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TURING MACHINES AS TRANSDUCERS Turing machine also provides us with a simple abstract model for digital computers in general. Since the primary purpose of a computer is to transform input into output, i.e. it acts as a transducer. TURING MACHINES AS TRANSDUCERS •Input - nonblank symbols on the tape at initial time •output- whatever is then on the tape at end A function f with domain D is said to be Turing -computable or just computable if there exists some Turing machine . M (Q , , , , q 0 , B , F ) such that q 0w M q f f ( w ), q f F , w D . Example Given two positive integers x and y, design a Turing machine that computes xy Assume w(x) and w(y) are on the tape in unary notation, separated by single blank, with read-write head on the leftmost symbol of w(x). After the computation, w(x+y) will be on the tape followed by blank and the read-write head will be positioned at the left end of the result. q 0 w ( x ) Bw ( y ) q f w ( x y ) B , q f final state q 0 111 B 11 1q 0 11 B 11 11 q 0 1 B 11 111 q 0 B 11 1111 q 1 11 11111 q 1 1 111111 q 1 B 11111 q 2 1 B 1111 q 3 1 BB 1q 3 1111 BB q 4 11111 B 111 q 3 11 BB q 3 11111 BB 11 q 3 111 BB q 3 B 11111 BB Q {q 0 , q 1 , q 2 , q 3 , q 4 } F {q 4 } ( q 0 ,1 ) ( q 0 ,1 , R ), ( q 0 , B ) ( q 1 ,1 , R ), ( q 1 ,1 ) ( q 1 ,1 , R ), ( q 1 , B ) ( q 2 , B , L ), ( q 2 ,1 ) ( q 3 , B , L ), ( q 3 ,1 ) ( q 3 ,1 , L ), ( q 3 , B ) ( q 4 , B , R ). Design a Turing machine that copies strings of 1’s. More precisely, find a machine that performs the computation q 0w q f ww , for any w {1} . 1. Replace every 1 by an x. 2. Find the rightmost x and replace it with 1. 3. Travel to the right end of the current nonblank region and create a 1 there. 4. Repeat Steps 2 and 3 until there are no more x’s. In computer programming , we use pseudocode (way of outlining a computation using descriptive phrases whose meaning we claim to understand) We use psedocode in Turing Machine also. Macroinstructions. Example : if a then q 0 else q k. In Turing q i , a q j 0 , a , R q i Q , q j0 , c q j , c , L c , Machine we write as q i , b q k 0 , b , R q i Q & all b - {a}, q k0 , c q k , c , L c , Replace macroinstructions with subprograms. Before A calls B, it writes the information needed by B on the tape in some region T. T # Workspace for A Region separator # Workspace for B A then passes control to B by making a transition to the start state of B. After transfer, B will use T to find its input. The workspace for B is separate from T and from the work space for A, so no interference can occur. When B is finished , it will return relevant results to region T, where A will expect to find it. Class Discussion Design a Turing machine to check whether a given Unary no. is divisible by “3” or not. Church/Turing Thesis - expanded 1. Any language that an algorithm can decide can be decided by a Turing machine and vicaa-versa. 2. Any language that an algorithm can recognize can be recognized by a Turing machine and vica-a-versa. 3 Any function that an algorithm can compute can be computed by a Turing machine and vica-a-versa 4. Time bounds, non-determinism also apply 1. Anything that can be done on any existing digital computer can also by done by a TM. 2. No one has yet been able to suggest a problem, solvable by what we intuitively consider an algorithm, for which a TM program cannot be written. 3. Alternative models have been proposed for mechanical computation, but none of them are more powerful than the TM model. Definition f : D R is a Turing An algorithm for a function machine M, which given as input any d D on its tape, eventually the correct answer f(d) R on its tape. Specifical ly, we can require halts with that q 0d M q f f ( d ), q f F d D .