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8 November – Day 22 Student Presentations
Continuation of Turing Machines: Example: Peter Linz Chapter 9 “addition” p. 242 [movement refers to head moving] “program” to add (control unit transition functions will add) “trick” is to use unary notation as it makes programming easier input: 1 1 1 0 1 1 output: 1 1 1 1 1 0 Transition functions () are the program “shorthand” q 0 111011 means: in state q 0 , tape appears as 111011 and read/write head will read the symbol just to the right of the state (q 0 ) next ( q 0 , 1) = (q 0 , 1, R)) q 0111011 1q 011011 11q 01011 111q 0011 ( q 0 , 0) = (q 1 , 1, R)) 1111q 111 ( q 1 , 1) = (q 1 , 1, R)) 11111q 11 111111q 1 ( q 1 , ) = (q 2 , , L) 11111q 21 ( q 2 , 1) = (q 3 , 0, L) 1111q 310
( q 3 , 1) = (q 3 , 1, L) 111q 3110 11q 31110 1q 311110 q 3111110 q 3 111110 ( q 3 , ) = (q 4 , , R)) q 4111110 q 4 is final; therefore, TM ends with correct answer “5” on the tape Notice how states are used to “remember” something … NTO Chapter 31 (review) [movement refers to tape moving] data representation numbers people: base 10 computers: base 2 to make “programming” a TM easier, we use unary (counting ‘sticks’) - a non place value system Multiplication example Fig 3.12 -- input tape at top (user’s request) = 4 * 3
- output (results) tape at bottom, TM calculated 12 “Program” on page 209 [transition state diagram, delta function] TM can solve any algorithm We believe that a TM is the simplest model of computation Dewdney’s quintuple for a TM (q, s, q’, s’, d) q = current state s = symbol under the tape q’ = state to transition to s’ = symbol to write on tape d = direction to move tape Note: TM programs are long because there is no real memory; they can’t count and remember a count in a variable; instead, states and temporary tape markers are used to ‘remember’ stuff Follow through multiplication and convince yourself how it works Multiplication is repeated addition, write 3 groups of 4 on the tape …
