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Models of Computation - Turing Machines

Department of Computer and Information Science, School of Science, IUPUI. Models of Computation - Turing Machines. CSCI 230. Dale Roberts, Lecturer Computer Science, IUPUI E-mail: droberts@cs.iupui.edu. . . b. b. 0. 1. 1. b. b. . 1. The Turing Machine (TM).

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Models of Computation - Turing Machines

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  1. Department of Computer and Information Science,School of Science, IUPUI Models of Computation - Turing Machines CSCI 230 Dale Roberts, Lecturer Computer Science, IUPUI E-mail: droberts@cs.iupui.edu

  2. . . b b 0 1 1 b b . 1 The Turing Machine (TM) • Turing Machine is a model of computing agent. • It is a pencil-and-paper type model that captures the essential features of a computing agent. • A Turing machine consists of • a tape that extends infinitely in both directions • the tape is divided into cells, each cell can carry a symbol. • the symbols comes from a finite set of symbols called the alphabet • the alphabet consists of symbols : b (blank), 0, 1, and special symbols X, Y, $ • the tape serve as memory • has a finite number of k states, 1, …k

  3. Turing Machine actions depend on two inputs: • The current state of the machine • content of cell currently being read (input) • A Turing machine can do only one operation at a time. Each time an operation is done, three actions may take place: • write a symbol to cell • go into a new state • move one cell left or right

  4. output input 0/1 1 2 W R R Your movement State: where you are now. . . . . b b b b 0 1 1 1 1 1 b b b b . . 12 1 The Turing Machine (TM) Example: Assume a Turing machine (TM) instruction: if you are in state 1 and you are reading symbol 0 then write symbol 1 onto tape go into state 2 move right State Transition: current state action (direction) Written as: (1,0,1,R,2) next state output input

  5. 1/0 1 0/1 The Turing Machine (TM) Example: Design a Turing Machine which will invert the string of binary digits • if the input string is 10110 then the output string should be 01001 • let us draw a state diagram • The TM instruction sets will be (1,0,1,R,1) (1,1,0,R,1) Let the initial configuration be: ... b 1 0 1 1 0 b ... ... b 1 0 1 1 0 b ... ... b 0 0 1 1 0 b ... ... b 0 1 1 1 0 b ... ... b 0 1 0 1 0 b ... ... b 0 1 0 0 0 b ... ... b 0 1 0 0 1 b ...

  6. 1/1 b/1 1 2 Halt R s1 s1 s1 s1 s2 Unary Representation for TM Unary representation is used in order to do any arithmetic operations on a TM. • Unary representation looks as follows 0  1 1  11 2  111 3  1111 4  11111 Example: Design a Turing Machine to add 1 to any number • start in state 1 • if the state is 1 and current input is 1, write 1 and move right and stay in state 1 • if the current state is 1 and current input is b, write 1 and move to state 2 and move right and HALT • TM instructions: (1,1,1,R,1) (1,b,1,R,2) state 2 does not exist, so halt Let the initial configuration be: ... b 1 1 1 b ... ... b 1 1 1 b ... ... b 1 1 1 b ... ... b 1 1 1 b ... ... b 1 1 1 b ... ... b 1 1 1 1 b ... 2 in unary representation 3 in unary representation

  7. 1/1 1/1 1/b 1/b b/1 b/b Halt 1 2 3 4 5 R R R R Example: Adding of two non-zero numbers (unary representation) Initial Setup . . b 1 1 1 b 1 1 b . . Answer should be: . . b b b 1 1 1 1 b . . (1,1,b,R,2): Erase leftmost 1 and move right (2,1,b,R,3): Erase second 1 and move right (3,1,1,R,3): pass over any 1’s until a blank is found (3,b,1,R,4): write 1 over the blank (4,1,1,R,4): pass over remaining 1’s (4,b,b,R,5): halt S1... b 1 1 1 b 1 1 b... S2... b b 1 1 b 1 1 b... S3... b b b 1 b 1 1 b... S3... b b b 1 b 1 1 b... S4... b b b 1 1 1 1 b... S4... b b b 1 1 1 1 b... S4... b b b 1 1 1 1 b...

  8. 1/1 1/1 R R b/1 b/b 1/b 1/b 1 2 3 4 5 R L L L Example: Add two numbers, 2 + 3 • initial setup would: a ‘b’ separate the two numbers. ... b 1 1 1 b 1 1 1 1 b ... 2 3 • and the expected answer is ... b 1 1 1 1 1 1 b ... 5 • The TM instruction sets will be (1,1,1,R,1) (1,b,1,R,2) (2,1,1,R,2) (2,b,b,L,3) (3,1,b,L,4) (4,1,b,L,5) Let the initial configuration be: S1... b 1 1 1 b 1 1 1 1 b... S1... b 1 1 1 b 1 1 1 1 b... S1... b 1 1 1 b 1 1 1 1 b... S1... b 1 1 1 b 1 1 1 1 b... S2... b 1 1 1 1 1 1 1 1 b... S2... b 1 1 1 1 1 1 1 1 b... S2... b 1 1 1 1 1 1 1 1 b... S2... b 1 1 1 1 1 1 1 1 b... S2... b 1 1 1 1 1 1 1 1 b... S3... b 1 1 1 1 1 1 1 1 b... S4... b 1 1 1 1 1 1 1 b b... S5... b 1 1 1 1 1 1 b b b...

  9. Example: Add two numbers, 2 + 3 try (1,1,b,R,2) (2,1,b,R,3) (3,1,1,R,3) (3,b,1,R,4) if an initial configuration is b b 1 1 1 b 1 1 1 1 b b .. 1/1 R 1/b 1/b b/1 1 2 3 4 R R R S1... b b 1 1 1 b 1 1 1 1 b b... S2... b b b 1 1 b 1 1 1 1 b b... S3... b b b b 1 b 1 1 1 1 b b... S3... b b b b 1 b 1 1 1 1 b b... S3... b b b b 1 1 1 1 1 1 b b... S4 (halt)

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