machines 17

18 Harder Problems? Incrementing a binary number You should try this! Adding numbers - need terminators # # # Convenient to write the result before the data. Multiplication - and so on! COMP 2600 Turing Machines 18

19 Incrementing a binary number Solution: 0 0,R Λ S 0 1 1,R Λ,L 0 1,S S 1 Λ 1,S 1 0,L H COMP 2600 Turing Machines 19

20 Decrementing a number is similar 0 0,R Λ S 0 1 1,R Λ,L 1 0,S S 1 0 1,L H If given number is 0 it fails (at state S 1 ). COMP 2600 Turing Machines 20

21 How to add two numbers? Input eg n {}}{ # m {}}{ Go back and forth between m and n, decrementing one (until this fails) and incrementing the other. We decrement m, and increment n, because n will expand leftwards. m gets changed to , n is replaced by the sum. Finally, delete the #11 1 on the right. COMP 2600 Turing Machines 21

22 How to add two numbers? ctd. 0 0,R 1 1,R # #,R Λ Λ,L S ,S Λ 1,S 1 Λ,R Λ S 4 # Λ,R Λ,S H 1 0,S S 1 S 2 0 # 1,L #,L S 3 0 0,L 1 1,L 1 0,L COMP 2600 Turing Machines 22