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Questions on Countability, Recognizability, and Decidability in Turing Machines

## One-to-One functions in Turing Machines

• Notation: for any string W and any symbol x , we write #x (W )to represent the number of occur-rences of symbol x in W Examples: #0(")=0, #1(")=0, #0(0010)=3, #1(0010)=1, #0(111)=0
• In this assignment, if you need to prove something is recognizable or decidable, you should be us-ing ‘Turing machine pseudocode’ (as in Lectures 49 and after) instead of trying to draw a transitiondiagram.
• Recall the L notation as it relates to regular expressions: for any regular expression R, we write L(R)to represent the set of strings matched by the regular expression R.The assignment questions start on the next page.
1. Let function D : {0, 1}∗ →Zbe defined by D(W )=#1(W )−#0(W ).
2. Let function C : {0, 1}∗ →{0, 1}∗ be defined by C (W )=1 ·W .